#  Riemann surfaces of second kind and effective finiteness theorems

Burglind Jöricke  IHES, 35 Route de Chartres
91440 Bures-sur-Yvette
France<EMAIL_ADDRESS>
###### Abstract.

The Geometric Shafarevich Conjecture and the Theorem of de Franchis state the
finiteness of the number of certain holomorphic objects on closed or punctured
Riemann surfaces. The analog for Riemann surfaces of second kind is an
estimate of the number of irreducible holomorphic objects up to homotopy (or
isotopy, respectively). This analog addresses the problem of the validity of
Gromov’s Oka principle.

For any finite open Riemann surface $X$ (maybe, of second kind) we give an
effective upper bound for the number of irreducible holomorphic mappings up to
homomotopy from $X$ to the twice punctured complex plane, and an effective
upper bound for the number of irreducible holomorphic torus bundles up to
isotopy on such Riemann surface.

If $X_{\sigma}$ is the $\sigma$-neighbourhood of a skeleton of a torus with a
hole, then the number of irreducible holomorphic mappings up to homomotopy
from $X_{\sigma}$ to the twice punctured complex plane grows exponentially in
$\frac{1}{\sigma}$. This can be interpreted as a quantitative statement of
Gromov’s Oka principle.

###### Key words and phrases:

finiteness theorems, Riemann surfaces of second kind, $3$-braids, torus
bundles, Oka principle.

###### 2020 Mathematics Subject Classification:

Primary 32G13; Secondary 20F36, 32H35, 32Q56, 57Mxx

September 3, 2024

##  1\. Introduction and statements of results

The following statement, known as Geometric Shafarevich conjecture, has been
conjectured by Shafarevich [27] in the case of compact base and fibers of type
$(\textsf{g},0)$.

Theorem A (Geometric Shafarevich Conjecture.) For a given compact or punctured
Riemann surface $X$ and given non-negative numbers g and m such that
$2\textsf{g}-2+\textsf{m}>0$ there are only finitely many locally
holomorphically non-trivial holomorphic fiber bundles over $X$ with fiber of
type $(\textsf{g},\textsf{m})$.

A connected closed Riemann surface (or a smooth connected closed surface) is
called of type $(\textsf{g},\textsf{m})$, if it has genus ${\sf{g}}$ and is
equipped with ${\sf{m}}$ distinguished points. Recall that a closed Riemann
surface with a finite number of points removed is called a punctured Riemann
surface. The removed points are called punctures. Sometimes it is convenient
to associate a punctured Riemann surface to a Riemann surface of type
$(\textsf{g},\textsf{m})$ by removing the distinguished points. A connected
Riemann surface is of first kind, if it is a closed or a punctured Riemann
surface, otherwise it is of second kind. A Riemann surface is called finite if
its fundamental group is finitely generated, and open if no connected
component is compact. Each finite connected open Riemann surface $X$ is
conformally equivalent to a domain (denoted again by $X$) on a closed Riemann
surface $X^{c}$ such that each connected component of the complement
$X^{c}\setminus X$ is either a point or a closed geometric disc [14]. A
geometric disc is a topological disc that lifts to a round disc in the
universal covering of the closed surface $X^{c}$ with respect to the standard
metric on the universal covering. The closed Riemann surface $X^{c}$ and the
embedding $X\subset X^{c}$ are unique up to conformal isomorphism. The
connected components of the complement will be called holes.

Theorem A was proved by Parshin [26] in the case of compact base and fibers of
type $(\textsf{g},0),\,\textsf{g}\geq 2,$ and by Arakelov [2] for punctured
Riemann surfaces as base and fibers of type $(\textsf{g},0)$. Imayoshi and
Shiga [16] gave a proof of the quoted version using Teichmüller theory.

The statement of Theorem A ”almost” contains the so called Finiteness Theorem
of Sections which is also called the Geometric Mordell conjecture (see [25]),
giving an important conceptional connection between geometry and number
theory. For more details we refer to the surveys by C.McMullen [25] and
B.Mazur [24].

There is an older finiteness theorem due to de Franchis [4] which can be
considered as a special case of the Geometric Shafarevich conjecture.

Theorem B (de Franchis). For closed connected Riemann surfaces $X$ and $Y$
with $Y$ of genus at least $2$ there are at most finitely many non-constant
holomorphic mappings from $X$ to $Y$.

There is a more comprehensive Theorem in this spirit.

Theorem C (de Franchis-Severi). For a closed connected Riemann surface $X$
there are (up to isomorphism) only finitely many non-constant holomorphic
mappings $f:X\to Y$ where $Y$ ranges over all closed Riemann surfaces of genus
at least $2$.

We may associate to any holomorphic mapping $f:X\to Y$ of Theorem B the bundle
over $X$ with fiber over $x\in X$ equal to $Y$ with distinguished point
$\\{f(x)\\}$. Thus, the fibers are of type $(\textsf{g},1)$. A holomorphic
self-isomorphism of a locally holomorphically non-trivial
$(\textsf{g},1)$-bundle may lead to a new holomorphic mapping from $X$ to $Y$,
but there are only finitely many different holomorphic self-isomorphisms.
Hence, Theorem B is a consequence of Theorem A, and Theorem B has analogs for
the source $X$ and the target $Y$ being punctured Riemann surfaces.

The Theorems A and B do not hold literally if the base $X$ is of second kind.
If the base is a Riemann surface of second kind the problem to be considered
is the finiteness of the number of irreducible isotopy classes containing
holomorphic objects. In case the base is a punctured Riemann surface this is
equivalent to the finiteness of the number of holomorphic objects. For more
detail see sections 2 and 3.

We will prove finiteness theorems with effective estimates for Riemann
surfaces of second kind. The estimates depend on a conformal invariant of the
base manifold. To define the invariant we recall Ahlfors’ definition of
extremal length (see [1]). For an annulus $A=\\{0\leq r<|z|<R\leq\infty\\}$
(and for any open set that is conformally equivalent to $A$) the extremal
length equals $\frac{2\pi}{\log\frac{R}{r}}$. For an open rectangle
$R=\\{z=x+iy:0<x<{\sf b},\,0<y<{\sf a}\,\\}$ in the plane with sides parallel
to the axes, and with horizontal side length b and vertical side length a the
extremal length equals $\lambda(R)=\frac{\sf a}{\sf b}$. For a conformal
mapping $\omega:R\to U$ of the rectangle $R$ onto a domain
$U\subset\mathbb{C}$ the image $U$ is called a curvilinear rectangle, if
$\omega$ extends to a continuous mapping on the closure $\bar{R}$, and the
restriction to each (closed) side of $R$ is a homeomorphism onto its image.
The images of the vertical (horizontal, respectively) sides of $R$ are called
the vertical (horizontal, respectively) curvilinear sides of the curvilinear
rectangle $\omega(R)$. The extremal length of the curvilinear rectangle $U$
equals the extremal length of $R$. (See [1]).

Let $X$ be a connected open Riemann surface of genus $g\geq 0$ with $m+1$
holes, $m\geq 0$, equipped with a base point $q_{0}$. The fundamental group
$\pi_{1}(X,q_{0})$ of $X$ is a free group in $2g+m$ generators. We choose a
system of generators as follows. The Riemann surface $X$ is conformally
equivalent to an open subset of a closed Riemann surface $X^{c}$ of genus $g$
([14]). Choose $2g$ generators $e_{j,0},\,j=1,\ldots,2g,$ of the fundamental
group $\pi_{1}(X,q_{0})$ whose images under the homomorphism induced by
inclusion $X\to X^{c}$ are generators of the fundamental group of $X^{c}$. We
may assume that each pair $e_{2j-1,0},e_{2j,0},\,j=1,\ldots g,$ corresponds to
a handle. Represent each pair of generators $e_{2j-1,0},\,e_{2j,0}$
corresponding to a handle by simple closed loops $\alpha_{j},\,\beta_{j}$ with
base point $q_{0}$.

Denote the connected components of $X^{c}\setminus X$ by
$\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{m+1}$. Each component is
either a point or a closed disc. For each component
$\mathcal{C}_{\ell},\,\ell=1,\ldots,m,$ we choose the generator
$e_{2g+\ell,0}$ of the fundamental group of $X$ which is represented by a
simple loop $\gamma_{2g+\ell}$ with base point $q_{0}$ that is contractible in
$X\cup\mathcal{C}_{\ell}$ and divides $X$ into two connected components, one
of them containing $\mathcal{C}_{\ell}$. We will say that $\gamma_{2g+\ell}$
surrounds $\mathcal{C}_{\ell}$.

We may choose the generators and the representatives so that the only
intersection point of any pair of loops is $q_{0}$, and when labeling the rays
of the loops emerging from the base point $q_{0}$ by
$\alpha_{j}^{-},\,\beta_{j}^{-}$ $\gamma_{j}^{-}$, and the incoming rays by
$\alpha_{j}^{+},\,\beta_{j}^{+}$ $\gamma_{j}^{+}$, then moving in
counterclockwise direction along a small circle around $q_{0}$ we meet the
rays in the order

$\displaystyle\ldots,\alpha^{-}_{j},\beta^{-}_{j},\alpha^{+}_{j},\beta^{+}_{j},\ldots,\gamma^{-}_{k},\gamma^{+}_{k},\ldots\;.$

We denote the described system of generators by $\mathcal{E}$, and fix it
throughout the paper.

Figure 1.

Let $\tilde{X}$ be the universal covering of $X$. For each element
$e\in\pi_{1}(X,q_{0})$ we consider the subgroup $\langle e\rangle$ of
$\pi_{1}(X,q_{0})$ generated by $e$. Let $\sigma(e_{0})$ be the covering
transformation corresponding to $e_{0}$, and $\langle\sigma(e_{0})\rangle$ the
group generated by $\sigma(e_{0})$. Denote by
$\mathcal{E}_{j},\,j=2,\ldots,8,$ the set of primitive elements of
$\pi_{1}(X,q_{0})$ which can be written as product of at most $j$ factors with
each factor being either an element of $\mathcal{E}$ or an element of
$\mathcal{E}^{-1}$, the set of inverses of elements of $\mathcal{E}$. Denote
by $\lambda_{j}=\lambda_{j}(X)$ the maximum over $e\in\mathcal{E}_{j}$ of the
extremal length of the annulus $\tilde{X}\diagup\langle\sigma(e_{0})\rangle$.
The quantity $\lambda_{4}(X)$ (for mappings to the twice punctured complex
plane), or $\lambda_{8}(X)$ (for $(1,1)$-bundles) is the mentioned conformal
invariant.

Let $E$ be a finite subset of the Riemann sphere $\mathbb{P}^{1}$ which
contains at least three points. Let $X$ be a finite Riemann surface with non-
trivial fundamental group. A continuous map $f:X\to\mathbb{P}^{1}\setminus E$
is reducible if it is homotopic (as a mapping to $\mathbb{P}^{1}\setminus E$)
to a mapping whose image is contained in $D\setminus E$ for an open
topological disc $D\subset\mathbb{P}^{1}$ with $E\setminus D$ containing at
least two points of $E$. Otherwise the mapping is called irreducible.

In the following theorem we take $E=\\{-1,1,\infty\\}$. We will often refer to
$\mathbb{P}^{1}\setminus\\{-1,1,\infty\\}$ as to the twice punctured complex
plane $\mathbb{C}\setminus\\{-1,1\\}$. Note that a continuous mapping from a
Riemann surface to the twice punctured complex plane is reducible, iff it is
homotopic to a mapping with image in a once punctured disc contained in
$\mathbb{P}^{1}\setminus E$. (The puncture may be equal to $\infty$.) There
are countably many non-homotopic reducible holomorphic mappings from any
finite open Riemann surface of second kind with non-trivial fundamental group
to the twice punctured complex plane (see the proof of Theorem 7 in [21]). On
the other hand the following theorem holds.

######  Theorem 1.

For each open connected Riemann surface $X$ of genus $g$ with $m+1$ holes
there are up to homotopy at most
$3(\frac{3}{2}e^{24\pi\lambda_{4}(X)})^{2g+m}$ irreducible holomorphic
mappings from $X$ into $Y\stackrel{{\scriptstyle
def}}{{=}}\mathbb{P}^{1}\setminus\\{-1,1,\infty\\}$.

Notice that the Riemann surface $X$ is allowed to be of second kind. If $X$ is
a torus with a hole, $\lambda_{4}(X)$ may be replaced by $\lambda_{3}(X)$.

A holomorphic $(1,0)$-bundle is also called a holomorphic torus bundle. A
holomorphic torus bundle equipped with a holomorphic section is also
considered as a holomorphic $(1,1)$ fiber bundle. The following lemma holds.

Lemma D. A smooth $(0,1)$-bundle admits a smooth section. A holomorphic torus
bundle is (smoothly) isotopic to a holomorphic torus bundle that admits a
holomorphic section.

For a proof see [21].

######  Theorem 2.

Let $X$ be an open connected Riemann surface of genus $g$ with $m+1$ holes. Up
to isotopy there are no more than $\big{(}2\cdot 3^{6}\cdot
5^{6}\cdot\exp(36\pi\lambda_{8}(X))\big{)}^{2g+m}$ irreducible holomorphic
$(1,1)$-bundles over $X$.

We wish to point out that reducible $(\sf{g,m})$-bundles over finite open
Riemann surfaces can be decomposed into irreducible bundle components, and
each reducible bundle is determined by its bundle components up to commuting
Dehn twists in the fiber over the base point.

Notice that Caporaso proved the existence of a uniform bound of the number of
objects in Theorem A in case $X$ is a closed Riemann surface of genus $g$ with
$m$ punctures, and the fibers are closed Riemann surfaces of genus
$\textsf{g}\geq 2$. The bound depends only on the numbers $g$, g and $m$.
Heier gave effective uniform estimates, but the constants are huge and depend
in a complicated way on the parameters.

Theorems 1 and 2 imply effective estimates for the number of locally
holomorphically non-trivial holomorphic $(1,1)$-bundles over punctured Riemann
surfaces, however, the constants depend also on the conformal type of the
base. More precisely, the following corollaries hold.

######  Corollary 1.

There are no more than $3(\frac{3}{2}e^{24\pi\lambda_{4}(X)})^{2g+m}$ non-
constant holomorphic mappings from a Riemann surface $X$ of type $(g,m+1)$ to
$\mathbb{P}^{1}\setminus\\{-1,1,\infty\\}$.

######  Corollary 2.

There are no more than $\big{(}2\cdot 3^{6}\cdot
5^{6}\cdot\exp(36\pi\lambda_{8}(X))\big{)}^{2g+m}$ locally holomorphically
non-trivial holomorphic $(1,1)$-bundles over a Riemann surface $X$ of type
$(g,m+1)$.

The following examples demonstrate the different nature of the problem in the
two cases, the case when the base is a punctured Riemann surface, and when it
is a Riemann surface of second kind. The first example considers holomorphic
mappings from a torus with a hole to the twice punctured complex plane. The
second example considers mappings from a planar domain to the twice punctured
plane. It shows that for Riemann surfaces $X$ of second kind we can not expect
an upper bound for the number of irreducible holomorphic objects up to
homotopy depending only on the topological type of the base manifold $X$ and
of the fibers.

Example 1. There are no non-constant holomorphic mappings from a torus with
one puncture to the twice punctured complex plane. Indeed, by Picard’s Theorem
each such mapping extends to a meromorphic mapping from the closed torus to
the Riemann sphere. This implies that the preimage of the set
$\\{-1,1,\infty\\}$ under the extended mapping must contain at least three
points, which is impossible.

The situation changes if $X$ is a torus with a large enough hole. Let
$\alpha\geq 1$ and $\sigma\in(0,1)$. Consider the torus with a hole
$T^{\alpha,\sigma}$ that is obtained from
$\mathbb{\mathbb{C}}\diagup(\mathbb{Z}+i\alpha\mathbb{Z}),$ (with $\alpha\geq
1$ being a real number) by removing a closed geometric rectangle of vertical
side length $\alpha-\sigma$ and horizontal side length $1-\sigma$ (i.e. we
remove a closed subset that lifts to such a closed rectangle in $\mathbb{C}$).
A fundamental domain for this Riemann surface is ”the golden cross on the
Swedish flag” turned by $\frac{\pi}{2}$ with width of the laths being $\sigma$
and length of the laths being $1$ and $\alpha$.

######  Proposition 1.

Up to homotopy there are at most $7e^{3\cdot
2^{6}\pi\frac{2\alpha+1}{\sigma}}$ irreducible holomorphic mappings from
$T^{\alpha,\sigma}$ to the twice punctured complex plane.

On the other hand, there are positive constants $c$, $C$, and $\sigma_{0}$
such that for any positive number $\sigma<\sigma_{0}$ there are at least
$ce^{C\frac{\alpha}{\sigma}}$ non-homotopic holomorphic mappings from
$T^{\alpha,\sigma}$ to the twice punctured complex plane.

Example 2. There are only finitely many holomorphic maps from a thrice
punctured Riemann sphere to another thrice punctured Riemann sphere. Indeed,
after normalizing both, the source and the target space, by a Möbius
transformation we may assume that both are equal to
$\mathbb{C}\setminus\\{-1,1\\}$. Each holomorphic map from
$\mathbb{C}\setminus\\{-1,1\\}$ to itself extends to a meromorphic map from
the Riemann sphere to itself, which maps the set $\\{-1,1,\infty\\}$ to itself
and maps no other point to this set. By the Riemann-Hurwitz formula the
meromorphic map takes each value exactly once. Indeed, suppose it takes each
value $l$ times for a natural number $l$. Then each point in
$\\{-1,1,\infty\\}$ has ramification index $l$. Apply the Riemann Hurwitz
formula for the branched covering $Y=\mathbb{P}^{1}\to X=\mathbb{P}^{1}$ of
multiplicity $l$

$\chi(Y)=l\cdot\chi(X)-\sum_{x\in X}(e_{x}-1).$

Here $e_{x}$ is the ramification index at the point $x$. For the Euler
characteristic we have $\chi(\mathbb{P}^{1})=2$, and $\sum_{x\in
X}(e_{x}-1)\geq\sum_{x=-1,1,\infty}(e_{x}-1)=3\,(l-1)$. We obtain $2\leq
2\,l-3\,(l-1)$ which is possible only if $l=1$. We saw that each non-constant
holomorphic mapping from $\mathbb{C}\setminus\\{-1,1\\}$ to itself extends to
a conformal mapping from the Riemann sphere to the Riemann sphere that maps
the set $\\{-1,1,\infty\\}$ to itself. There are only finitely may such maps,
each a Möbius transformation commuting the three points.

A domain in the plane is called doubly connected, if its fundamental group is
isomorphic to the free group in two generators. We consider now doubly
connected planar domains contained in $\mathbb{C}\setminus\\{-1,1\\}$. We take
domains of a special form. Consider a skeleton $S$ of
$\mathbb{C}\setminus\\{-1,1\\}$ which is the union of two simple closed curves
$\gamma_{1}$ and $\gamma_{2}$ with base point zero, that surround $-1$ and
$+1$, respectively, and are piecewise smooth, and smooth and pairwise disjoint
outside the base point. Suppose $\gamma_{2}$ is contained in the cone $\\{\rho
e^{\theta}:\;\rho\geq 0,\,|\theta|\leq\kappa<\pi\\}$ and $\gamma_{1}$ is
contained in the cone $\\{\rho e^{\theta}:\;\rho\geq
0,\,|\theta-\pi|\leq\kappa<\pi\\}$. Let $\sigma$ be a small positive number
and let $\Omega_{\sigma}$ be the $\sigma$-neighbourhood of $S$.

######  Proposition 2.

There are positive constants $\sigma_{0}$, $C_{1}$ and $C_{2}$, depending only
on $\gamma_{1}$ and $\gamma_{2}$ such that for each positive
$\sigma<\sigma_{0}$ there are up to homotopy no more than
$C_{1}e^{\frac{C_{2}}{\sigma}}$ irreducible holomorphic mappings from
$\Omega_{\sigma}$ to the twice punctured complex plane.

Vice versa, there are positive constants $C_{1}^{\prime}$ and $C_{2}^{\prime}$
such that for each $\sigma<\sigma_{0}$ there are at least
$C_{1}^{\prime}e^{\frac{C_{2}^{\prime}}{\sigma}}$ non-homotopic irreducible
holomorphic mappings from $\Omega_{\sigma}$ to the twice punctured complex
plane.

The present results may be understood as quantitative statements with regard
to Gromov’s Oka principle. A mapping from a Stein manifold to a complex
manifold $Y$ is said to satisfy Gromov’s Oka principle if it is homotopic to a
holomorphic map. The target spaces $Y$, for which each continuous mapping from
a Stein manifold to $Y$ satisfies the Oka principle, are called Oka manifolds
and received a lot of attention ([12], [9], [10]).

The twice punctured complex plane is not an Oka manifold. Theorem 1 and
Propositions 1 and 2 give an estimate of the number of homotopy classes of
mappings from particular Stein manifolds to the twice punctured complex plane,
that satisfy Gromov’s Oka principle. The estimate depends on the conformal
structure of the Stein manifold.

The author is grateful to B.Farb who suggested to use the concept of conformal
module and extremal length for a proof of finiteness theorems, and to
B.Berndtsson for proposing the kernel for solving the $\bar{\partial}$-problem
that arises in the proof of Proposition 1. The work on the paper was started
while the author was visiting the Max-Planck-Institute and was finished during
a stay at IHES. The author would like to thank these institutions for the
support. The author is also indebted to Fanny Dufour for drawing the figures.

##  2\. Holomorphic mappings into the twice punctured plane

In this section we will prove Theorem 1. We first need some preparation.

The change of the base point. Let $\mathcal{X}$ be a connected smooth manifold
and let $\alpha$ be an arc in $\mathcal{X}$ with initial point $x_{0}$ and
terminating point $x$. Change the base point $x_{0}\in\mathcal{X}$ along a
curve $\alpha$ to the point $x\in\mathcal{X}$. This leads to an isomorphism
$\mbox{Is}_{\alpha}:\pi_{1}(\mathcal{X},x_{0})\to\pi_{1}(\mathcal{X},x)$ of
fundamental groups induced by the correspondence
$\gamma\to\alpha^{-1}\gamma\alpha$ for any loop $\gamma$ with base point
$x_{0}$ and the arc $\alpha$ with initial point $x_{0}$ and terminating point
$x$. We will denote the correspondence $\gamma\to\alpha^{-1}\gamma\alpha$
between curves also by $\mbox{Is}_{\alpha}$.

We call two homomorphisms $h_{j}:G_{1}\to G_{2},\,j=1,2,\,$ from a group
$G_{1}$ to a group $G_{2}$ conjugate if there is an element $g^{\prime}\in
G_{2}$ such that for each $g\in G_{1}$ the equality
$h_{2}(g)={g^{\prime}}^{-1}h_{1}(g)g^{\prime}$ holds. For two arcs
$\alpha_{1}$ and $\alpha_{2}$ with initial point $x_{0}$ and terminating point
$x$ we have
$\alpha_{2}^{-1}\gamma\alpha_{2}=(\alpha_{1}^{-1}\alpha_{2})^{-1}\alpha_{1}^{-1}\gamma\alpha_{1}(\alpha_{1}^{-1}\alpha_{2})$.
Hence, the two isomorphisms $\mbox{Is}_{\alpha_{1}}$ and
$\mbox{Is}_{\alpha_{2}}$ differ by conjugation with the element of
$\pi_{1}(\mathcal{X},x)$ represented by $\alpha_{1}^{-1}\alpha_{2}$.

Free homotopic curves are related by homotopy with fixed base point and an
application of a homomorphism $\mbox{Is}_{\alpha}$ that is defined up to
conjugation. Hence, free homotopy classes of curves can be identified with
conjugacy classes of elements of the fundamental group
$\pi_{1}(\mathcal{X},x_{0})$ of $\mathcal{X}$.

For two smooth manifolds $\mathcal{X}$ and $\mathcal{Y}$ with base points
$x_{0}\in\mathcal{X}$ and $y_{0}\in\mathcal{Y}$ and a continuous mapping
$F:\mathcal{X}\to\mathcal{Y}$ with $F(x_{0})=y_{0}$ we denote by
$F_{*}:\pi_{1}(\mathcal{X},x_{0})\to\pi_{1}(\mathcal{Y},y_{0})$ the induced
map on fundamental groups. For each element
$e_{0}\in\pi_{1}(\mathcal{X},x_{0})$ the image $F_{*}(e_{0})$ is called the
monodromy along $e_{0}$, and the homomorphism $F_{*}$ is called the monodromy
homomorphism corresponding to $F$. The homomorphism $F_{*}$ determines the
homotopy class of $F$ with fixed base point in the source and fixed value at
the base point. Consider a free homotopy $F_{t},\,t\in(0,1)$, of
homeomorphisms from $\mathcal{X}$ to $\mathcal{Y}$ such that the value
$F_{t}(x_{0})$ at the base point $x_{0}$ of the source space varies along a
loop. The homomorphisms $(F_{0})_{*}$ and $(F_{1})_{*}$ are related by
conjugation with the element of the fundamental group of $\mathcal{Y}$
represented by the loop. Moreover, since the fundamental group
$\pi_{1}(\mathcal{Y},y)$ with base point $y$ is related to the fundamental
group $\pi_{1}(\mathcal{Y},y_{0})$ with base point $y_{0}$ by an isomorphism
determined up to conjugation we obtain the following theorem (see [13],[28]).

Theorem E. The free homotopy classes of continuous mappings from $\mathcal{X}$
to $\mathcal{Y}$ are in one-to-one correspondence to the set of conjugacy
classes of homomorphisms between the fundamental groups of $\mathcal{X}$ and
$\mathcal{Y}$.

Extremal length. The fundamental group $\pi_{1}\stackrel{{\scriptstyle
def}}{{=}}\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},0)$ is canonically isomorphic
to the fundamental group $\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$
for an arbitrary point $q^{\prime}\in(-1,1)$. For the arc $\alpha$ defining
the isomorphism we take the unique arc contained in $(-1,1)$ that joins
$q^{\prime}$ and $0$. The fundamental group
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},0)$ is a free group in two generators.
We choose standard generators $a_{1}$ and $a_{2}$, where $a_{1}$ is
represented by a simple closed curve with base point $0$ which surrounds $-1$
counterclockwise, and $a_{2}$ is represented by a simple closed curve with
base point $0$ which surrounds $1$ counterclockwise. For $q^{\prime}\in(-1,1)$
we also denote by $a_{j}$ the generator of
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ which is obtained from the
respective standard generator of $\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},0)$ by
the standard isomorphism between fundamental groups with base point on
$(-1,1)$. More detailed, $a_{1}$ is the generator of
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ which is represented by a
loop with base point $q^{\prime}$ that surrounds $-1$ counterclockwise, and
$a_{2}$ is the generator of
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ which is represented by a
loop with base point $q^{\prime}$ that surrounds $1$ counterclockwise. We
refer to $a_{1}$ and $a_{2}$ as to the standard generators of
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$.

Further, the group $\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},0)$ is canonically
isomorphic to the relative fundamental group
$\pi_{1}^{tr}(\mathbb{C}\setminus\\{-1,1\\})\stackrel{{\scriptstyle
def}}{{=}}\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},(-1,1))$ whose elements are
homotopy classes of (not necessarily closed) curves in
$\mathbb{C}\setminus\\{-1,1\\}$ with end points on the interval $(-1,1)$. We
refer to $\pi_{1}^{tr}(\mathbb{C}\setminus\\{-1,1\\})$ as fundamental group
with totally real horizontal boundary values ($tr$-boundary values for short).
For an element $w\in\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ with
base point $q^{\prime}\in(-1,1)$ we denote by $w_{tr}$ the element of the
relative fundamental group $\pi_{1}^{tr}(\mathbb{C}\setminus\\{-1,1\\})$ with
totally real boundary values, corresponding to $w$. For more details see [18].

Each element of a free group can be written uniquely as a reduced word in the
generators. (A word is reduced if neighbouring terms are powers of different
generators.) The degree (or word length) $d(w)$ of a reduced word $w$ in the
generators of a free group is the sum of the absolute values of the powers of
generators in the reduced word. If the word is the identity its degree is
defined to be zero. We will identify elements of a free group with reduced
words in generators of the group.

For a rectangle $R$ let $f:R\to\mathbb{C}\setminus\\{-1,1\\}$ be a mapping
which admits a continuous extension to the closure $\bar{R}$ (denoted again by
$f$) which maps the (open) horizontal sides to $(-1,1)$. We say that the
mapping $f$ represents an element
$w_{tr}\in\pi_{1}^{tr}(\mathbb{C}\setminus\\{-1,1\\})$ if for each maximal
vertical line segment contained in $R$ (i.e. $R$ intersected with a vertical
line in $\mathbb{C}$) the restriction of $f$ to the closure of the line
segment represents $w_{tr}$.

The extremal length $\;\;\Lambda(w_{tr})\;\;$ of an element $\;\;w_{tr}\;\;$
in the relative fundamental group
$\;\pi_{1}^{tr}(\mathbb{C}\setminus\\{-1,1\\})\;$ is defined as

$\displaystyle\Lambda(w_{tr})\stackrel{{\scriptstyle def}}{{=}}$
$\displaystyle\inf\\{\lambda(R):R\,\mbox{ a rectangle which admits a
holomorphic map to}\;\mathbb{C}\setminus\\{-1,1\\}$ $\displaystyle\,\mbox{
that represents}\;w_{tr}\\}\,.$ (1)

For an element $w\in\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ and the
associated element $w_{tr}$ we will also write $\Lambda_{tr}(w)$ instead of
$\Lambda(w_{tr})$. In [18] (see also [19]) a natural syllable decomposition of
any reduced word $\;w\;$ in
$\;\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})\;$ is given. Let $d_{k}$
be the degree of the $k$-th syllable from the left. (We consider each syllable
as a reduced word in the elements of the fundamental group.) Put

$\mathcal{L}_{-}(w)\stackrel{{\scriptstyle
def}}{{=}}\sum\log(3d_{k}),\quad\mathcal{L}_{+}(w)\stackrel{{\scriptstyle
def}}{{=}}\sum\log(4d_{k})\,.$ (2)

Notice that $\mathcal{L}_{\pm}(w^{-1})=\mathcal{L}_{\pm}(w)$. We define
$\mathcal{L}_{-}({\rm Id})=\mathcal{L}_{+}({\rm Id})=0$ for the identity ${\rm
Id}$. We need the following theorem which is proved in [18] (see Theorem 1′
there).

Theorem F. If $w$ is not equal to a (trivial or non-trivial) power of $a_{1}$
or of $a_{2}$ then

$\frac{1}{2\pi}\mathcal{L}_{-}(w)\leq\Lambda(w_{tr})\leq
300\mathcal{L}_{+}(w)\,$ (3)

where the sum runs over the degrees of all syllables of $w_{tr}$.

Regular zero sets. We need some facts concerning regular zero sets of smooth
real valued functions. We will call a subset of a smooth manifold
$\mathcal{X}$ a simple relatively closed curve if it is the connected
component of a regular level set of a smooth real-valued function on
$\mathcal{X}$.

Let $\mathcal{X}$ be a connected finite open Riemann surface. Suppose the zero
set $L$ of a non-constant smooth real valued function on $\mathcal{X}$ is
regular. Each component of $L$ is either a simple closed curve or it can be
parameterized by a continuous mapping $\ell:(-\infty,\infty)\to\mathcal{X}$.
We call a component of the latter kind a simple relatively closed arc in
$\mathcal{X}$.

A relatively closed curve $\gamma$ in a connected finite open Riemann surface
$\mathcal{X}$ is said to be contractible to a hole of $\mathcal{X}$, if the
following holds. Consider $\mathcal{X}$ as domain
$\mathcal{X}^{c}\setminus\cup\mathcal{C}_{j}$ on a closed Riemann surface
$\mathcal{X}^{c}$. Here the $\mathcal{C}_{j}$ are the holes, each is either a
closed disc or a point. The condition is the following. For each pair $U_{1}$,
$U_{2}$ of open subsets of $\mathcal{X}^{c}$, $\cup\mathcal{C}_{j}\subset
U_{1}\Subset U_{2}$, there exists a homotopy of $\gamma$ that fixes
$\gamma\cap U_{1}$ and moves $\gamma$ into $U_{2}$. Taking for $U_{2}$ small
enough neighbourhoods of $\cup\mathcal{C}_{j}$ we see that the homotopy moves
$\gamma$ into an annulus adjacent to one of the holes.

For each relatively compact domain $\mathcal{X}^{\prime}\Subset\mathcal{X}$ in
$\mathcal{X}$ there is a finite cover of
$L\cap\overline{\mathcal{X}^{\prime}}$ by open subsets $U_{k}$ of
$\mathcal{X}$ such that each $L\cap U_{k}$ is connected. Each set $L\cap
U_{k}$ is contained in a component of $L$. Hence, only finitely many connected
components of $L$ intersect $\mathcal{X}^{\prime}$. Let $L_{0}$ be a connected
component of $L$ which is a simple relatively closed arc parameterized by
$\ell_{0}:\mathbb{R}\to\mathcal{X}$. Since each set $L_{0}\cap U_{k}$ is
connected it is the image of an interval under $\ell_{0}$. Take real numbers
$t_{0}^{-}$ and $t_{0}^{+}$ such that all these intervals are contained in
$(t_{0}^{-},t_{0}^{+})$. Then the images
$(\ell\big{(}(-\infty,t_{0}^{-})\big{)}$ and
$(\ell\big{(}(t_{0}^{+},+\infty)\big{)}$ are contained in
$\mathcal{X}\setminus\mathcal{X}^{\prime}$, maybe, in different components.
Such parameters $t_{0}^{-}$ and $t_{0}^{+}$ can be found for each relatively
compact deformation retract $\mathcal{X}^{\prime}$ of $\mathcal{X}$. Hence for
each relatively closed arc $L_{0}\subset L$ the set of limit points
$L_{0}^{+}$ of $\ell_{0}(t)$ for $t\to\infty$ is contained in a boundary
component of $\mathcal{X}$. Also, the set of limit points $L_{0}^{-}$ of
$\ell_{0}(t)$ for $t\to-\infty$ is contained in a boundary component of
$\mathcal{X}$. The boundary components may be equal or different. Moreover, if
$\mathcal{X}^{\prime}\Subset\mathcal{X}$ is a relatively compact domain in
$\mathcal{X}$ which is a deformation retract of $\mathcal{X}$, and a connected
component $L_{0}$ of $L$ does not intersect $\overline{\mathcal{X}^{\prime}}$
then $L_{0}$ is contractible to a hole of $\mathcal{X}$. Indeed,
$\mathcal{X}\setminus\overline{\mathcal{X}^{\prime}}$ is the union of disjoint
annuli, each of which is adjacent to a boundary component of $\mathcal{X}$,
and the union of the limit sets $L_{0}^{+}\cup L_{0}^{-}$ is contained in a
single boundary component of $\mathcal{X}$.

Further, denote by $L^{\prime}$ the union of all connected components of $L$
that are simple relatively closed arcs. Consider those components $L_{j}$ of
$L^{\prime}$ that intersect $\mathcal{X}^{\prime}$. There are finitely many
such $L_{j}$. Parameterize each $L_{j}$ by a mapping
$\ell_{j}:\mathbb{R}\to\mathcal{X}$. For each $j$ we let
$[t_{j}^{-},t_{j}^{+}]$ be a compact interval for which

$\ell_{j}(\mathbb{R}\setminus[t_{j}^{-},t_{j}^{+}])\subset\mathcal{X}\setminus\overline{\mathcal{X}^{\prime}}\,.$
(4)

Let $\mathcal{X}^{\prime\prime}$,
$\mathcal{X}^{\prime}\Subset\mathcal{X}^{\prime\prime}\Subset\mathcal{X}$, be
a domain which is a deformation retract of $\mathcal{X}$ such that
$\ell_{j}([t_{j}^{-},t_{j}^{+}])\subset\mathcal{X}^{\prime\prime}$ for each
$j$. Then all connected components of
$L^{\prime}\cap\mathcal{X}^{\prime\prime}$, that do not contain a set
$\ell_{j}([t_{j}^{-},t_{j}^{+}])$, are contractible to a hole of
$\mathcal{X}^{\prime\prime}$. Indeed, each such component is contained in the
union of annuli
$\mathcal{X}^{\prime\prime}\setminus\overline{\mathcal{X}^{\prime}}$.

Some remarks on coverings. Let $X$ be a finite open Riemann surface with base
point $q_{0}$ and let $P:\tilde{X}\to X$ be the universal covering map. By a
covering $P:\mathcal{Y}\to\mathcal{X}$ we mean a continuous map $P$ from a
topological space $\mathcal{X}$ to a topological space $\mathcal{Y}$ such that
for each point $x\in\mathcal{X}$ there is a neighbourhood $V(x)$ of $x$ such
that the mapping $P$ maps each connected component of the preimage of $V(x)$
homeomorphically onto $V(x)$. (Note that in function theory sometimes these
objects are called unlimited unramified coverings to reserve the notion
”covering” for more general objects.) Recall that a homeomorphism
$\varphi:\tilde{X}\to\tilde{X}$ for which $P\circ\varphi=P$ is called a
covering transformation (or deck transformation). The covering transformations
form a group, denoted by ${\rm Deck}(\tilde{X},X)$. For each pair of points
$\tilde{x}_{1},\tilde{x}_{2}\in\tilde{X}$ with
$P(\tilde{x}_{1})=P(\tilde{x}_{2})$ there exists exactly one covering
transformation that maps $\tilde{x}_{1}$ to $\tilde{x}_{2}$. (See e.g. [8]).

Throughout the paper we will fix a base point $q_{0}\in X$ and a base point
$\tilde{q}_{0}\in{\sf P}^{-1}(q_{0})\subset\tilde{X}$. The group of covering
transformations of $\tilde{X}$ can be identified with the fundamental group
$\pi_{1}(X,q_{0})$ of $X$ by the following correspondence. (See e.g. [8]).

Take a covering transformation $\sigma\in{\rm Deck}(\tilde{X},X)$. Let
$\tilde{\gamma}_{0}$ be an arc in $\tilde{X}$ with initial point
$\sigma(\tilde{q}_{0})$ and terminating point $\tilde{q}_{0}$. Denote by ${\rm
Is}^{\tilde{q}_{0}}(\sigma)$ the element of $\pi_{1}(X,q_{0})$ represented by
the loop $P(\tilde{\gamma}_{0})$. The mapping ${\rm
Deck}(\tilde{X},X)\ni\sigma\to{\rm
Is}^{\tilde{q}_{0}}(\sigma)\in\pi_{1}(X,q_{0})$ is a group homomorphism. The
homomorphism ${\rm Is}^{\tilde{q}_{0}}$ is injective and surjective, hence it
is a group isomorphism. The inverse $({\rm Is}^{\tilde{q}_{0}})^{-1}$ is
obtained as follows. Represent an element $e_{0}\in\pi_{1}(X,q_{0})$ by a loop
$\gamma_{0}$. Consider the lift $\tilde{\gamma}_{0}$ of $\gamma_{0}$ to
$\tilde{X}$ that has initial point $\tilde{q}_{0}$. Then $({\rm
Is}^{\tilde{q}_{0}})^{-1}(e_{0})$ is the covering transformation that maps the
endpoint of $\tilde{\gamma}_{0}$ to $\tilde{q}_{0}$.

For another point $\tilde{q}$ of $\tilde{X}$ and the point
$q\stackrel{{\scriptstyle def}}{{=}}{\sf P}(\tilde{q})\in X$ the isomorphism
${\rm Is}^{\tilde{q}}:{\rm Deck}(\tilde{X},X))\to\pi_{1}(X,q)$ assigns to each
$\sigma\in{\rm Deck}(\tilde{X},X)$ the element of $\pi_{1}(X,q)$ that is
represented by ${\sf P}(\tilde{\gamma})$ for a curve $\tilde{\gamma}$ in
$\tilde{X}$ that joins $\sigma(\tilde{q})$ with $\tilde{q}$. ${\rm
Is}^{\tilde{q}}$ is related to ${\rm Is}^{\tilde{q}_{0}}$ as follows. Let
$\tilde{\alpha}$ be an arc in $\tilde{X}$ with initial point $\tilde{q}_{0}$
and terminating point $\tilde{q}$. Put $q={\sf P}(\tilde{q})$ and $\alpha={\sf
P}(\tilde{\alpha})$. Then for the isomorphism ${\rm
Is}_{\alpha}:\pi_{1}(X,q_{0})\to\pi_{1}(X,q)$ the equation

${\rm Is}^{\tilde{q}}(\sigma)={\rm Is}_{\alpha}\circ{\rm
Is}^{\tilde{q}_{0}}(\sigma),\;\;\sigma\in{\rm Deck}(\tilde{X},X),$ (5)

holds, i.e. the following diagram Figure 2 is commutative.

---
$\textstyle{{\pi_{1}(X,q_{0})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
Is}_{\alpha}}$$\textstyle{{{\rm
Deck}(\tilde{X},X)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
Is}^{\tilde{q}_{0}}}$$\scriptstyle{{\rm
Is}^{\tilde{q}}}$$\textstyle{\pi_{1}(X,q)}$ Figure 2.

Indeed, let $\tilde{\alpha}^{-1}$ denote the curve that is obtained from
$\tilde{\alpha}$ by inverting the direction on $\tilde{\alpha}$, i.e. moving
from $\tilde{q}$ to $\tilde{q}_{0}$. The curve $\sigma((\tilde{\alpha})^{-1})$
has initial point $\sigma(\tilde{q})$ and terminating point
$\sigma(\tilde{q}_{0})$. Hence, for a curve $\tilde{\gamma}_{0}$ in
$\tilde{X}$ that joins $\sigma(\tilde{q}_{0})$ with $\tilde{q}_{0}$, the curve
$\sigma((\tilde{\alpha})^{-1})\;\tilde{\gamma}_{0}\;\tilde{\alpha}$ in
$\tilde{X}$ has initial point $\sigma(\tilde{q})$ and terminating point
$\tilde{q}$. Therefore ${\sf
P}(\sigma((\tilde{\alpha})^{-1})\;\tilde{\gamma}_{0}\;\tilde{\alpha})$
represents ${\rm Is}^{\tilde{q}}(\sigma)$. On the other hand

${\sf P}(\sigma(\tilde{\alpha}^{-1})\;\tilde{\gamma}_{0}\;\tilde{\alpha})={\sf
P}(\sigma(\tilde{\alpha}^{-1}))\;{\sf P}(\tilde{\gamma}_{0})\;{\sf
P}(\tilde{\alpha})=\alpha^{-1}\gamma_{0}\,\alpha$ (6)

represents ${\rm Is}_{\alpha}(e_{0})$. In particular, if
$\tilde{q}^{\prime}_{0}\in{\sf P}^{-1}(q_{0})$ is another preimage of the base
point $q_{0}$ under the projection $\sf P$, then the associated isomorphisms
to the fundamental group $\pi_{1}(X,q_{0})$ are conjugate, i.e. ${\rm
Is}^{\tilde{q}^{\prime}_{0}}(e_{0})=(e^{\prime}_{0})^{-1}{\rm
Is}^{\tilde{q}_{0}}(e_{0})e^{\prime}_{0}$ for each $e_{0}\in\pi_{1}(X,q_{0})$.
The element $e^{\prime}_{0}$ is represented by the projection of an arc in
$\tilde{X}$ with initial point $\tilde{q}_{0}$ and terminating point
$\tilde{q}^{\prime}_{0}$.

Keeping fixed $\tilde{q}_{0}$ qnd $q_{0}$ we will say that a point
$\tilde{q}\in\tilde{X}$ and a curve $\alpha$ in $X$ are compatible if the
diagram Figure 2 is commutative, equivalently, if equation (5) holds. We may
also start with choosing a curve $\alpha$ in $X$ with initial point $q_{0}$
and terminating point $q$. Then there is a point
$\tilde{q}=\tilde{q}(\alpha)$, such that $\tilde{q}$ and $\alpha$ are
compatible. Indeed, let $\tilde{\alpha}$ be the lift of $\alpha$, that has
initial point $\tilde{q}_{0}$. Denote the terminating point of
$\tilde{\alpha}$ by $\tilde{q}(\alpha)$, and repeat the previous arguments.

Let $N$ be a subgroup of $\pi_{1}(X,q_{0})$. Denote by $X(N)$ the quotient
$\tilde{X}\diagup({\rm Is}^{\tilde{q}_{0}})^{-1}(N)$. We obtain a covering
$\omega^{N}_{\mbox{Id}}:\tilde{X}\to X(N)$ with group of covering
transformations isomorphic to $N$. The fundamental group of $X(N)$ with base
point $(q_{0})_{N}\stackrel{{\scriptstyle def}}{{=}}\omega_{\rm
Id}^{N}(\tilde{q}_{0})$ can be identified with $N$.

If $N_{1}$ and $N_{2}$ are subgroups of $\pi_{1}(X,q_{0})$ and $N_{1}$ is a
subgroup of $N_{2}$ (we write $N_{1}\leq N_{2}$) then there is a covering map
$\omega^{N_{2}}_{N_{1}}:\tilde{X}\diagup({\rm
Is}^{\tilde{q}_{0}})^{-1}(N_{1})\to\tilde{X}\diagup({\rm
Is}^{\tilde{q}_{0}})^{-1}(N_{2})$, such that
$\omega^{N_{2}}_{N_{1}}\circ\omega^{N_{1}}_{\mbox{Id}}=\omega^{N_{2}}_{\mbox{Id}}$.
Moreover, the diagram Figure 3 below is commutative.

Figure 3.

Indeed, take any point $x_{1}\in\tilde{X}\diagup({\rm
Is}^{\tilde{q}_{0}})^{-1}(N_{1})$ and a preimage $\tilde{x}$ of $x_{1}$ under
$\omega^{N_{1}}_{\mbox{\footnotesize{Id}}}$. There exists a neighbourhood
$V(\tilde{x})$ of $\tilde{x}$ in $\tilde{X}$ such that
$V(\tilde{x})\cap\sigma(V(\tilde{x}))=\emptyset$ for all covering
transformations $\sigma\in{\rm Deck}(\tilde{X},X)$. Then for $j=1,2$ the
mapping
$\omega^{N_{j},\tilde{x}}_{\mbox{\footnotesize{Id}}}\stackrel{{\scriptstyle
def}}{{=}}\omega^{N_{j}}_{\mbox{\footnotesize{Id}}}\mid V(\tilde{x})$ is a
homeomorphism from $V(\tilde{x})$ onto its image denoted by $V_{j}$. Put
$x_{2}=\omega^{N_{j},\tilde{x}}_{\mbox{\footnotesize{Id}}}(\tilde{x})$. The
set $V_{j}\subset\tilde{X}\diagup({\rm Is}^{\tilde{q}_{0}})^{-1}(N_{j})$ is a
neighbourhood of $x_{j}$ for $j=1,2$.

For each preimage
$\tilde{x}^{\prime}\in(\omega^{N_{1}}_{\mbox{\footnotesize{Id}}})^{-1}(x_{1})$
there is a covering transformation $\varphi_{\tilde{x},\tilde{x}^{\prime}}$ in
$({\rm Is}^{\tilde{q}_{0}})^{-1}(N_{1})$ which maps the neighbourhood
$V(\tilde{x}^{\prime})$ of $\tilde{x}^{\prime}$ conformally onto the
neighbourhood $V(\tilde{x})$ of $\tilde{x}$ so that on $V(\tilde{x}^{\prime})$
the equality
$\omega^{N_{1},\tilde{x}^{\prime}}_{\mbox{\footnotesize{Id}}}=\omega^{N_{1},\tilde{x}}_{\mbox{\footnotesize{Id}}}\circ\varphi_{\tilde{x},\tilde{x}^{\prime}}$
holds. Choose
$\tilde{x}\in(\omega^{N_{1}}_{\mbox{\footnotesize{Id}}})^{-1}(x_{1})$ and
define
$\omega^{N_{2}}_{N_{1}}(y)=\omega^{N_{2},\tilde{x}}_{N_{1}}(y)\stackrel{{\scriptstyle
def}}{{=}}\omega^{N_{2},\tilde{x}}_{\mbox{\footnotesize{Id}}}((\omega^{N_{1},\tilde{x}}_{\mbox{\footnotesize{Id}}})^{-1}(y))$
for each $y\in V_{1}$. We get a correctly defined mapping from $V_{1}$ onto
$V_{2}$. Indeed, since $N_{1}$ is a subgroup of $N_{2}$, the covering
transformation $\varphi_{\tilde{x},\tilde{x}^{\prime}}$ is contained in $({\rm
Is}^{\tilde{q}_{0}})^{-1}(N_{2})$, and we get for another point
$\tilde{x}^{\prime}\in(\omega^{N_{1}}_{\mbox{\footnotesize{Id}}})^{-1}(x_{1})$
the equality
$\omega^{N_{2},\tilde{x}^{\prime}}_{\mbox{\footnotesize{Id}}}=\omega^{N_{2},\tilde{x}}_{\mbox{\footnotesize{Id}}}\circ\varphi_{\tilde{x},\tilde{x}^{\prime}}$.
Hence, for $y\in V_{1}(x_{1})$

$\omega^{N_{2},\tilde{x}^{\prime}}_{\mbox{\footnotesize{Id}}}\circ(\omega^{N_{1},\tilde{x}^{\prime}}_{\mbox{\footnotesize{Id}}})^{-1}(y)=(\omega^{N_{2},\tilde{x}}_{\mbox{\footnotesize{Id}}}\circ\varphi_{\tilde{x},\tilde{x}^{\prime}})\circ(\omega^{N_{1},\tilde{x}}_{\mbox{\footnotesize{Id}}}\circ\varphi_{\tilde{x},\tilde{x}^{\prime}})^{-1}(y)=\omega^{N_{2},\tilde{x}}_{\mbox{\footnotesize{Id}}}\circ(\omega^{N_{1},\tilde{x}}_{\mbox{\footnotesize{Id}}})^{-1}(y).$

Since the mappings $\omega^{N_{j},\tilde{x}}_{\mbox{\footnotesize{Id}}}$,
$j=1,2,$ are homeomorphisms from $V(\tilde{x})$ onto its image, the mapping
$\omega^{N_{2}}_{N_{1}}$ is a homeomorphism from $V(x_{1})$ onto $V(x_{2})$.
The same holds for all preimages of $V(x_{2})$ under $\omega^{N_{2}}_{N_{1}}$.
Hence, $\omega^{N_{2}}_{N_{1}}$ is a covering map. The commutativity of the
part of the diagram that involves the mappings
$\omega^{N_{1}}_{\mbox{\footnotesize{Id}}}$,
$\omega^{N_{2}}_{\mbox{\footnotesize{Id}}}$, and $\omega_{N_{1}}^{N_{2}}$ is
clear by construction.

The existence of $\omega_{N_{1}}^{\pi_{1}(X,q_{0})}$ and the equality
$P=\omega_{N_{1}}^{\pi_{1}(X,q_{0})}\circ\omega^{N_{1}}_{\mbox{\footnotesize{Id}}}$
follows by applying the above arguments with $N_{2}=\pi_{1}(X,q_{0})$. The
equality
$P=\omega_{N_{2}}^{\pi_{1}(X,q_{0})}\circ\omega^{N_{2}}_{\mbox{\footnotesize{Id}}}$
follows in the same way. Since

$\displaystyle P=$
$\displaystyle\omega_{N_{2}}^{\pi_{1}(X,q_{0})}\circ\omega_{N_{1}}^{N_{2}}\circ\omega^{N_{1}}_{\mbox{\footnotesize{Id}}}$
$\displaystyle=$
$\displaystyle\omega_{N_{1}}^{\pi_{1}(X,q_{0})}\quad\quad\;\;\,\circ\omega^{N_{1}}_{\mbox{\footnotesize{Id}}},$

we have

$\omega_{N_{2}}^{\pi_{1}(X,q_{0})}\circ\omega_{N_{1}}^{N_{2}}=\omega_{N_{1}}^{\pi_{1}(X,q_{0})}$

We will also use the notation $\omega^{N}\stackrel{{\scriptstyle
def}}{{=}}\omega^{N}_{\mbox{\footnotesize{Id}}}$ and
$\omega_{N}\stackrel{{\scriptstyle def}}{{=}}\omega_{N}^{\pi_{1}(X,q_{0})}$
for a subgroup $N$ of $\pi_{1}(X,q_{0})$.

In our situation $N$ will usually be a subgroup
$\langle\tilde{\mathcal{E}}\rangle$ of $\pi_{1}(X,q_{0})$ generated by a
system $\tilde{\mathcal{E}}$ of elements of
$\mathcal{E}\subset\pi_{1}(X,q_{0})$.

Consider a set $\mathcal{E}^{\prime}\subset\pi_{1}(X,x_{0})$ that consists of
$2g^{\prime}+m^{\prime}$ elements of $\mathcal{E}$, such that among them are
exactly $g^{\prime}$ of the chosen pairs in $\mathcal{E}$ that correspond to
$g^{\prime}$ of the handles of $X$, and from all remaining pairs of elements
of $\mathcal{E}$ corresponding to a handle at most one element is contained in
$\mathcal{E}^{\prime}$. Then $X(\langle\mathcal{E}^{\prime}\rangle)$ is a
surface of genus $g^{\prime}$ with $m^{\prime}$ holes. Indeed, consider the
chosen loops $\alpha_{j},\,\beta_{j}$, and $\gamma_{2g+\ell}$, that represent
the elements of $\mathcal{E}$. The union of those of them that represent
elements of $\mathcal{E}^{\prime}$ can be identified with a deformation
retract of $X(\langle\mathcal{E}^{\prime}\rangle)$. To see this, we lift the
collection of all $\alpha_{j},\,\beta_{j}$, and $\gamma_{j}$, representing
elements of $\mathcal{E}$ to arcs in the universal covering $\tilde{X}$ of $X$
with initial point $\tilde{q}_{0}$, and take the quotient
$X(\langle\mathcal{E}^{\prime}\rangle)$ of $\tilde{X}$ by the action of ${\rm
Is}^{\tilde{q}_{0}}(\langle\mathcal{E}^{\prime}\rangle)$. The lifts of the
loops representing elements of $\mathcal{E}\setminus\mathcal{E}^{\prime}$
project to arcs in $X(\langle\mathcal{E}^{\prime}\rangle)$. The lifts of the
loops representing elements of $\mathcal{E}^{\prime}$ project to loops in
$X(\langle\mathcal{E}^{\prime}\rangle)$, the union of which is a bouquet of
circles which is a deformation retract of
$X(\langle\mathcal{E}^{\prime}\rangle)$.

Let again $N_{1}\leq N_{2}$ be subgroups of $\pi_{1}(X,q_{0})$. Consider the
covering $\omega_{N_{1}}^{N_{2}}:\tilde{X}\diagup({\rm
Is}^{{\tilde{q}}_{0}})^{-1}(N_{1})\to\tilde{X}\diagup({\rm
Is}^{{\tilde{q}}_{0}})^{-1}(N_{2})$. Let $\beta$ be a simple relatively closed
curve in $\tilde{X}\diagup({\rm Is}^{\tilde{q}_{0}})^{-1}(N_{2})$. Then
$(\omega_{N_{1}}^{N_{2}})^{-1}(\beta)$ is the union of simple relatively
closed curves in $\tilde{X}\diagup({\rm Is}^{{\tilde{q}}_{0}})^{-1}(N_{1})$
and $\omega_{N_{1}}^{N_{2}}:(\omega_{N_{1}}^{N_{2}})^{-1}(\beta)\to\beta$ is a
covering. Indeed, we cover $\beta$ by small discs $U_{k}$ in
$\tilde{X}\diagup({\rm Is}^{{\tilde{q}}_{0}})^{-1}(N_{1})$ such that for each
$k$ the restriction of $\omega_{N_{1}}^{N_{2}}$ to each connected component of
$(\omega_{N_{1}}^{N_{2}})^{-1}(U_{k})$ is a homeomorphism onto $U_{k}$, and
$U_{k}$ intersects $\beta$ along a connected set. Take any $k$ with
$U_{k}\cap\beta\neq\emptyset$. Consider the preimages
$(\omega_{N_{1}}^{N_{2}})^{-1}(U_{k})$. Restrict $(\omega_{N_{1}}^{N_{2}})$ to
the intersection of each such preimage with
$(\omega_{N_{1}}^{N_{2}})^{-1}(\beta)$. We obtain a homeomorphism onto
$U_{k}\cap\beta$.

The extremal length of monodromies. Let as before $X$ be a connected finite
open Riemann surface with base point $q_{0}$, and $\tilde{q}_{0}$ a point in
the universal covering $\tilde{X}$ for which ${\sf P}(\tilde{q}_{0})=q_{0}$
for the covering map ${\sf P}:\tilde{X}\to X$.

Recall that for an arbitrary point $q\in X$ the free homotopy class of an
element $e$ of the fundamental group $\pi_{1}(X,q)$ can be identified with the
conjugacy of elements of $\pi_{1}(X,q)$ containing $e$ and is denoted by
$\widehat{e}$. Notice that for $e_{0}\in\pi_{1}(X,q_{0})$ and a curve $\alpha$
in $X$ with initial point $q_{0}$ and terminating point $q$ the free homotopy
classes of $e_{0}$ and of $e={\rm Is}_{\alpha}(e_{0})$ coincide, i.e.
$\widehat{e}=\widehat{e}_{0}$. Consider a simple smooth relatively closed
curve $L$ in $X$. We will say that a free homotopy class of curves
$\widehat{e_{0}}$ intersects $L$ if each representative of $\widehat{e_{0}}$
intersects $L$. Choose an orientation of $L$. The intersection number of
$\widehat{e_{0}}$ with the oriented curve $L$ is the intersection number with
$L$ of some (and, hence, of each) smooth loop representing $\widehat{e_{0}}$
that intersects $L$ transversally. This intersection number is the sum of the
intersection numbers over all intersection points. The intersection number at
an intersection point equals $+1$ if the orientation determined by the tangent
vector to $L$ followed by the tangent vector to the curve representing
$\widehat{e_{0}}$ is the orientation of $X$, and equals $-1$ otherwise.

Let $A$ be an annulus. A continuous mapping $\omega:A\to X$ is said to
represent a conjugacy class $\widehat{e}$ of elements of the fundamental group
$\pi_{1}(X,q)$ for a point $q\in X$, if for each simple closed curve $\gamma$
in $A$ that is homologous to a boundary component of $A$ the restriction of
the mapping to $\gamma$ represents $\widehat{e}$ or the inverse
$\widehat{e^{-1}}$.

Let $A$ be an annulus with base point $p$, and let $\omega$ be a continuous
mapping from $A$ to a finite Riemann surface ${X}$ with base point $q$ such
that $\omega(p)=q$. We write $\omega:(A,p)\to({X},q)$. The mapping is said to
represent the element $e$ of the fundamental group $\pi_{1}({X},q)$ if the
preimage $(\omega_{*})^{-1}(e)$ under the induced mapping
$\omega_{*}:\pi_{1}(A,p)\to\pi_{1}({X},q)$ between fundamental groups is a
generator of $\pi_{1}(A,p)$. There are two generators and they are inverse to
each other.

We associate to each element $e_{0}\in\pi_{1}(X,q_{0})$ of the free group
$\pi_{1}(X,q_{0})$ the annulus $X(\langle e_{0}\rangle)=\tilde{X}\diagup({\rm
Is}^{\tilde{q}_{0}})^{-1}(\langle e_{0}\rangle)$ with base point $q_{\langle
e_{0}\rangle}=\omega_{\rm Id}^{\langle e_{0}\rangle}(\tilde{q}_{0})$ and the
covering map $\omega_{\langle e_{0}\rangle}\stackrel{{\scriptstyle
def}}{{=}}\omega^{\pi_{1}(X,q_{0})}_{\langle e_{0}\rangle}:X(\langle
e_{0}\rangle)\to X$. Then $\omega_{\langle e_{0}\rangle}(q_{\langle
e_{0}\rangle})=q_{0}$. The mapping $\omega^{\pi_{1}(X,q_{0})}_{\langle
e_{0}\rangle}:(X(\langle e_{0}\rangle),q_{\langle e_{0}\rangle})\to(X,q_{0})$
represents $e_{0}$. Indeed, let $\gamma$ be a closed curve in $X$ with base
point $q_{0}$ which represents $e_{0}$. Consider the lift of $\gamma$ to a
curve $\tilde{\gamma}$ in $\tilde{X}$ with terminating point $\tilde{q}_{0}$.
Its initial point equals $({\rm
Is}^{\tilde{q}_{0}})^{-1}(e_{0})(\tilde{q}_{0})$. Hence, the projection
$\omega^{\langle e_{0}\rangle}(\tilde{\gamma})$ of $\tilde{\gamma}$ to
$\tilde{X}\diagup({\rm Is}^{\tilde{q}_{0}})^{-1}(\langle e_{0}\rangle)$ is a
closed curve denoted by $\gamma_{\langle e_{0}\rangle}$ which is a generator
of the fundamental group $\pi_{1}(X(\langle e_{0}\rangle),(q_{0})_{\langle
e_{0}\rangle})$. Since $\omega_{\langle e_{0}\rangle}(\gamma_{\langle
e_{0}\rangle})=\omega_{\langle e_{0}\rangle}(\omega^{\langle
e_{0}\rangle}(\tilde{\gamma}))={\sf P}(\tilde{\gamma})=\gamma$, we proved,
that the locally conformal mapping $\omega_{\langle e_{0}\rangle}:((X(\langle
e_{0}\rangle),(q_{0})_{\langle e_{0}\rangle})\to(X,q_{0})$ represents $e_{0}$.

Take a curve $\alpha$ in $X$ that joins $q_{0}$ and $q$, and a point
$\tilde{q}=\tilde{q}({\alpha})\in\tilde{X}$ such that $\alpha$ and $\tilde{q}$
are compatible, i.e. ${\rm Is}^{\tilde{q}}={\rm Is}_{\alpha}\circ{\rm
Is}^{\tilde{q}_{0}}$ (see equation (5)). Put $e={\rm Is}_{\alpha}(e_{0})$. By
equation (5)) $({\rm Is}^{\tilde{q}})^{-1}(e)=({\rm
Is}^{\tilde{q}_{0}})^{-1}(e_{0})$, hence, $\tilde{X}\diagup({\rm
Is}^{\tilde{q}})^{-1}(\langle e\rangle)=\tilde{X}\diagup({\rm
Is}^{\tilde{q}_{0}})^{-1}(\langle e_{0}\rangle)=X(\langle e_{0}\rangle)$. The
locally conformal mapping $\omega_{\langle e_{0}\rangle}:X(\langle
e_{0}\rangle)\to X$ takes the point $q_{\langle
e\rangle}\stackrel{{\scriptstyle def}}{{=}}\omega^{\langle
e_{0}\rangle}(\tilde{q})$ to $q\in X$. Moreover, $\omega_{\langle
e_{0}\rangle}:(X(\langle e_{0}\rangle),q_{\langle e\rangle})\to(X,q)$
represents $e\in\pi_{1}(X,q)$. This can be seen by repeating the previous
arguments.

Let $\alpha$ be an arbitrary curve in $X$ joining $q_{0}$ with $q$, and
$\tilde{q}\in{\sf P}^{-1}(q)$ be arbitrary (i.e., $\alpha$ and $\tilde{q}$ are
not required to be compatible). Let $e\in\pi_{1}(X,q)$. Denote the projection
$\tilde{X}\to\tilde{X}\diagup({\rm Is}^{\tilde{q}})^{-1}(\langle e\rangle)$ by
$\omega^{\langle e\rangle,\tilde{q}}$, and the projection
$\tilde{X}\diagup({\rm Is}^{\tilde{q}})^{-1}(\langle e\rangle)\to X$ by
$\omega_{\langle e\rangle,\tilde{q}}$. Put $q_{\langle
e\rangle,\tilde{q}}=\omega^{\langle e\rangle,\tilde{q}}(\tilde{q})$. Then the
mapping $\omega_{\langle e\rangle,\tilde{q}}:\tilde{X}\diagup\big{(}({\rm
Is}^{\tilde{q}})^{-1}(\langle e\rangle),q_{\langle
e\rangle,\tilde{q}}\big{)}\to(X,q)$ represents $e$.

The mapping $\omega_{\langle e_{0}\rangle}:X(\langle e_{0}\rangle)\to X$
represents the free homotopy class $\widehat{e_{0}}$. Moreover, $X(\langle
e_{0}\rangle)$ is the ”thickest” annulus which admits a holomorphic mapping to
$X$, that represents the conjugacy class $\widehat{e}_{0}$. This means, that
the annulus has smallest extremal length among annuli with the mentioned
property. Indeed, let $A\overset{\omega}{-\\!\\!\\!\longrightarrow}X$ be a
holomorphic mapping of an annulus that represents $\widehat{e_{0}}$. $A$ is
conformally equivalent to a round annulus in the plane, hence, we may assume
that $A$ has the form $A=\\{z\in\mathbb{C}:\,r<|z|<R\\}$ for positive real
numbers $r<R$.

Take a simple closed curve $\gamma^{A}$ in $A$ that is homologous to a
boundary component of $A$. After maybe, reorienting $\gamma^{A}$, its image
under $\omega$ represents the class $\widehat{e_{0}}$. Choose a point $q^{A}$
in $\gamma^{A}$, and put $q=\omega(q^{A})$. Then $\gamma^{A}$ represents a
generator of $\pi_{1}(A,q^{A})$ and $\gamma$ represents an element $e$ of
$\pi_{1}(X,q)$ in the conjugacy class $\widehat{e_{0}}$. Choose a curve
$\alpha$ in $X$ with initial point $q_{0}$ and terminating point $q$, and a
point $\tilde{q}$ in $\tilde{X}$ so that $\alpha$ and $\tilde{q}$ are
compatible, and, hence, for $e={\rm Is}_{\alpha}(e_{0})$ the equality $({\rm
Is}^{\tilde{q}_{0}})^{-1}(e_{0})=({\rm Is}^{\tilde{q}})^{-1}(e)$ holds. Let
$L$ be the relatively closed arc $(p\cdot\mathbb{R})\cap A$ in $A$ through $p$
along the radius. After a homotopy of $\gamma^{A}$ with fixed base point, we
may assume that its base point $q^{A}$ is the only point of $\gamma^{A}$ that
is contained in $L$. The restriction $\omega|(A\setminus L)$ lifts to a
mapping $\tilde{\omega}:(A\setminus L)\to\tilde{X}$, that extends continuously
to the two strands $L_{\pm}$ of $L$. (Here $L_{-}$ contains the initial point
of $\gamma^{A}$.) Let $p_{\pm}$ be the copies of $p$ on the two strands
$L_{\pm}$. We choose the lift $\tilde{\omega}$ so that
$\tilde{\omega}(p_{+})=\tilde{q}$. Since the mapping $(A,q^{A})\to(X,q)$
represents $e$, we obtain $\tilde{\omega}(p_{-})=\sigma(\tilde{q})$ for
$\sigma=({\rm Is}^{\tilde{q}})^{-1}(e)$. Then for each $z\in L$ the covering
transformation $\sigma$ maps the point $\tilde{z}_{+}\in\tilde{\omega}(L_{+})$
for which ${\sf P}(\tilde{z}_{+})=z$ to the point
$\tilde{z}_{-}\in\tilde{\omega}(L_{-})$ for which ${\sf P}(\tilde{z}_{-})=z$.
Hence $\omega$ lifts to a holomorphic mapping $\iota:A\to X(\langle
e_{0}\rangle)$. By Lemma 7 of [18] $\lambda(A)\geq\lambda(X(\langle
e_{0}\rangle))$.

For each point $q\in X$ and each element $e\in\pi_{1}(X,q)$ we denote by
$A(\widehat{e})$ the conformal class of the ”thickest” annulus that admits a
holomorphic mapping into $X$ that represents $\widehat{e}$. For each
$\tilde{q}^{\prime}\in\tilde{X}$ and each element $e\in\pi_{1}(X,{\sf
P}(\tilde{q}^{\prime}))$ the mapping $\omega_{\langle
e\rangle,\tilde{q}^{\prime}}:\tilde{X}\diagup({\rm
Is}^{\tilde{q}^{\prime}})^{-1}(\langle e\rangle)\to X$ represents the
conjugacy class $\widehat{e}$ of $e$. Hence, if $\widehat{e}=\widehat{e_{0}}$,
then $\lambda(\tilde{X}\diagup({\rm Is}^{\tilde{q}^{\prime}})^{-1}(\langle
e\rangle))\leq\lambda(A(\widehat{e}))$. Interchanging the role of the
representatives $e$ and $e_{0}$ of $\widehat{e_{0}}$ and of the base points
$\tilde{q}^{\prime}$ and $\tilde{q}_{0}$ of $\tilde{X}$, we see that in fact
$\lambda(\tilde{X}\diagup({\rm Is}^{\tilde{q}^{\prime}})^{-1}(\langle
e\rangle))=\lambda(A(\widehat{e}))$ for each $\tilde{q}^{\prime}\in\tilde{X}$
and each element $e\in\pi_{1}(X,{\sf P}(\tilde{q}^{\prime}))$. Notice that
$A(\widehat{e^{-1}})=A(\widehat{e})$.

The following lemma will be crucial for the estimate of the
$\mathcal{L}_{-}$-invariant of the monodromies of holomorphic mappings from a
finite open Riemann surface to $\mathbb{C}\setminus\\{-1,1\\}$.

######  Lemma 1.

Let $f:X\to\mathbb{C}\setminus\\{-1,1\\}$ be a non-contractible holomorphic
function on a connected finite open Riemann surface $X$, such that $0$ is a
regular value of ${\rm Im}f$. Assume that $L_{0}$ is a simple relatively
closed curve in $X$ such that $f(L_{0})\subset(-1,1)$. Let $q\in L_{0}$ and
$q^{\prime}=f(q)$.

If for an element $e\in\pi_{1}(X,q)$ the free homotopy class $\widehat{e}$
intersects $L_{0}$, then either the reduced word
$f_{*}(e)\in\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ is a non-zero
power of a standard generator of
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ or the inequality

$\mathcal{L}_{-}(f_{*}(e))\leq 2\pi\lambda(A(\widehat{e}))$ (7)

holds.

Notice that we make a normalization in the statement of the Lemma by requiring
that $f$ maps $L_{0}$ into the interval $(-1,1)$, not merely into
$\mathbb{R}\setminus\\{-1,1\\}$.

Lemma 1 will be a consequence of the following lemma.

######  Lemma 2.

Let $f$, $L_{0}$, $q\in L_{0}$ be as in Lemma 1, and $e\in\pi_{1}(X,q)$. Let
$\tilde{q}$ be an arbitrary point in ${\sf P}^{-1}(q)$. Consider the annulus
$A\stackrel{{\scriptstyle def}}{{=}}\tilde{X}\diagup({\rm
Is}^{\tilde{q}})^{-1}(\langle e\rangle)$ and the holomorphic projection
$\omega_{A}\stackrel{{\scriptstyle def}}{{=}}\omega_{\langle
e\rangle,\tilde{q}}$. Put $q_{A}\stackrel{{\scriptstyle
def}}{{=}}\omega^{\langle e\rangle,\tilde{q}}(\tilde{q})$ and let
$L_{A}\subset A$ be the connected component of $(\omega_{A})^{-1}(L_{0})$ that
contains $q_{A}$. Then the mapping $\omega_{A}:(A,q_{A})\to(X,q)$ represents
$e$.

If $\widehat{e}$ intersects $L_{0}$, then $L_{A}$ is a relatively closed curve
in $A$ that has limit points on both boundary components of $A$, and the lift
$f\circ\omega_{A}$ is a holomorphic function on $A$ that maps $L_{A}$ into
$(-1,1)$.

Proof of Lemma 2. Let $\gamma:[0,1]\to X$ be a curve with base point $q$ in
$X$ that represents $e$, and let $\tilde{\gamma}$ be the lift of $\gamma$ to
$\tilde{X}$ with terminating point $\tilde{\gamma}(1)$ equal to $\tilde{q}$.
Put $\sigma\stackrel{{\scriptstyle def}}{{=}}({\rm Is}^{\tilde{q}})^{-1}(e)$.
Then the initial point $\tilde{\gamma}(0)$ equals $\sigma(\tilde{q})$.

All connected components of ${\sf P}^{-1}(L_{0})$ are relatively closed curves
in $\tilde{X}\cong\mathbb{C}_{+}$ (where $\mathbb{C}_{+}$ denotes the upper
half-plane) with limit points on the boundary of $\tilde{X}$. Indeed, the lift
$f\circ{\sf P}$ of $f$ to $\tilde{X}$ takes values in $(-1,1)$ on ${\sf
P}^{-1}(L_{0})$. Hence, by the maximum principle applied to $\exp(f\circ{\sf
P})$ the latter cannot contain compact connected components since $f$ is not
constant.

Let $\tilde{L}_{\tilde{q}}$ be the connected component of ${\sf
P}^{-1}(L_{0})$ that contains $\tilde{q}$. The point $\sigma(\tilde{q})$
cannot be contained in $\tilde{L}_{\tilde{q}}$. Indeed, otherwise there exists
a curve in $\tilde{X}$ that is homotopic to $\tilde{\gamma}$ with fixed
endpoints and is contained in $\tilde{L}_{\tilde{q}}$. A small translation of
the latter curve gives a curve in $\tilde{X}$ that does not intersect ${\sf
P}^{-1}(L_{0})$ and projects under $\sf P$ to a curve that is free homotopic
to $\gamma$. This contradicts the fact that $\widehat{e}$ intersects $L_{0}$.

Each of the two connected components $\tilde{L}_{\tilde{q}}$ and
$\sigma(\tilde{L}_{\tilde{q}})$ divides $\tilde{X}$. Let $\Omega$ be the
domain on $\tilde{X}$ that is bounded by $\tilde{L}_{\tilde{q}}$ and
$\sigma(\tilde{L}_{\tilde{q}})$ and parts of the boundary of $\tilde{X}$.
After a homotopy of $\tilde{\gamma}$ that fixes the endpoints we may assume
that $\tilde{\gamma}((0,1))$ is contained in $\Omega$. Indeed, for each
connected component of $\tilde{\gamma}((0,1))\setminus\Omega$ there is a
homotopy with fixed endpoints that moves the connected component to an arc on
$\tilde{L}_{\tilde{q}}$ or $\sigma(\tilde{L}_{\tilde{q}})$. A small
perturbation yields a curve $\tilde{\gamma}^{\prime}$ which is homotopic with
fixed endpoints to $\tilde{\gamma}$ and has interior contained in $\Omega$.
Notice that $\tilde{\gamma}^{\prime}((0,1))$ does not meet any
$\sigma^{k}(\tilde{L}_{\tilde{q}})$.

The curve $\omega^{\langle e\rangle,\tilde{q}}(\tilde{\gamma}^{\prime})$ is a
closed curve on $A$ that represents a generator of the fundamental group of
$A$ with base point $q_{A}$. Moreover, $\omega_{A}\circ\omega^{\langle
e\rangle,\tilde{q}}(\tilde{\gamma}^{\prime})=\omega_{\langle
e\rangle,\tilde{q}}\circ\omega^{\langle
e\rangle,\tilde{q}}(\tilde{\gamma}^{\prime})={\sf P}(\tilde{\gamma}^{\prime})$
represents $e$. Hence, the mapping $\omega_{A}:(A,q_{A})\to(X,q)$ represents
$e$.

The curve $\omega^{\langle e\rangle,\tilde{q}}(\tilde{\gamma}^{\prime})$
intersects $L_{\langle e\rangle}=\omega^{\langle
e\rangle,\tilde{q}}(\tilde{L}_{\tilde{q}})$ exactly once. Hence, $L_{\langle
e\rangle}$ has limit points on both boundary circles of $A$ for otherwise
$L_{\langle e\rangle}$ would intersect one of the components of
$A\setminus\omega^{\langle e\rangle,\tilde{q}}(\tilde{\gamma}^{\prime})$ along
a set which is relatively compact in $A$, and $\tilde{\gamma}^{\prime}$ would
have intersection number zero with $L_{\langle e\rangle}$. It is clear that
$f\circ\omega_{A}(L_{A})=f(L_{0})\subset(-1,1)$. The lemma is proved. $\Box$

Proof of Lemma 1. Let $\omega_{A}:(A,q_{A})\to(X,q)$ be the holomorphic
mapping from lemma 2 that represents $e$, and let $L_{A}\ni q_{A}$ be the
relatively closed curve in $A$ with limit set on both boundary components of
$A$. Consider a loop ${\gamma}_{A}:[0,1]\to A$ with base point
$\gamma(0)=\gamma(1)=q_{A}$ such that $\gamma_{A}((0,1))\subset A\setminus
L_{A}$. Then $\gamma=\omega_{A}(\gamma_{A})$ represents $e$. The mapping
$f\circ\omega_{A}$ is holomorphic on $A$ and
$f\circ\omega_{A}(\gamma_{A})=f(\gamma)$ represents
$f_{*}(e)\in\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ with
$q^{\prime}=f\circ\omega_{A}(q_{A})=f(q)\in(-1,1)$. Hence,
$f\circ\omega_{A}(\gamma_{A})$ also represents the element
$(f_{*}(e))_{tr}\in\pi_{1}^{tr}(\mathbb{C}\setminus\\{-1,1\\})$ in the
relative fundamental group fundamental group
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},(-1,1))=\pi_{1}^{tr}(\mathbb{C}\setminus\\{-1,1\\})$
corresponding to $f_{*}(e)$.

We prove now that $\Lambda_{tr}(f_{*}(e))\leq\lambda(A)$. Let $A_{0}\Subset A$
be any relatively compact annulus in $A$ with smooth boundary such that
$q_{A}\in A_{0}$. If $A_{0}$ is sufficiently large, then the connected
component $L_{A_{0}}$ of $L_{A}\cap A_{0}$ that contains $q_{A}$ has endpoints
on different boundary components of $A_{0}$. The set $A_{0}\setminus
L_{A_{0}}$ is a curvilinear rectangle. The horizontal curvilinear sides are
the strands of the cut that are reachable from the curvilinear rectangle
moving counterclockwise, or clockwise, respectively. The vertical curvilinear
sides are obtained from the boundary circles of $A_{0}$ by removing an
endpoint of the arc $L_{A_{0}}$. Since $f\circ\omega_{A}$ maps $L_{A}$ to
$(-1,1)$, the restriction of $f\circ\omega_{A}$ to $A_{0}\setminus L_{A_{0}}$
represents $(f_{*}(e))_{tr}$. Hence,

$\Lambda_{tr}(f_{*}(e))\leq\lambda(A_{0}\setminus L_{A_{0}})\,.$ (8)

Moreover,

$\lambda(A_{0}\setminus L_{A_{0}})\leq\lambda(A_{0})\,.$ (9)

This is a consequence of the following facts. First, $\lambda(A_{0}\setminus
L_{A_{0}})$ is equal to the extremal length $\lambda(\Gamma(A_{0}\setminus
L_{A_{0}}))$ in the sense of Ahlfors [1] of the family $\Gamma(A_{0}\setminus
L_{A_{0}})$ of curves in the curvilinear rectangle $A_{0}\setminus L_{A_{0}}$
that join the two horizontal sides of the curvilinear rectangle. Further,
$\lambda(A_{0})$ is equal to the extremal length $\lambda(\Gamma(A_{0}))$ [1]
of the family $\Gamma(A_{0})$ of curves in $A_{0}$ that are free homotopic to
representatives of the positive generator of $A_{0}$. Finally, by [1], Ch.1
Theorem 2, the inequality

$\displaystyle\lambda(\Gamma(A_{0}\setminus
L_{A_{0}}))\leq\lambda(\Gamma(A_{0}))\,$ (10)

holds. We obtain the inequality $\Lambda_{tr}(f_{*}(e))\leq\lambda(A_{0})\,$
for each annulus $A_{0}\Subset A$, hence, since $A$ belongs to the class
$A(\widehat{e})$ of conformally equivalent annuli,

$\displaystyle\Lambda_{tr}(f_{*}(e))\leq\lambda(A(\widehat{e}))\,,$ (11)

and the Lemma follows from Theorem F. $\Box$

The monodromies along two generators. In the following Lemma we combine the
information on the monodromies along two generators of the fundamental group
$\pi_{1}(X,q)$. We allow the situation when the monodromy along one generator
or along both generators of the fundamental group of $X$ is a power of a
standard generator of $\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},f(q))$.

######  Lemma 3.

Let $f:X\to\mathbb{C}\setminus\\{-1,1\\}$ be a holomorphic function on a
connected open Riemann surface $X$ such that $0$ is a regular value of the
imaginary part of $f$. Suppose $f$ maps a simple relatively closed curve
$L_{0}$ in $X$ to $(-1,1)$, and $q$ is a point in $L_{0}$. Let $e^{(1)}$ and
$e^{(2)}$ be primitive elements of $\pi_{1}(X,q)$. Suppose that for each
$e=e^{(1)},\;e=e^{(2)}$, and $e=e^{(1)}\,e^{(2)}$, the free homotopy class
$\widehat{e}$ intersects $L_{0}$. Then either $f_{*}(e^{(j)}),\,j=1,2,\,$ are
(trivial or non-trivial) powers of the same standard generator of
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ with
$q^{\prime}=f(q)\in(-1,1)$, or each of them is the product of at most two
elements $w_{1}$ and $w_{2}$ of
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ with

$\mathcal{L}_{-}(w_{j})\leq 2\pi\lambda_{e^{(1)},e^{(2)}},\,j=1,2,$

where

$\lambda_{e^{(1)},e^{(2)}}\stackrel{{\scriptstyle
def}}{{=}}\max\\{\lambda(A(\widehat{e^{(1)}})),\,\lambda(A(\widehat{e^{(2)}})),\,\lambda(A(\widehat{e^{(1)}\,e^{(2)}}))\\}.$

Hence,

$\mathcal{L}_{-}(f_{*}(e^{(j)})\leq 4\pi\lambda_{e^{(1)},e^{(2)}},\,j=1,2.$
(12)

Proof. If the monodromies $f_{*}(e^{(1)})$ and $f_{*}(e^{(2)})$ are not powers
of a single standard generator (the identity is considered as zeroth power of
a standard generator) we obtain the following. At most two of the elements,
$f_{*}(e^{(1)})$, $f_{*}(e^{(2)})$, and
$f_{*}(e^{(1)}\,e^{(2)})=f_{*}(e^{(1)})\,f_{*}(e^{(2)})$, are powers of a
standard generator, and if two of them are powers of a standard generator,
then they are non-zero powers of different standard generators. If two of them
are non-zero powers of standard generators, then the third has the form
$a_{\ell}^{k}a_{\ell^{\prime}}^{k^{\prime}}$ with $a_{\ell}$ and
$a_{\ell^{\prime}}$ being different generators and $k$ and $k^{\prime}$ being
non-zero integers. By Lemma 1 the $\mathcal{L}_{-}$ of the third element does
not exceed $2\pi\lambda_{e^{(1)},e^{(2)}}$. On the other hand it equals
$\log(3|k^{\prime}|)+\log(3|k|)$. Hence,
$\mathcal{L}_{-}(a_{\ell}^{k})=\log(3|k|)\leq 2\pi\lambda_{e^{(1)},e^{(2)}}$
and $\mathcal{L}_{-}(a_{\ell^{\prime}}^{k^{\prime}})=\log(3|k^{\prime}|)\leq
2\pi\lambda_{e^{(1)},e^{(2)}}$.

If two of the elements $f_{*}(e^{(1)})$, $f_{*}(e^{(2)})$, and
$f_{*}(e^{(1)}\,e^{(2)})=f_{*}(e^{(1)})\,f_{*}(e^{(2)})$, are not powers of a
standard generator, then the $\mathcal{L}_{-}$ of each of the two elements
does not exceed $2\pi\lambda_{e^{(1)},e^{(2)}}$. Since the $\mathcal{L}_{-}$
of an element coincides with the $\mathcal{L}_{-}$ of its inverse, the third
element is the product of two elements with $\mathcal{L}_{-}$ not exceeding
$2\pi\lambda_{e^{(1)},e^{(2)}}$. Since for $x,x^{\prime}\geq 3$ the inequality
$\log(x+x^{\prime})\leq\log x+\log x^{\prime}$ holds, the $\mathcal{L}_{-}$ of
the product does not exceed the sum of the $\mathcal{L}_{-}$ of the factors.
Hence the $\mathcal{L}_{-}$ of the third element does not exceed
$4\pi\lambda_{e^{(1)},e^{(2)}}$. Hence, inequality (12) holds. $\Box$

The following proposition states the existence of suitable connected
components of the zero set of the imaginary part of certain analytic functions
on tori with a hole and on planar domains. For any subset
$\mathcal{E}^{\prime}$ of $\pi_{1}(X;q_{0})$ we denote by
$(\mathcal{E}^{\prime})^{-1}$ the set of all elements that are inverse to
elements in $\mathcal{E}^{\prime}$. Recall that $\mathcal{E}_{j}$ is the set
of primitive elements of $\pi_{1}(X,q_{0})$ which can be written as product of
at most $j$ elements of $\mathcal{E}\cup(\mathcal{E})^{-1}$ for the set
$\mathcal{E}$ of generators of $\pi_{1}(X,q_{0})$ chosen in the introduction.

######  Proposition 3.

Let $X$ be a torus with a hole or a planar domain with base point $q_{0}$ and
fundamental group $\pi_{1}(X,q_{0})$, and let $\mathcal{E}$ be the set of
generators of $\pi_{1}(X,q_{0})$ that was chosen in Section 1. Let
$f:X\to\mathbb{C}\setminus\\{-1,1\\}$ be a non-contractible holomorphic
mapping such that $0$ is a regular value of ${\rm{Im}}f$. Then there exist a
simple relatively closed curve $L_{0}\subset X$ such that
$f(L_{0})\subset\mathbb{R}\setminus\\{-1,1\\},$ and a set
$\mathcal{E}_{2}^{\prime}\subset\mathcal{E}_{2}\subset\pi_{1}(X,q_{0})$ of
primitive elements of $\pi_{1}(X,q_{0})$, such that the following holds. Each
element $e_{j,0}\in\mathcal{E}\subset\pi_{1}(X,q_{0})$ is the product of at
most two elements of
$\mathcal{E}_{2}^{\prime}\cup(\mathcal{E}_{2}^{\prime})^{-1}$. Moreover, for
each $e_{0}\in\pi_{1}(X,q_{0})$ which is the product of one or two elements
from $\mathcal{E}_{2}^{\prime}$ the free homotopy class $\widehat{e_{0}}$ has
positive intersection number with $L_{0}$ (after suitable orientation of
$L_{0}$).

Notice the following fact. If $f$ is irreducible, then it is not contractible,
and, hence, the preimage $f^{-1}(\mathbb{R})$ is not empty.

Denote by $M_{1}$ a Möbius transformation which permutes the points
$-1,\,1,\,\infty$ and maps the interval $(-\infty,-1)$ onto $(-1,1)$, and let
$M_{2}$ be a Möbius transformation which permutes the points $-1,\,1,\,\infty$
and maps the interval $(1,\infty)$ onto $(-1,1)$. Let
$M_{0}\stackrel{{\scriptstyle def}}{{=}}\mbox{Id}$.

The main step for the proof of Theorem 1 is the following Proposition 4.

Recall that $\lambda_{j}(X)$ was defined in the introduction. Since for
$e_{0}\in\pi_{1}(X,q_{0})$ the equality $\lambda(\tilde{X}\diagup({\rm
Is}^{\tilde{q}_{0}})^{-1}(\langle e_{0}\rangle))=\lambda(A(\widehat{e_{0}}))$,
$\lambda_{j}(X)$ is the maximum over $e_{0}\in\mathcal{E}_{j}$ of
$\lambda(A(\widehat{e_{0}}))$.

######  Proposition 4.

Let $X$ be a connected finite open Riemann surface with base point $q_{0}$,
and let $\mathcal{E}$ be the set of generators of $\pi_{1}(X,q_{0})$ that was
chosen in Section 1. Suppose $f:X\to\mathbb{C}\setminus\\{-1,1\\}$ is an
irreducible holomorphic mapping, such that $0$ is a regular value of
${\rm{Im}}f$. Then for one of the functions $M_{l}\circ f,\,l=0,1,2,\,$ which
we denote by $F$, there exists a point $q\in X$ (depending on $f$), such that
the point $q^{\prime}\stackrel{{\scriptstyle def}}{{=}}F(q)$ is contained in
$(-1,1)$, and a curve $\alpha$ in $X$ joining $q_{0}$ with $q$, such that the
following holds. For each element $e_{j}\in\mbox{Is}_{\alpha}(\mathcal{E})$
the monodromy $F_{*}(e_{j})$ is the product of at most four elements of
$\pi_{1}(\mathbb{(}C)\setminus\\{-1,1\\},q^{\prime})$ of $\mathcal{L}_{-}$ not
exceeding $2\pi\lambda_{4}(X)$ and, hence,

$\mathcal{L}_{-}(F_{*}(e_{j}))\leq 8\pi\lambda_{4}(X)\;\;\mbox{for
each}\;\,j.$ (13)

Notice that $F$ is irreducible iff $f$ is irreducible. By Theorem E and Lemma
4 of [21] the non-contractible mapping $F$ is irreducible if and only if the
image $F_{*}(\pi_{1}(X,q))$ of the monodromy homomorohism is not generated by
a single element of the fundamental group
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$, $q^{\prime}\in(-1,1)$,
that is conjugate to a standard generator of
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$.

Notice, that all monodromies of contractible mappings are equal to zero, hence
the inequality (13) holds automatically for contractible mappings.

We postpone the proof of the two propositions and prove first the Theorem 1.

Proof of Theorem 1. Let $X$ be a connected finite open Riemann surface
(possibly of second kind) with base point $q_{0}$. Using Theorem E and
Proposition 4 we want to bound the number of irreducible free homotoy classes
of mappings from $X$ to $\mathbb{C}\setminus\\{-1,1\\}$ that contain a
holomorphic mapping. Consider a free homotopy class that is represented by an
irreducible holomorphic mapping ${\sf f}:X\to\mathbb{C}\setminus\\{-1,1\\}$.
Recall that irreducible mappings are not contractible. Consider an arbitrary
open Riemann surface $X^{0}\Subset X$ which is relatively compact in $X$ and
is a deformation retract of $X$. Let $\varepsilon$ be a small enough real
number $\varepsilon$, such that the function $({\sf f}-i\varepsilon)\mid
X^{0}$ takes values in $\mathbb{C}\setminus\\{-1,1\\}$ and $0$ is a regular
value of its imaginary part. Put $f=({\sf f}-i\varepsilon)\mid X^{0}$. Notice
that the irreducible mapping $f$ on $X^{0}$ is free homotopic to ${\sf f}\mid
X^{0}$. We identify the fundamental groups of $X$ and of $X^{0}$ by the
inclusion mapping from $X^{0}$ to $X$.

The statement of Proposition 4 applies to the function $f$ on $X^{0}$.
Associate to $f$ the function $F=M_{l}\circ f$ on $X^{0}$, and the points $q$
and $q^{\prime}$ from Proposition 4. The proposition provides information
about the monodromies of the mapping $F$ along the elements
$\mbox{Is}_{\alpha}(\mathcal{E})$ for some curve $\alpha$ that was chosen
depending on $f$, and has initial point $q_{0}$ and terminating point $q$. We
have to relate this information to the monodromies of the mapping $F$ along
the elements of the system $\mathcal{E}$ of generators of the fundamental
group $\pi_{1}(X,q_{0})\cong\pi_{1}(X^{0},q_{0})$ with base point $q_{0}$,
that was chosen in Section 1 and is independent on the function $f$. Write
$e_{j}=\mbox{Is}_{\alpha}(e_{j,0})\in\pi_{1}(X,q)$ for
$e_{j,0}\in\pi_{1}(X,q_{0})$. The image of $\alpha$ under the mapping $F$ is
the curve $\beta=F\circ\alpha$ in $\mathbb{C}\setminus\\{-1,1\\}$ with initial
point $F(q_{0})$ and terminating point $q^{\prime}$. We have
$F_{*}(e_{j,0})=(\mbox{Is}_{\beta})^{-1}(F_{*}(e_{j}))$. Choose a homotopy
$F_{t},\,t\in[0,1]$, that joins the mapping $F_{0}\stackrel{{\scriptstyle
def}}{{=}}F$ with a (smooth) mapping $F_{1}$ denoted by $\tilde{F}$, so that
the value $F_{t}(q_{0})$ moves from the point $F(q_{0})$ to $q^{\prime}$ along
the curve $\beta$. Then $\tilde{F}(q_{0})=q^{\prime}$ and
$\tilde{F}_{*}(e_{j,0})=F_{*}(e_{j})$.

Inequality (13) gives for each $j$ the inequality

$\mathcal{L}_{-}(\tilde{F}_{*}(e_{j,0}))\leq 8\pi\lambda_{4}(X^{0}).$ (14)

Notice that if $f$ is a contractible function, we may associate to it the
function $\tilde{F}$ on $X^{0}$ which is equal $f\mid X^{0}$ and the
inequality (14) is automatically satisfied for the monodromies of $\tilde{F}$.

By Lemma 1 of [20] there are at most
$\frac{1}{2}e^{24\pi\lambda_{4}(X^{0})}+1\leq\frac{3}{2}e^{24\pi\lambda_{4}(X^{0})}$
different reduced words $w\in\pi_{1}(\mathcal{C}\setminus\\{-1,1\\}),0)$
(including the identity) with $\mathcal{L}_{-}(w)\leq 8\pi\lambda_{4}(X^{0})$.
Identify standard generators of
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ with standard generators
of $\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},0)$ by the canonical isomorphism. We
obtain, that $\tilde{F}_{*}$ is contained in a set of at most
$(\frac{3}{2}e^{24\pi\lambda_{4}(X^{0})})^{2g+m}$ different homomorphisms
$\pi_{1}(X^{0},q_{0})\cong\pi_{1}(X,q_{0})\to\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})\cong\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},0)$.
By Theorem E there is a one-to-one correspondence between free homotopy
classes of functions from $X^{0}$ to $\mathbb{C}\setminus\\{-1,1\\}$ and
conjugacy classes of homomorphisms
$\pi_{1}(X^{0},q_{0})\to\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})\cong\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},0)$.
Since $F$ is free homotopic to $\tilde{F}$, the function $F$ represents one of
at most $(\frac{3}{2}e^{24\pi\lambda_{4}(X^{0})})^{2g+m}$ different free
homotopy classes from $X^{0}$ to the twice punctured complex plane.

The function $f$ is equal to $M_{l}^{-1}\circ F$ for one of the numbers $0$,
$1$, or $2$, hence it represents one of
$3(\frac{3}{2}e^{24\pi\lambda_{4}(X^{0})})^{2g+m}$ free homotopy classes of
mappings from $X^{0}$ to the twice punctured complex plane. Since the
considered set of mappings $f$ on $X$ is equal to the set of mappings $({\sf
f}-i\varepsilon)|X^{0}$ for a holomorphic map $\sf f$ on $X$, there are no
more than $3(\frac{3}{2}e^{24\pi\lambda_{4}(X^{0})})^{2g+m}$ free homotopy
classes of irreducible mappings $X\to\mathbb{C}\setminus\\{-1,1\\}$ containing
a holomorphic mapping $\sf f$. Theorem 1 is proved with the upper bound
$3(\frac{3}{2}e^{24\pi\lambda_{4}(X^{0})})^{2g+m}$ for an arbitrary relatively
compact domain $X^{0}\subset X$ that is a deformation retract of $X$.

It remains to prove that
$\lambda_{4}(X)=\inf\\{\lambda_{4}(X^{0}):X^{0}\Subset X\mbox{is\, a\,
deformation\, retract\, of\,}X\,\\}$. We have to prove that for each
$e_{0}\in\pi_{1}(X,x_{0})$ the number
$\lambda(A(\widehat{e_{0}}))=\lambda(\widetilde{X}\diagup({\rm
Is}^{\tilde{q}_{0}})^{-1}(\langle e_{0}\rangle))$ is equal to the infimum of
$\lambda(\widetilde{X^{0}}\diagup({\rm Is}^{\tilde{q}_{0}})^{-1}(\langle
e_{0}\rangle))$ over all $X^{0}$ being open relatively compact subsets of $X$
which are deformation retracts of $X$. Here $\widetilde{X^{0}}$ is the
universal covering of $X^{0}$, and the fundamental groups of $X$ and $X^{0}$
are identified. $\widetilde{X^{0}}$ ($\widetilde{X}$, respectively) can be
defined as set of homotopy classes of arcs in $X^{0}$ (in $X$, respectively)
joining $q_{0}$ with a point $q\in X^{0}$ (in $X$ respectively) equipped with
the complex structure induced by the projection to the endpoint of the arcs,
and the point $\tilde{q}_{0}$ corresponds to the class of the constant curve.
The isomorphism $({\rm Is}^{\tilde{q}_{0}})^{-1}$ from $\pi_{1}(X^{0},q_{0})$
to the group of covering transformations on $\widetilde{X^{0}}$ is defined in
the same way as it was done for $X$ instead of $X^{0}$. We see that there is a
holomorphic mapping from $\widetilde{X}^{0}\diagup({\rm
Is}^{\tilde{q}_{0}})^{-1}(\langle e_{0}\rangle)$ into
$\widetilde{X}\diagup({\rm Is}^{\tilde{q}_{0}})^{-1}(\langle e_{0}\rangle)$.
Hence, the extremal length of the first set is not smaller than the extremal
length of the second set.

Vice versa, take any annulus $A^{0}$ which is a relatively compact subset of
$A(\widehat{e_{0}})$ and is a deformation retract of $A(\widehat{e_{0}})$. Its
projection to $X$ is relatively compact in $X$, hence, it is contained in a
relatively compact deformation retract $X^{0}$ of $X$. Hence, $A^{0}$ can be
considered as subset of $\widetilde{X^{0}}\diagup({\rm
Is}^{\tilde{q}_{0}})^{-1}(\langle e_{0}\rangle)$, and, hence,
$\lambda(\widetilde{X^{0}}\diagup({\rm Is}^{\tilde{q}_{0}})^{-1}(\langle
e_{0}\rangle))\leq\lambda(A^{0})$. Since
$\lambda(A(\widehat{e_{0}}))=\inf\\{\lambda(A^{0}):A^{0}\Subset
A(\widehat{e_{0}})\mbox{\,is\, a\, deformation\, retract\,
of\,}A(\widehat{e_{0}})\\}$ we are done. $\Box$

We proved a slightly stronger statement, namely, the number of homotopy
classes of mappings $X\to\mathbb{C}\setminus\\{-1,1\\}$ that contain a
contractible holomorphic mapping or an irreducible holomorphic mapping does
not exceed $(\frac{3}{2}e^{24\pi\lambda_{4}(X)})^{2g+m}$.

Proof of Proposition 3. Denote the zero set $\\{x\in X:\mbox{Im}f(x)=0\\}$ by
$L$. Since $f$ is not contractible, $L\neq\emptyset$.

1\. A torus with a hole. Assume first that $X$ is a torus with a hole with
base point $q_{0}$. For notational convenience we denote by $e_{0}$ and
${e^{\prime}_{0}}$ the two elements of the set of generators $\mathcal{E}$ of
$\pi_{1}(X,q_{0})$ that was chosen in Section 1. We claim that there is a
connected component $L_{0}$ of $L$ which has (after suitable orientation)
positive intersection number with the free homotopy class of one of the
elements of $\mathcal{E}$, say with $\widehat{e_{0}}$, and $L_{0}$ has
positive intersection number with one of the classes
$\widehat{e^{\prime}_{0}}$, or $\widehat{(e^{\prime}_{0})^{-1}}$, or
$\widehat{e_{0}\,e^{\prime}_{0}}$.

The claim is easy to prove in the case when there is a component of $L_{0}$
which is a simple closed curve that is not contractible and not contractible
to the hole of $X$. Indeed, consider the inclusion of $X$ into a closed torus
$X^{c}$ and the homomorphism on fundamental groups
$\pi_{1}(X,q_{0})\to\pi_{1}(X^{c},q_{0})$ induced by the inclusion. Denote by
$e_{0}^{c}$ and ${e^{\prime}_{0}}^{c}$ the images of $e_{0}$ and
$e_{0}^{\prime}$ under this homomorphism. Notice that $e_{0}^{c}$ and
${e^{\prime}_{0}}^{c}$ commute. The (image under the inclusion of the) curve
$L_{0}$ is a simple closed non-contractible curve in $X^{c}$. It represents
the free homotopy class of an element
$(e_{0}^{c})^{j}({e^{\prime}_{0}}^{c})^{k}$ for some integers $j$ and $k$
which are not both equal to zero. Hence, $L_{0}$ is not null-homologous in
$X^{c}$, and by the Poincaré Duality Theorem for one of the generators, say
for $e_{0}^{c}$, the representatives of the free homotopy class
$\widehat{e_{0}^{c}}$ have non-zero intersection number with $L_{0}$. After
suitable orientation of $L_{0}$, we may assume that this intersection number
is positive. There is a representative of the class $\widehat{e_{0}^{c}}$
which is contained in $X$, hence, $\widehat{e_{0}}$ has positive intersection
number with $L_{0}$.

Suppose all compact connected components of $L$ are contractible or
contractible to the hole of $X$. Consider a relatively compact domain
${X}^{\prime\prime}\Subset{X}$ in ${X}$ with smooth boundary which is a
deformation retract of ${X}$ such that for each connected component of $L$ at
most one component of its intersection with ${X}^{\prime\prime}$ is not
contractible to the hole of ${X}^{\prime\prime}$. (See the paragraph on
”Regular zero sets”.) There is at least one component of
$L\cap{X}^{\prime\prime}$ that is not contractible to the hole of
${X}^{\prime\prime}$. Indeed, otherwise the free homotopy class of each
element of $\mathcal{E}$ could be represented by a loop avoiding $L$, and,
hence, the monodromy of $f$ along each element of $\mathcal{E}$ would be
conjugate to the identity, and, hence, equal to the identity, i.e. contrary to
the assumption, $f:X\to\mathbb{C}\setminus\\{-1,1\\}$ would be free homotopic
to a constant.

Take a component $L_{0}^{\prime\prime}$ of $L\cap{X}^{\prime\prime}$ that is
not contractible to the hole of ${X}^{\prime\prime}$. There is an arc of
$\partial{X}^{\prime\prime}$ between the endpoints of $L_{0}^{\prime\prime}$
such that the union $\tilde{L}_{0}$ of the component $L_{0}^{\prime\prime}$
with this arc is a closed curve in ${X}$ that is not contractible and not
contractible to the hole. Hence, for one of the elements of $\mathcal{E}$, say
for $e_{0}$, the intersection number of the free homotopy class
$\widehat{e_{0}}$ with the closed curve $\tilde{L}_{0}$ is positive after
orienting the closed curve suitably. We may take a representative $\gamma$ of
$\widehat{e_{0}}$ that is contained in ${X}^{\prime\prime}$. Then $\gamma$ has
positive intersection number with $L_{0}^{\prime\prime}$. Denote the connected
component of $L$ that contains $L_{0}^{\prime\prime}$ by $L_{0}$. All
components of $L_{0}\cap{X}^{\prime\prime}$ different from
$L_{0}^{\prime\prime}$ are contractible to the hole of ${X}^{\prime\prime}$.
Hence, $\gamma$ has intersection number zero with each of these components.
Hence, $\gamma$ has positive intersection number with $L_{0}$ since
$\tilde{\gamma}\subset{X}^{\prime\prime}$. We proved that the class
$\widehat{e_{0}}$ has positive intersection number with $L_{0}$.

If $\widehat{e^{\prime}_{0}}$ also has non-zero intersection number with
$L_{0}$ we define $e^{\prime\prime}_{0}=(e^{\prime}_{0})^{\pm 1}$ so that the
intersection number of $\widehat{e^{\prime\prime}_{0}}$ with $L_{0}$ is
positive. If $\widehat{e^{\prime}_{0}}$ has zero intersection number with
$L_{0}$ we put $e^{\prime\prime}_{0}=e_{0}\,e^{\prime}_{0}$. Then again the
intersection number of $\widehat{e^{\prime\prime}_{0}}$ with $L_{0}$ is
positive. Also, the intersection number of
$\widehat{e_{0}\,e^{\prime\prime}_{0}}$ with $L_{0}$ is positive. The set
$\mathcal{E}_{2}^{\prime}\stackrel{{\scriptstyle
def}}{{=}}\\{e_{0},e_{0}^{\prime\prime}\\}$ satisfies the condition required
in the proposition. We obtained Proposition 3 for a torus with a hole.

2\. A planar domain. Let $X$ be a planar domain. The domain $X$ is conformally
equivalent to a disc with $m$ smoothly bounded holes, equivalently, to the
Riemann sphere with $m+1$ smoothly bounded holes,
$\mathbb{P}^{1}\setminus\bigcup_{j=1}^{m+1}\mathcal{C}_{j}$, where
$\mathcal{C}_{m+1}$ contains the point $\infty$. As before the base point of
$X$ is denoted by $q_{0}$, and for each $j=1,\ldots,m,$ the generator
$e_{j,0}\in\mathcal{E}\subset\pi_{1}(X,q_{0})$ is represented by a curve
surrounding $\mathcal{C}_{j}$ once counterclockwise. We claim that there
exists a component $L_{0}$ of $L$ with limit points on the boundary of two
components $\partial{\mathcal{C}}_{j^{\prime}}$ and
$\partial{\mathcal{C}}_{j^{\prime\prime}}$ for some
$j^{\prime},j^{\prime\prime}\in\\{1,\ldots,m+1\\}$ with $j^{\prime\prime}\neq
j^{\prime}$.

Indeed, assume the contrary. Then, if a component of $L$ has limit points on
$\partial{\mathcal{C}}_{j},\,j\leq m,$ then all its limit points are on
$\partial{\mathcal{C}}_{j}$. Take a smoothly bounded simply connected domain
${\mathcal{C}}^{\prime}_{j}\Subset X\cup\overline{{\mathcal{C}}_{j}}$ that
contains the closure $\overline{{\mathcal{C}}_{j}}$ , so that its boundary
$\partial{\mathcal{C}}^{\prime}_{j}$ represents $\widehat{e_{j,0}}$. Then all
components $L^{\prime}_{k}$ of $L\setminus{\mathcal{C}}^{\prime}_{j}$ with an
endpoint on $\partial\mathcal{C}^{\prime}_{j}$ have both endpoints on this
circle. Fix a point
$p^{\prime}_{j}\in\partial\mathcal{C}^{\prime}_{j}\setminus L$. Assign to each
component $L_{k}$ of $L\setminus{\mathcal{C}}^{\prime}_{j}$ with both
endpoints on $\partial\mathcal{C}^{\prime}_{j}$ the closed arc $\alpha_{k}$ in
$\partial\mathcal{C}^{\prime}_{j}\setminus\\{p^{\prime}_{j}\\}$ with the same
endpoints as $L_{k}$.

The arcs $\alpha_{k}$ are partially ordered by inclusion. For an arc
$\alpha_{k}$ which contains no other of the arcs (a minimal arc) the curve
$f\circ\alpha_{k}$ except its endpoints is contained in
$\mathbb{C}\setminus\mathbb{R}$. Moreover, since $L_{k}$ is connected, the
endpoints of $f\circ\alpha_{k}$ lie on $\overline{f(L_{k})}$ which is
contained in one connected component of $\mathbb{R}\setminus\\{-1,1\\}$.
Hence, the curve $f\circ\alpha_{k}$ is homotopic in
$\mathbb{C}\setminus\\{-1,1\\}$ (with fixed endpoints) to a curve in
$\mathbb{R}\setminus\\{-1,1\\}$. The function $f$ either maps all points on
$\partial\mathcal{C}^{\prime}_{j}\setminus\alpha_{k}$ that are close to
$\alpha_{k}$ to the open upper half-plane or maps them all to the open lower
half-plane. (Recall, that zero is a regular value of $\mbox{Im}f$.) Hence, for
an open arc
$\alpha^{\prime}_{k}\subset\partial\mathcal{C}^{\prime}_{j}\setminus
p^{\prime}_{j}$ that contains $\alpha_{k}$ the curve
$f\circ\alpha^{\prime}_{k}$ is homotopic in $\mathbb{C}\setminus\\{-1,1\\}$
(with fixed endpoints) to a curve in $\mathbb{C}\setminus\mathbb{R}$.

Consider the arcs $\alpha_{k}$ with the following property. For an open arc
$\alpha^{\prime}_{k}$ in
$\partial\mathcal{C}^{\prime}_{j}\setminus\\{p^{\prime}_{j}\\}$ which contains
the closed arc $\alpha_{k}$ the mapping $f\circ\alpha^{\prime}_{k}$ is
homotopic in $\mathbb{C}\setminus\\{-1,1\\}$ (with fixed endpoints) to a curve
contained in $\mathbb{C}\setminus\mathbb{R}$. Induction on the arcs by
inclusion shows that this property is satisfied for all maximal arcs among the
$\alpha_{k}$ and, hence, $f\mid\partial\mathcal{C}^{\prime}_{j}$ is
contractible in $\mathbb{C}\setminus\\{-1,1\\}$. We saw that for each hole
$\mathcal{C}_{j},\,j\leq m$, whose boundary contains limit points of a
connected component of $L$, the monodromy along the curve
$\mathcal{C}^{\prime}_{j}$ with base point $p^{\prime}_{j}$ that repesents
$\widehat{e_{0,j}}$ is trivial. The contradiction proves the claim.

With $j^{\prime}$ and $j^{\prime\prime}$ being the numbers of the claim we
consider the set $\mathcal{E}_{2}^{\prime}\subset\mathcal{E}_{2}$ which
consists of the following primitive elements: $e_{j^{\prime},0}$, the element
$(e_{j^{\prime\prime},0})^{-1}$ provided $j^{\prime\prime}\neq m+1$, and
$e_{j^{\prime},0}\,e_{j,0}$ for all $j=1,\ldots,m,\,j\neq j^{\prime},j\neq
j^{\prime\prime}$. The free homotopy class of each element of
$\mathcal{E}_{2}^{\prime}$ has intersection number $1$ with $L_{0}$ after
suitable orientation of the curve $L_{0}$. Each product of at most two
different elements of $\mathcal{E}_{2}^{\prime}$ is a primitive element of
$\pi_{1}(X,q)$ and is contained in $\mathcal{E}_{4}$. Moreover, the
intersection number with $L_{0}$ of the free homotopy class of any such
element equals $1$ or $2$. Each element of $\mathcal{E}$ is the product of at
most two elements of
$\mathcal{E}_{2}^{\prime}\cup(\mathcal{E}_{2}^{\prime})^{-1}$.

The proposition is proved for the case of planar domains $X$. $\Box$

Proof of Proposition 4.

1\. A torus with a hole. Consider the curve $L_{0}$ and the set
$\mathcal{E}_{2}^{\prime}\subset\pi_{1}(X,q_{0})$ obtained in Proposition 3.
For one of the functions $M_{l}\circ f$, denoted by $F$, the image $F(L_{0})$
is contained in $(-1,1)$. Move the base point $q_{0}$ to a point $q\in L_{0}$
along a curve $\alpha$ in $X$, and consider the generators
$e=\mbox{Is}_{\alpha}(e_{0})$ and
$e^{\prime}=\mbox{Is}_{\alpha}({e^{\prime}_{0}})$ of $\pi_{1}(X,q)$, and the
set $\mbox{Is}_{\alpha}(\mathcal{E}_{2}^{\prime})\subset\pi_{1}(X,q)$. Then
$e$ and $e^{\prime}$ are products of at most two elements of
$\mbox{Is}_{\alpha}(\mathcal{E}_{2}^{\prime})$. Since the free homotopy class
of an element of $\pi_{1}(X,q_{0})$ coincides with the free homotopy class of
the element of $\pi_{1}(X,q)$ obtained by applying $\mbox{Is}_{\alpha}$, the
free homotopy class of each product of one or two elements of
$\mbox{Is}_{\alpha}(\mathcal{E}_{2}^{\prime})$ intersects $L_{0}$. We may
assume as in the proof of Proposition 3 that
$\mbox{Is}_{\alpha}(\mathcal{E}_{2}^{\prime})$ consists of the elements $e$
and $e^{\prime\prime}$, where $e^{\prime\prime}$ is the product of at most two
elements among the $e$ and $e^{\prime}$ and their inverses. Lemma 3 applies to
the pair $e$, $e^{\prime\prime}$, the function $F$, and the curve $L_{0}$. By
the conditions of Proposition 4 and Lemma 4 of [21] the monodromies of $F$
along $e$ and $e^{\prime\prime}$ are not powers of a single standard generator
of the fundamental group of
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$. Since for each
$\tilde{e}_{0}\in\pi_{1}(X,q_{0})$ the equality
$A(\widehat{\tilde{e_{0}}})=A(\mbox{Is}_{\alpha}(\widehat{\tilde{e_{0}}}))$
holds, this implies that the monodromy along $e$ and $e^{\prime\prime}$ is the
product of at most two elements of $\mathcal{L}_{-}$ not exceeding
$2\pi\lambda_{3}(X)$, and, hence, it has $\mathcal{L}_{-}$ not exceeding
$4\pi\lambda_{3}(X)$. Since $e^{\prime}$ is the product of at most two
elements among the $e$ and $e^{\prime\prime}$ and their inverses, we obtain
Proposition 4 for $e$ and $e^{\prime}$, in particular
$\mathcal{L}_{-}(F_{*}(e))$ and $\mathcal{L}_{-}(F_{*}(e^{\prime}))$ do not
exceed $8\pi\lambda_{3}(X)$. Proposition 4 is proved for tori with a hole.

2\. A planar domain. Consider the curve $L_{0}$ and the set
$\mathcal{E}_{2}^{\prime}$ of Proposition 3. Move the base point $q_{0}$ along
an arc $\alpha$ to a point $q\in L_{0}$. Then
$f(q)\in\mathbb{R}\setminus\\{-1,1\\}$ and for one of the mappings $f$,
$M_{1}\circ f$, or $M_{2}\circ f$, denoted by $F$,
$q^{\prime}\stackrel{{\scriptstyle def}}{{=}}F(q)$ is contained in $(-1,1)$.
Denote $e_{j}=\mbox{Is}_{\alpha}(e_{j,0})$ for each $j$. The $e_{j}$ form the
basis $\mbox{Is}_{\alpha}(\mathcal{E})$ of $\pi_{1}(X,q)$. The set
$\mbox{Is}_{\alpha}(\mathcal{E}_{2}^{\prime})$ consists of primitive elements
of $\pi_{1}(X,q)$ such that the free homotopy class of each product of one or
two elements of $\mbox{Is}_{\alpha}(\mathcal{E}_{2}^{\prime})$ intersects
$L_{0}$. Moreover, each element of $\mbox{Is}_{\alpha}(\mathcal{E})$ is the
product of one or two elements of
$\mbox{Is}_{\alpha}(\mathcal{E}_{2}^{\prime})\cup(\mbox{Is}_{\alpha}(\mathcal{E}_{2}^{\prime}))^{-1}$.

By the condition of the proposition not all monodromies
$F_{*}(e),e\in\mbox{Is}_{\alpha}(\mathcal{E}_{2}^{\prime}),$ are (trivial or
non-trivial) powers of the same standard generator of
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$. Apply Lemma 3 to all
pairs of elements of $\mbox{Is}_{\alpha}(\mathcal{E}_{2}^{\prime})$ whose
monodromies are not (trivial or non-trivial) powers of the same standard
generator of $\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$. Since the
product of at most two different elements of
$\mbox{Is}_{\alpha}(\mathcal{E}_{2}^{\prime})$ is contained in
$\mbox{Is}_{\alpha}(\mathcal{E}_{4})$ Lemma 3 shows that the monodromy
$F_{*}(e)$ along each element $e\in\mathcal{E}_{2}^{\prime}$ is the product of
at most two factors, each with $\mathcal{L}_{-}$ not exceeding
$2\pi\lambda_{4}(X)$. Since each element of $\mbox{Is}_{\alpha}(\mathcal{E})$
is a product of at most two factors in
$\mathcal{E}_{2}^{\prime}\cup(\mathcal{E}_{2}^{\prime})^{-1}$, the monodromy
$F_{*}(e_{j})$ along each generator $e_{j}$ of $\pi_{1}(X,q)$ is the product
of at most $4$ factors of $\mathcal{L}_{-}$ not exceeding
$2\pi\lambda_{4}(X)$, and, hence, each monodromy $F_{*}(e_{j})$ has
$\mathcal{L}_{-}$ not exceeding $8\pi\lambda_{4}(X)$. Proposition 4 is proved
for planar domains.

3\. The general case. Reduction to the case of non-trivial monodromies. We
first provide a reduction to the case when the monodromy along each element of
$\mathcal{E}\subset\pi_{1}(X,q_{0})$ is different from the identity. In other
words, we will prove that provided Proposition 4 is true for this case, it is
true in the general case.

Let $X$, $f$ and $\mathcal{E}$ be as in the statement of Proposition 4. We let
$\mathring{\mathcal{E}}$ be the set of elements from the chosen system of
generators $\mathcal{E}$ of $\pi_{1}(X,q_{0})$ along which the monodromy is
different from the identity. Keeping the same base point $\tilde{q}_{0}$ of
$\tilde{X}$ as before we consider the quotient
$X(\langle\mathring{\mathcal{E}}\rangle)=\tilde{X}\diagup\langle({\rm
Is}^{\tilde{q}_{0}})^{-1}(\mathring{\mathcal{E}})\rangle$ and the covering
$\omega_{\langle\mathring{\mathcal{E}}\rangle}:X(\langle\mathring{\mathcal{E}}\rangle)\to
X$, and the lift
$f_{\langle\mathring{\mathcal{E}}\rangle}=f\circ\omega_{\langle\mathring{\mathcal{E}}\rangle}$
of $f$ to $X(\langle\mathring{\mathcal{E}}\rangle)$. Put
$(q_{0})_{\langle\mathring{\mathcal{E}}\rangle}=\omega^{\langle\mathring{\mathcal{E}}\rangle}(\tilde{q}_{0})$.
The fundamental group of the Riemann surface
$X(\langle\mathring{\mathcal{E}}\rangle)$ with base point
$(q_{0})_{\langle\mathring{\mathcal{E}}\rangle}$ can be identified with
$\langle\mathring{\mathcal{E}}\rangle$. For each
$e_{j,0}\in\mathring{\mathcal{E}}$ we will denote by
$(e_{j,0})_{\langle\mathring{\mathcal{E}}\rangle}$ the element of the
fundamental group
$\pi_{1}(X\langle\mathring{\mathcal{E}}\rangle),(q_{0})_{\langle\mathring{\mathcal{E}}\rangle})$
that corresponds to
$(\omega_{\langle\mathring{\mathcal{E}}\rangle})_{*}((e_{j,0})_{\langle\mathring{\mathcal{E}}\rangle})=e_{j,0}$.
The lift $f_{\langle\mathring{\mathcal{E}}\rangle}$ to
$X(\langle\mathring{\mathcal{E}}\rangle)$ has non-trivial monodromy along each
element of $\mathring{\mathcal{E}}$. Our assumption implies that
$f_{\langle\mathring{\mathcal{E}}\rangle}:X(\langle\mathring{\mathcal{E}}\rangle)\to\mathbb{C}\setminus\\{-1,1\\}$
is irreducible. Moreover, zero is a regular value of
$f_{\langle\mathring{\mathcal{E}}\rangle}$. Apply to the Riemann surface
$X(\langle\mathring{\mathcal{E}}\rangle)$ and the function
$f_{\langle\mathring{\mathcal{E}}\rangle}$ the case of Proposition 4 when all
monodromies are non-trivial. We obtain a point
$q_{\langle\mathring{\mathcal{E}}\rangle}\in
X(\langle\mathring{\mathcal{E}}\rangle)$ and a Möbius transformation $M_{l}$,
such that the function $F_{\langle\mathring{\mathcal{E}}\rangle}=M_{l}\circ
f_{\langle\mathring{\mathcal{E}}\rangle}$ maps
$q_{\langle\mathring{\mathcal{E}}\rangle}$ to a point $q^{\prime}\in(-1,1)$,
and a curve $\alpha_{\langle\mathring{\mathcal{E}}\rangle}$ in
$X(\langle\mathring{\mathcal{E}}\rangle)$ with initial point
$(q_{0})_{\langle\mathring{\mathcal{E}}\rangle}$ and terminating point
$q_{\langle\mathring{\mathcal{E}}\rangle}$, such that the following holds. For
each
$(e_{j})_{\langle\mathring{\mathcal{E}}\rangle}=\mbox{Is}_{\alpha_{\langle\mathring{\mathcal{E}}\rangle}}((e_{j,0})_{\langle\mathring{\mathcal{E}}\rangle})\in\mbox{Is}_{\alpha_{\langle\mathring{\mathcal{E}}\rangle}}(\mathring{\mathcal{E}})$
the monodromy
$(F_{\langle\mathring{\mathcal{E}}\rangle})_{*}((e_{j})_{\langle\mathring{\mathcal{E}}\rangle})$
is the product of at most four elements of
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ of $\mathcal{L}_{-}$ not
exceeding $2\pi\lambda_{4}(X(\langle\mathring{\mathcal{E}}\rangle))$. In
particular,
$\mathcal{L}_{-}((F_{\langle\mathring{\mathcal{E}}\rangle})_{*}((e_{j})_{\langle\mathring{\mathcal{E}}\rangle}))\leq
8\pi\lambda_{4}(X(\langle\mathring{\mathcal{E}}\rangle))$.

Consider the projection
$\omega_{\langle\mathring{\mathcal{E}}\rangle}:X(\langle\mathring{\mathcal{E}}\rangle)\to
X$. Put $F=M_{l}\circ f,\,$ for the Möbius transformation mentioned above, so
that
$F_{\langle\mathring{\mathcal{E}}\rangle}=F\circ\omega_{\langle\mathring{\mathcal{E}}\rangle}$.
Take
$\alpha=\omega_{\langle\mathring{\mathcal{E}}\rangle}\circ\alpha_{\langle\mathring{\mathcal{E}}\rangle}$,
and $q\stackrel{{\scriptstyle
def}}{{=}}\omega_{\langle\mathring{\mathcal{E}}\rangle}(q_{\langle\mathring{\mathcal{E}}\rangle})$.
For each $e_{j,0}\in\mathring{\mathcal{E}}$ we obtain
$(F_{\langle\mathring{\mathcal{E}}\rangle})_{*}((e_{j,0})_{\langle\mathring{\mathcal{E}}\rangle})=(M_{l}\circ
f\circ\omega_{\langle\mathring{\mathcal{E}}\rangle})_{*}((e_{j,0})_{\langle\mathring{\mathcal{E}}\rangle})=F_{*}\circ(\omega_{\langle\mathring{\mathcal{E}}\rangle})_{*}((e_{j,0})_{\langle\mathring{\mathcal{E}}\rangle})=F_{*}(e_{j,0})$.
Further, for
$(e_{j})_{\langle\mathring{\mathcal{E}}\rangle}=\mbox{Is}_{\alpha_{\langle\mathring{\mathcal{E}}\rangle}}((e_{j,0})_{\langle\mathring{\mathcal{E}}\rangle})$
we get
$(\omega_{\langle\mathring{\mathcal{E}}\rangle})_{*}((e_{j})_{\langle\mathring{\mathcal{E}}\rangle})={\rm
Is}_{\alpha}(e_{j,0})$. This can be checked by applying
$\omega_{\langle\mathring{\mathcal{E}}\rangle}$ to curves representing
$(e_{j,0})_{\langle\mathring{\mathcal{E}}\rangle}$. Hence,
$(F_{\langle\mathring{\mathcal{E}}\rangle})_{*}((e_{j})_{\langle\mathring{\mathcal{E}}\rangle})=F_{*}({\rm
Is}_{\alpha}(e_{j,0}))=F_{*}(e_{j})$ for each $e_{j}\in{\rm
Is}_{\alpha}(\mathring{\mathcal{E}})$. We obtain the inequality

$\mathcal{L}_{-}(F_{*}(e_{j}))\leq
8\pi\lambda_{4}(X(\langle\mathring{\mathcal{E}}\rangle))$ (15)

for each $e_{j}\in{\rm Is}_{\alpha}(\mathring{\mathcal{E}})$, and, hence, for
each $e_{j}\in{\rm Is}_{\alpha}({\mathcal{E}})$, since
$\mathcal{L}_{-}(e_{j})=0$ for $e_{j}\in{\rm
Is}_{\alpha}({\mathcal{E}}\setminus\mathring{\mathcal{E}})$.

It remains to prove the inequality
$\lambda_{4}(X(\langle\mathring{\mathcal{E}}\rangle))\leq\lambda_{4}(X)$. The
quantity $\lambda_{4}(X)$ is equal to the maximum of the extremal length of
the annuli $\tilde{X}\diagup({\rm Is}^{\tilde{q}_{0}})^{-1}(\langle
e_{0}\rangle)$ over primitive elements $e_{0}$ of the fundamental group
$\pi_{1}(X,q_{0})$ of $X$ that are products of at most four elements of
$\mathcal{E}\cup(\mathcal{E})^{-1}$ for the chosen set of generators
$\mathcal{E}$ of $\pi_{1}(X,q_{0})$. Recall that we represented the Riemann
surface $X$ as quotient $\tilde{X}\diagup({\rm
Is}^{\tilde{q}_{0}})^{-1}(\langle\mathcal{E}\rangle)$.

The Riemann surface $X(\langle\mathring{\mathcal{E}}\rangle)$ is equal to the
quotient $\tilde{X}\diagup({\rm
Is}^{\tilde{q}_{0}})^{-1}(\langle\mathring{\mathcal{E}}\rangle)$. The chosen
set $\mathring{\mathcal{E}}$ corresponds to the set
$\mathring{\mathcal{E}}_{\langle\mathring{\mathcal{E}}\rangle}$ of generators
of the fundamental group
$\pi_{1}(X(\langle\mathring{\mathcal{E}}\rangle),q_{\langle\mathring{\mathcal{E}}\rangle})$.
The quantity $\lambda_{4}(X(\langle\mathring{\mathcal{E}}\rangle))$ is the
maximum of the extremal length of $\tilde{X}\diagup({\rm
Is}^{\tilde{q}_{0}})^{-1}(\langle e_{0}\rangle)$ over elements
$e_{0}\in\pi_{1}(X,q_{0})$, such that
$(e_{0})_{\langle\mathring{\mathcal{E}}\rangle}$ is a primitive element of the
fundamental group
$\pi_{1}(X(\langle\mathring{\mathcal{E}}\rangle),q_{\langle\mathring{\mathcal{E}}\rangle})$
of $X(\langle\mathring{\mathcal{E}}\rangle)$ which is the product of at most
four elements of
$\mathring{\mathcal{E}}_{\langle\mathring{\mathcal{E}}\rangle}\cup(\mathring{\mathcal{E}}_{\langle\mathring{\mathcal{E}}\rangle})^{-1}$.
This quantity does not exceed $\lambda_{4}(X)$.

We proved Proposition 4 for the general case provided it is proved in the case
when all monodromies are non-trivial.

Positive intersection number. We assume now that the monodromies along all
elements of the chosen set of generators $\mathcal{E}$ of $\pi_{1}(X,q_{0})$
are non-trivial and the Riemann surface is not a torus with a hole nor a
planar domain, and prove the proposition for this case. In this case $X$
contains a handle that corresponds to generators, denoted by $e_{0}$ and
$e^{\prime}_{0}$, contained in $\mathcal{E}\subset\pi_{1}(X,q_{0})$ so that
none of the two monodromies $f_{*}(e_{0})$ or $f_{*}(e^{\prime}_{0})$ is the
identity. There may be several choices of such handles. If there is a handle
so that the monodromies of $f$ along the two elements from $\mathcal{E}$
corresponding to the handle are not powers of the same element of
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},f(q_{0}))$, we take such a handle.

Consider the Riemann surface $X(\langle e_{0},e^{\prime}_{0}\rangle)$. It is a
torus with a hole and admits a holomorphic covering $\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}:X(\langle e_{0},e^{\prime}_{0}\rangle)\to X$ of
$X$. Put ${q_{0}}_{\langle e_{0},e^{\prime}_{0}\rangle}=\omega^{\langle
e_{0},e^{\prime}_{0}\rangle}(\tilde{q}_{0})$. The fundamental group of
$X(\langle e_{0},e^{\prime}_{0}\rangle)$ with base point ${q_{0}}_{\langle
e_{0},e^{\prime}_{0}\rangle}$ can be identified with $\langle
e_{0},e^{\prime}_{0}\rangle$. For an element $\tilde{e}_{0}\in\langle
e_{0},e^{\prime}_{0}\rangle$ we denote by $(\tilde{e}_{0})_{\langle
e_{0},e^{\prime}_{0}\rangle}$ the element of $\pi_{1}(X(\langle
e_{0},e^{\prime}_{0}\rangle),{q_{0}}_{\langle e_{0},e^{\prime}_{0}\rangle})$
that projects to $\tilde{e}_{0}$ under $\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}$. By our conditions the lift $f_{\langle
e_{0},e^{\prime}_{0}\rangle}$ of $f$ to the Riemann surface $X(\langle
e_{0},e^{\prime}_{0}\rangle)$ is not contractible and $0$ is a regular value
of its imaginary part. Let $L\subset X$ be the set where $f$ is real. The set
$\omega_{\langle e_{0},e^{\prime}_{0}\rangle}^{-1}(L)$ coincides with the set
of points on which the lift $f_{\langle
e_{0},e^{\prime}_{0}\rangle}=f\circ\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}$ of the mapping $f$ is real. According to
Proposition 3 for the case of a torus with a hole there is a connected
component $L_{\langle e_{0},e^{\prime}_{0}\rangle}$ of $\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}^{-1}(L)$ such that the free homotopy class of one
of the elements $(e_{0})_{\langle e_{0},e^{\prime}_{0}\rangle}$ or
$(e_{0}^{\prime})_{\langle e_{0},e^{\prime}_{0}\rangle}$, say of the class
$\widehat{(e_{0})_{\langle e_{0},e^{\prime}_{0}\rangle}}$ of loops in
$X_{\langle e_{0},e^{\prime}_{0}\rangle}$, that are free homotopic to loops
representing $(e_{0})_{\langle e_{0},e^{\prime}_{0}\rangle}$, has positive
intersection number with $L_{\langle e_{0},e^{\prime}_{0}\rangle}$ (after
suitably orienting $L_{\langle e_{0},e^{\prime}_{0}\rangle}$).

Take a point $q_{\langle e_{0},e^{\prime}_{0}\rangle}\in L_{\langle
e_{0},e^{\prime}_{0}\rangle}$. The function $f\circ\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}$ takes a real value at $q_{\langle
e_{0},e^{\prime}_{0}\rangle}$. Put $q=\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}(q_{\langle e_{0},e^{\prime}_{0}\rangle})$. Choose
a point $\tilde{q}\in\tilde{X}$, for which $\omega^{\langle
e_{0},e^{\prime}_{0}\rangle}(\tilde{q})=q_{\langle
e_{0},e^{\prime}_{0}\rangle}$. Let $\tilde{\alpha}$ be a curve in $\tilde{X}$
with initial point $\tilde{q}_{0}$ and terminating point $\tilde{q}$. Then
$\alpha_{\langle e_{0},e^{\prime}_{0}\rangle}\stackrel{{\scriptstyle
def}}{{=}}\omega^{\langle e_{0},e^{\prime}_{0}\rangle}(\tilde{\alpha})$ is a
curve in $X(\langle e_{0},e^{\prime}_{0}\rangle)$ with initial point
${q_{0}}_{\langle e_{0},e^{\prime}_{0}\rangle}$ and terminating point
$q_{\langle e_{0},e^{\prime}_{0}\rangle}$. Note that with this choice
$\alpha_{\langle e_{0},e^{\prime}_{0}\rangle}$ and $\tilde{q}$ are compatible
(as a curve in $X(\langle e_{0},e^{\prime}_{0}\rangle)$ and a point in the
universal covering of $X(\langle e_{0},e^{\prime}_{0}\rangle)$ that projects
to the terminating point of the curve).

Let $\tilde{\mathcal{E}}\subset\mathcal{E}$ be any set, that contains $e_{0}$
and $e^{\prime}_{0}$, for which the quotient $X\diagup({\rm
Is}^{\tilde{q}_{0}})^{-1}(\langle\tilde{\mathcal{E}}\rangle)=X(\langle\tilde{\mathcal{E}}\rangle)$
is a Riemann surface of genus $1$ with $m^{\prime}+1$ holes. (Here
$m^{\prime}\geq 0$. We allow the case $\tilde{\mathcal{E}}=\pi_{1}(X,q_{0})$,
if $X$ is of genus $1$ with $m+1$ holes.) We consider the projection
$\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}^{\langle\tilde{\mathcal{E}}\rangle}:X(\langle
e_{0},e^{\prime}_{0}\rangle)\to X(\langle\tilde{\mathcal{E}}\rangle)$. Put
$L_{\langle\tilde{\mathcal{E}}\rangle}\stackrel{{\scriptstyle
def}}{{=}}\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}^{\langle\tilde{\mathcal{E}}\rangle}(L_{{\langle
e_{0},e^{\prime}_{0}\rangle}})$. We write also $L_{0}\stackrel{{\scriptstyle
def}}{{=}}L_{\pi_{1}(X,q_{0})}=\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}^{\pi_{1}(X,q_{0})}(L_{\langle
e_{0},e^{\prime}_{0}\rangle})$. Each $L_{\langle\tilde{\mathcal{E}}\rangle}$
is connected. By the commutative diagram Figure 3 the equality
$\omega^{\pi_{1}(X,q_{0})}_{\langle\tilde{\mathcal{E}}\rangle}\circ\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}^{\langle\tilde{\mathcal{E}}\rangle}=\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}^{\pi_{1}(X,q_{0})}$ holds. Hence,

$\omega^{\pi_{1}(X,q_{0})}_{\langle\tilde{\mathcal{E}}\rangle}(L_{\langle\tilde{\mathcal{E}}})=L_{0}\subset
L\,.$ (16)

In other words,
$L_{\langle\tilde{\mathcal{E}}\rangle}\subset(\omega^{\pi_{1}(X,q_{0})}_{\langle\tilde{\mathcal{E}}\rangle})^{-1}(L)$,
and therefore $L_{\langle\tilde{\mathcal{E}}\rangle}$ has no self-
intersection.

We want to show that there exists an element $e_{0}^{*}$ of
$\tilde{\mathcal{E}}$ such that the class
$\widehat{(e_{0}^{*})_{\langle\tilde{\mathcal{E}}\rangle}}$ of loops on
$X({\langle\tilde{\mathcal{E}}\rangle})$, that are free homotopic to
representatives of $(e_{0}^{*})_{\langle\tilde{\mathcal{E}}\rangle}$, has
positive intersection number with $L_{\langle\tilde{\mathcal{E}}\rangle}$
(with a suitable orientation of $L_{\langle\tilde{\mathcal{E}}\rangle}$).

The free homotopy class
$\widehat{(e_{0})_{\langle\tilde{\mathcal{E}}\rangle}}$ in
$X(\langle\tilde{\mathcal{E}}\rangle)$ that is related to $e_{0}$ intersects
$L_{\langle\tilde{\mathcal{E}}\rangle}$. Indeed, consider any loop
$\gamma^{\prime}_{\langle\tilde{\mathcal{E}}\rangle}$ in
$X(\langle\tilde{\mathcal{E}}\rangle)$ with some base point
$q^{\prime}_{\langle\tilde{\mathcal{E}}\rangle}$, that represents
$\widehat{(e_{0})_{\langle\tilde{\mathcal{E}}\rangle}}$. There exists a loop
${\gamma}^{\prime}_{\langle e_{0},e^{\prime}_{0}\rangle}$ in $X(\langle
e_{0},e^{\prime}_{0}\rangle)$ in $X(\langle e_{0},e^{\prime}_{0}\rangle)$
which represents $\widehat{(e_{0})_{\langle e_{0},e^{\prime}_{0}\rangle}}$
such that $\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}^{\langle\tilde{\mathcal{E}}\rangle}({\gamma}^{\prime}_{\langle
e_{0},e^{\prime}_{0}\rangle})=\gamma^{\prime}_{\langle\tilde{\mathcal{E}}\rangle}$.
Such a curve ${\gamma}^{\prime}_{\langle e_{0},e^{\prime}_{0}\rangle}$ can be
obtained as follows. There is a loop
$\gamma^{\prime\prime}_{\langle\tilde{\mathcal{E}}\rangle}$ in
$X(\langle\tilde{\mathcal{E}}\rangle)$ with base point
${q_{0}}_{\langle\tilde{\mathcal{E}}\rangle}$ that represents
$(e_{0})_{\langle\tilde{\mathcal{E}}\rangle}$, and a curve
$\alpha^{\prime}_{\langle\tilde{\mathcal{E}}\rangle}$ in
$X(\langle\tilde{\mathcal{E}}\rangle)$, such that
$\gamma^{\prime}_{\langle\tilde{\mathcal{E}}\rangle}$ is homotopic with fixed
endpoint to
$(\alpha^{\prime}_{\langle\tilde{\mathcal{E}}\rangle})^{-1}\,\gamma^{\prime\prime}_{\langle\tilde{\mathcal{E}}\rangle}\,\alpha^{\prime}_{\langle\tilde{\mathcal{E}}\rangle}$.
Consider the lift $\tilde{\gamma}^{\prime\prime}$ of
$\gamma^{\prime\prime}_{\langle\tilde{\mathcal{E}}\rangle}$ to $\tilde{X}$
with terminating point $\tilde{q}_{0}$, and the lift $\tilde{\alpha}^{\prime}$
of $\alpha^{\prime}_{\langle\tilde{\mathcal{E}}\rangle}$ with initial point
$\tilde{q}_{0}$. The initial point of
$\gamma^{\prime\prime}_{\langle\tilde{\mathcal{E}}\rangle}$ equals
$\sigma(\tilde{q}_{0})$ for the covering transformation $\sigma=({\rm
Is}^{\tilde{q}_{0}})^{-1}(e_{0})$. The initial point of the curve
$\sigma((\tilde{\alpha}^{\prime})^{-1})\tilde{\gamma}^{\prime\prime}\tilde{\alpha}^{\prime}$
is obtained from its terminating point by applying the covering transformation
$\sigma$. Hence, $\omega^{\langle
e_{0},e^{\prime}_{0}\rangle}(\sigma((\tilde{\alpha}^{\prime})^{-1})\tilde{\gamma}^{\prime\prime}\tilde{\alpha}^{\prime})$
is a closed curve in $X(\langle e_{0},e^{\prime}_{0}\rangle$ that represents
$\widehat{(e_{0})_{\langle e_{0},e^{\prime}_{0}\rangle}}$ and projects to
$(\alpha^{\prime}_{\langle\tilde{\mathcal{E}}\rangle})^{-1}\,\gamma^{\prime\prime}_{\langle\tilde{\mathcal{E}}\rangle}\,\alpha^{\prime}_{\langle\tilde{\mathcal{E}}\rangle}$
under $\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}^{\langle\tilde{\mathcal{E}}\rangle}$. Since
$\gamma^{\prime}_{\langle\tilde{\mathcal{E}}\rangle}$ is homotopic to
$(\alpha^{\prime}_{\langle\tilde{\mathcal{E}}\rangle})^{-1}\,\gamma^{\prime\prime}_{\langle\tilde{\mathcal{E}}\rangle}\,\alpha^{\prime}_{\langle\tilde{\mathcal{E}}\rangle}$
with fixed base point, it also has a lift to $X(\langle
e_{0},e^{\prime}_{0}\rangle)$ which represents $\widehat{(e_{0})_{\langle
e_{0},e^{\prime}_{0}\rangle}}$.

Since $\widehat{(e_{0})_{\langle e_{0},e^{\prime}_{0}\rangle}}$ intersects
$L_{\langle e_{0},e^{\prime}_{0}\rangle}$, the loop ${\gamma}_{\langle
e_{0},e^{\prime}_{0}\rangle}$ has an intersection point $p^{\prime}_{\langle
e_{0},e^{\prime}_{0}\rangle}$ with $L_{\langle e_{0},e^{\prime}_{0}\rangle}$.
The point $p^{\prime}_{\langle\tilde{\mathcal{E}}\rangle}=\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}^{\langle\tilde{\mathcal{E}}\rangle}(p^{\prime}_{\langle
e_{0},e^{\prime}_{0}\rangle})$ is contained in
$\gamma_{\langle\tilde{\mathcal{E}}\rangle}$ and in
$L_{\langle\tilde{\mathcal{E}}\rangle}$. We proved that the free homotopy
class $\widehat{(e_{0})_{\langle\tilde{\mathcal{E}}\rangle}}$ in
$X(\langle\tilde{\mathcal{E}}\rangle)$ intersects
$L_{\langle\tilde{\mathcal{E}}\rangle}$.

There are several possibilities for the behaviour of the curve
$L_{\langle\tilde{\mathcal{E}}\rangle}$. Identify
$L_{\langle\tilde{\mathcal{E}}\rangle}$ with its image under the embedding
$X(\langle\tilde{\mathcal{E}}\rangle)\hookrightarrow
X(\langle\tilde{\mathcal{E}}\rangle)^{c}$ into the closed torus obtained by
filling the holes of $X(\langle\tilde{\mathcal{E}}\rangle)$, and identify the
elements of the fundamental group
$\pi_{1}(X(\langle\tilde{\mathcal{E}}\rangle),(q_{0})_{\langle\tilde{\mathcal{E}}\rangle})$
with their images in the fundamental group of the closed torus
$X(\langle\tilde{\mathcal{E}}\rangle)^{c}$.

If $L_{\langle\tilde{\mathcal{E}}\rangle}$ is closed, then it does not divide
$X(\langle\tilde{\mathcal{E}}\rangle)^{c}$. Indeed, if
$L_{\langle\tilde{\mathcal{E}}\rangle}$ did divide the closed torus
$X(\langle\tilde{\mathcal{E}}\rangle)^{c}$, it would be null-homologous in the
closed torus, hence, it would be homotopic to a constant in the closed torus,
since the fundamental group of the closed torus is commutative. Then for both,
$e_{0}$ and $e^{\prime}_{0}$, there would exist loops representing their free
homotopy classes $\widehat{(e_{0})^{c}_{\langle\tilde{\mathcal{E}}\rangle}}$
and $\widehat{({e^{\prime}_{0}})^{c}_{\langle\tilde{\mathcal{E}}\rangle}}$ in
$X(\langle\tilde{\mathcal{E}}\rangle)^{c}$ and not intersecting
$L_{\langle\tilde{\mathcal{E}}\rangle}$.

Since $L_{\langle\tilde{\mathcal{E}}\rangle}$ does not divide the closed torus
$X(\langle\tilde{\mathcal{E}}\rangle)^{c}$, the intersection number with
$L_{\langle\tilde{\mathcal{E}}\rangle}$ of one of the free homotopy classes,
$\widehat{(e_{0})^{c}_{\langle\tilde{\mathcal{E}}\rangle}}$ or
$\widehat{({e^{\prime}_{0}})^{c}_{\langle\tilde{\mathcal{E}}\rangle}}$, on
$X(\langle\tilde{\mathcal{E}}\rangle)^{c}$ is non-zero, hence one of the
classes, $\widehat{(e_{0})_{\langle\tilde{\mathcal{E}}\rangle}}$ or
$\widehat{(e_{0}^{\prime})_{\langle\tilde{\mathcal{E}}\rangle}}$, on
$X(\langle\tilde{\mathcal{E}}\rangle)$ has positive intersection number with
$L_{\langle\tilde{\mathcal{E}}\rangle}$ after orienting
$L_{\langle\tilde{\mathcal{E}}\rangle}$ suitably.

Suppose $L_{\langle\tilde{\mathcal{E}}\rangle}$ is a simple relatively closed
arc in $X(\langle\tilde{\mathcal{E}}\rangle)$ with the whole limit set on the
boundary of a single hole $\mathcal{C}$. Then by the same reasoning as above,
$L_{\langle\tilde{\mathcal{E}}\rangle}\cup\mathcal{C}$ does not divide the
closed torus $X(\langle\tilde{\mathcal{E}}\rangle)^{c}$ and again the
intersection number with $L_{\langle\tilde{\mathcal{E}}\rangle}$ of one of the
classes, $\widehat{(e_{0})_{\langle\tilde{\mathcal{E}}\rangle}}$ or
$\widehat{(e_{0}^{\prime})_{\langle\tilde{\mathcal{E}}\rangle}}$, is non-zero.

If $L_{\langle\tilde{\mathcal{E}}\rangle}$ is a relatively closed arc in
$X(\langle\tilde{\mathcal{E}}\rangle)$ with limit sets on the boundary of
different holes. Then one of these holes, $\mathcal{C}_{j}$, has label $j$ not
exceeding $m$. The free homotopy class
$\widehat{(e_{j,0})_{\langle\tilde{\mathcal{E}}\rangle}}$ corresponding to the
generator $e_{j,0}\in\mathcal{E}\subset\pi_{1}(X,q_{0})$ whose representives
surround $\mathcal{C}_{j}$, has non-vanishing intersection number with
$L_{\langle\tilde{\mathcal{E}}\rangle}$. After orienting
$L_{\langle\tilde{\mathcal{E}}\rangle}$ suitably the intersection number of
$\widehat{(e_{j,0})_{\langle\tilde{\mathcal{E}}\rangle}}$ with
$L_{\langle\tilde{\mathcal{E}}\rangle}$ is positive.

End of Proof. Let again the monodromies of $f$ along all elements of
$\mathcal{E}$ be non-trivial, and let $\tilde{\mathcal{E}}$ be a subset of
$\mathcal{E}$ such that the quotient $X(\langle\tilde{\mathcal{E}}\rangle)$ is
a torus with $m^{\prime}+1$ holes. We obtained the following objects related
to $X(\langle e_{0},e^{\prime}_{0}\rangle)$, the point ${q_{0}}_{{\langle
e_{0},e^{\prime}_{0}\rangle}}$, the curve $L_{{\langle
e_{0},e^{\prime}_{0}\rangle}}$, the point ${q}_{{\langle
e_{0},e^{\prime}_{0}\rangle}}\in L_{{\langle e_{0},e^{\prime}_{0}\rangle}}$,
and the curve $\alpha_{{\langle e_{0},e^{\prime}_{0}\rangle}}$. Let
${q_{0}}_{{\langle\tilde{\mathcal{E}}\rangle}}$,
$L_{{\langle\tilde{\mathcal{E}}\rangle}}$, the point
${q}_{{\langle\tilde{\mathcal{E}}\rangle}}$ in
$L_{{\langle\tilde{\mathcal{E}}\rangle}}$, and
$\alpha_{{\langle\tilde{\mathcal{E}}\rangle}}$ be the respective images under
$\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}^{\langle\tilde{\mathcal{E}}\rangle}$. Recall that
$f_{\langle\tilde{\mathcal{E}}\rangle}=f\circ\omega_{\langle\tilde{\mathcal{E}}\rangle}^{\pi_{1}(X,q_{0})}$
and, hence
$f_{\langle\tilde{\mathcal{E}}\rangle}(L_{\langle\tilde{\mathcal{E}}\rangle})=f\circ\omega_{\langle\tilde{\mathcal{E}}\rangle}^{\pi_{1}(X,q_{0})}(L_{\langle\tilde{\mathcal{E}}\rangle})=f(L_{0})$
(see (16)) is contained in the connected component of
$\mathbb{R}\setminus\\{-1,1\\}$ that contains $f(L_{0})$. Choose the Möbius
transformation $M_{l}$ for which the mapping $F=M_{l}\circ f$ takes $L_{0}$ to
$(-1,1)$. The mapping $F_{\langle\tilde{\mathcal{E}}\rangle}=M_{l}\circ
f_{\langle\tilde{\mathcal{E}}\rangle}$ takes
$L_{\langle\tilde{\mathcal{E}}\rangle}$ to $(-1,1)$, and the point
$F_{\langle\tilde{\mathcal{E}}\rangle}(q_{\langle\tilde{\mathcal{E}}\rangle})=F(q)$,
denoted by $q^{\prime}$, is contained in $(-1,1)$. Let $\alpha=\omega_{\langle
e_{0},e^{\prime}_{0}\rangle}^{\pi_{1}(X,q_{0})}\circ\alpha_{\langle
e_{0},e^{\prime}_{0}\rangle}$. For $e_{j,0}\in\tilde{\mathcal{E}}$ we put
$(e_{j})_{\langle\tilde{\mathcal{E}}\rangle}=\mbox{Is}_{\alpha_{\langle\tilde{\mathcal{E}}\rangle}}((e_{j,0})_{\langle\tilde{\mathcal{E}}\rangle})$
and $e_{j}=\mbox{Is}_{\alpha}(e_{j,0})$. Since $\alpha_{\langle
e_{0},e^{\prime}_{0}\rangle}$ and $\tilde{q}$ are compatible, also
$\alpha_{\langle\tilde{\mathcal{E}}\rangle}$ and $\tilde{q}$ are compatible,
and $\alpha$ and $\tilde{q}$ are compatible, and the equalities
$F_{*}(e_{j})=(F_{\langle\tilde{\mathcal{E}}\rangle})_{*}((e_{j})_{\langle\tilde{\mathcal{E}}\rangle}),\,e_{j_{0}}\in\tilde{\mathcal{E}},$
on monodromies hold.

Take now an arbitrary element $\tilde{e}_{0}$ of ${\mathcal{E}}$. We want to
prove the statement of Proposition 4 for the monodromy of $F$ along
$\tilde{e}_{0}$. For each element $\tilde{e}_{0}$ of ${\mathcal{E}}$ we may
choose a set $\tilde{\mathcal{E}}\subset\mathcal{E}$ which contains
$\tilde{e}_{0}$, $e_{0}$, and $e^{\prime}_{0}$ such that $\tilde{X}\diagup{\rm
Is}^{\tilde{q}_{0}}(\langle\tilde{\mathcal{E}}\rangle)$ is a Riemann surface
of genus $1$ with $m^{\prime}+1$ holes for some $m^{\prime},\,0\leq
m^{\prime}\leq 1,$ and the monodromies of $F$ along the elements of
$\mbox{Is}_{\alpha}(\tilde{\mathcal{E}})$ are not powers of a single $a_{j}$.
Indeed, if the monodromy along the three elements
$\tilde{e}\stackrel{{\scriptstyle
def}}{{=}}\mbox{Is}_{\alpha}(\tilde{e}_{0})$, ${e}\stackrel{{\scriptstyle
def}}{{=}}\mbox{Is}_{\alpha}({e}_{0})$, and
${e}^{\prime}\stackrel{{\scriptstyle
def}}{{=}}\mbox{Is}_{\alpha}({e}^{\prime}_{0})$ are not powers of a single
$a_{j}$ then we may take the set
$\tilde{\mathcal{E}}=\\{e_{0},e^{\prime}_{0},\tilde{e}_{0}\\}$. If all three
monodromies are powers of a single $a_{j}$, then there exists some element
$\tilde{e}^{\prime}$ of
$\mbox{Is}_{\alpha}(\mathcal{E})\setminus\\{e,e^{\prime},\tilde{e}\\}$ such
that the monodromy of $f$ along this element is not a power of $a_{j}$. Then
the monodromies along $e_{0}$ and $e^{\prime}_{0}$ are powers of a single
element, hence the same is true for each pair of elements of $\mathcal{E}$
corresponding to a handle, and the elements $(\tilde{e})_{0}$ and
$\tilde{e}^{\prime}_{0}$ cannot form a pair of elements from $\mathcal{E}$
that correspond to a handle. Hence, for
$\tilde{\mathcal{E}}=\\{e_{0}^{\prime},e_{0},\tilde{e}_{0},\tilde{e}_{0}^{\prime}\\}$
the quotient $\tilde{X}\diagup{\rm
Is}^{\tilde{q}_{0}}(\tilde{\langle{\mathcal{E}}\rangle})$ has genus $1$.

It remains to prove that for each of the chosen $\tilde{\mathcal{E}}$ each
monodromy
$(F_{\langle\tilde{\mathcal{E}}\rangle})_{*}((e_{j})_{\langle\tilde{\mathcal{E}}\rangle})$,
$e_{j}\in\mbox{Is}_{\alpha}({\tilde{\mathcal{E}}})$, is the product of at most
four elements of $\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ of
$\mathcal{L}_{-}$ not exceeding
$2\pi\lambda_{4}(X(\langle\tilde{\mathcal{E}}\rangle)$ and
$\mathcal{L}_{-}((F_{\langle\tilde{\mathcal{E}}\rangle})_{*}((e_{j})_{\langle\tilde{\mathcal{E}}\rangle}))\leq
8\pi\lambda_{4}(X(\langle\tilde{\mathcal{E}}\rangle))$. This follows along the
same lines as the previous proofs. We proved the existence of the element
$e_{0}^{*}\in\tilde{\mathcal{E}}$ such that the free homotopy class
$\widehat{(e_{0}^{*})_{\langle\tilde{\mathcal{E}}\rangle}}$ in
$X(\langle\tilde{\mathcal{E}}\rangle)$ has positive intersection number with
$L_{\langle\tilde{\mathcal{E}}\rangle}$ (oriented suitably). Then for each
other element $e_{j,0}\in\tilde{\mathcal{E}}$ the free homotopy class in
$X(\langle\tilde{\mathcal{E}}\rangle)$ corresponding to one of the elements
$(e_{j,0})^{\pm 1}$ or $e_{0}^{*}\,e_{j,0}$, denoted by $e_{j,0}^{\prime}$,
has positive intersection number with $L_{\langle\tilde{\mathcal{E}}\rangle}$.
Let $\tilde{\mathcal{E}}_{2}^{\prime}\subset\mathcal{E}_{2}$ be the set
consisting of $e_{0}^{*}$ and any other element $e_{j,0}$ of
$\tilde{\mathcal{E}}$ replaced by $e_{j,0}^{\prime}$. Then
$\tilde{\mathcal{E}}_{2}^{\prime}$ is a subset of the set
$\tilde{\mathcal{E}}_{2}$ of primitive elements that are products of one or
two elements of $\mathcal{E}\cup\mathcal{E}^{-1}$. The free homotopy class of
the product of one or two elements of $(\tilde{\mathcal{E}})_{2}^{\prime}$
intersects $L_{\langle\tilde{\mathcal{E}}\rangle}$ positively.

For each element of $\tilde{\mathcal{E}}_{2}^{\prime}$ there is another
element of $\tilde{\mathcal{E}}_{2}^{\prime}$ such that the monodromies of
$F_{\langle\tilde{\mathcal{E}}\rangle}$ along the images of the two elements
under $\mbox{Is}_{\alpha}$ are not powers of the same $a_{j}$. Apply Lemma 3
to such pairs of elements of $\tilde{\mathcal{E}}_{2}^{\prime}$ and notice
that each element of $\tilde{\mathcal{E}}$ is the product of at most two
elements of $\tilde{\mathcal{E}}_{2}^{\prime}$. We see that each monodromy
$(F_{\langle\tilde{\mathcal{E}}\rangle})_{*}((e_{j})_{\langle\tilde{\mathcal{E}}\rangle})$,
$e_{j}\in\mbox{Is}_{\alpha}({\tilde{\mathcal{E}}})$, is the product of at most
four elements of $\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ of
$\mathcal{L}_{-}$ that does not exceed
$2\pi\lambda_{4}(X(\langle\tilde{\mathcal{E}}\rangle))$. Since it can be seen
as above that
$\lambda_{4}(X(\langle\tilde{\mathcal{E}}\rangle))\leq\lambda_{4}(X)$, we
obtain that the monodromy along each element $e_{j}\in{\rm
Is}_{\alpha}(\tilde{\mathcal{E}})$ is the product of at most four elements of
$\pi_{1}(\mathbb{C}\setminus\\{-1,1\\},q^{\prime})$ of $\mathcal{L}_{-}$ not
exceeding $2\pi\lambda_{4}(X)$, and the estimate
$\mathcal{L}_{-}(F_{*}(e_{j}))\leq 8\pi\lambda_{4}(X)$ holds. The proposition
is proved. $\Box$

##  3\. ($\sf{g},\sf{m}$)-bundles over Riemann surfaces

We will consider bundles whose fibers are smooth surfaces or Riemann surfaces
of type $(\sf{g},\sf{m})$.

######  Definition 1.

(Smooth oriented $(\textsf{g},\textsf{m})$ fiber bundles.) Let $X$ be a smooth
oriented manifold of dimension $k$, let ${\mathcal{X}}$ be a smooth (oriented)
manifold of dimension $k+2$ and ${\mathcal{P}}:{\mathcal{X}}\to X$ an
orientation preserving smooth proper submersion such that for each point $x\in
X$ the fiber ${\mathcal{P}}^{-1}(x)$ is a closed oriented surface of genus
$\sf{g}$. Let $\mathbold{E}$ be a smooth submanifold of $\mathcal{X}$ that
intersects each fiber $\mathcal{P}^{-1}(x)$ along a set $E_{x}$ of $\sf{m}$
distinguished points. Then the tuple
${\mathfrak{F}}_{\textsf{g},\textsf{m}}=({\mathcal{X}},{\mathcal{P}},\mathbold{E},X)$
is called a smooth (oriented) fiber bundle over $X$ with fiber a smooth closed
oriented surface of genus g with m distinguished points (for short, a smooth
oriented $(\textsf{g},\textsf{m})$-bundle).

If ${\sf m}=0$ the set $\mathbold E$ is the empty set and we will often denote
the bundle by $({\mathcal{X}},{\mathcal{P}},X)$. If ${\sf{m}}>0$ the mapping
$x\to E_{x}$ locally defines $\sf m$ smooth sections. $(\textsf{g},0)$-bundles
will also be called genus g fiber bundles. For $\textsf{g}=1$ and
$\textsf{m}=0$ the bundle is also called an elliptic fiber bundle.

In the case when the base manifold is a Riemann surface, a holomorphic
$(\textsf{g}$,$\textsf{m})$ fiber bundle over $X$ is defined as follows.

######  Definition 2.

Let $X$ be a Riemann surface, let ${\mathcal{X}}$ be a complex surface, and
${\mathcal{P}}$ a holomorphic proper submersion from ${\mathcal{X}}$ onto $X$,
such that each fiber $\mathcal{P}^{-1}(x)$ is a closed Riemann surface of
genus $\sf g$. Suppose $\mathbold{E}$ is a complex one-dimensional submanifold
of $\mathcal{X}$ that intersects each fiber $\mathcal{P}^{-1}(x)$ along a set
$E_{x}$ of $m$ distinguished points. Then the tuple
${\mathfrak{F}}_{\textsf{g},\textsf{m}}=({\mathcal{X}},{\mathcal{P}},\mathbold{E},X)$
is called a holomorphic $(\textsf{g}$,$\textsf{m})$ fiber bundle over $X$.

Notice that for ${\sf{m}}>0$ the mapping $x\to E_{x}$ locally defines $\sf{m}$
holomorphic sections.

Two smooth oriented (holomorphic, respectively) $(\textsf{g},\,\textsf{m})$
fiber bundles,
${\mathfrak{F}}^{0}=({\mathcal{X}}^{0},{\mathcal{P}}^{0},\mathbold{E}^{0},X)$
and
${\mathfrak{F}}^{1}=({\mathcal{X}}^{1},{\mathcal{P}}^{1},\mathbold{E}^{1},X)$,
are called smoothly isomorphic (holomorphically isomorphic, respectively,) if
there are smooth (holomorphic, respectively) homeomorphisms
$\Phi:\mathcal{X}^{0}\to\mathcal{X}^{1}$ and $\phi:X^{0}\to X^{1}$ such that
for each $x\in X^{0}$ the mapping $\Phi$ maps the fiber
$(\mathcal{P}^{0})^{-1}(x)$ onto the fiber $(\mathcal{P}^{1})^{-1}(\phi(x))$
and the set of distinguished points in $(\mathcal{P}^{0})^{-1}(x)$ to the set
of distinguished points in $(\mathcal{P}^{1})^{-1}(\phi(x))$. Holomorphically
isomorphic bundles will be considered the same holomorphic bundles.

Two smooth (oriented) $(\textsf{g},\textsf{m})$ fiber bundles over the same
oriented smooth base manifold $X$,
${\mathfrak{F}}^{0}=({\mathcal{X}}^{0},{\mathcal{P}}^{0},\mathbold{E}^{0},X)$,
and
${\mathfrak{F}}^{1}=({\mathcal{X}}^{1},{\mathcal{P}}^{1},\mathbold{E}^{1},X)$,
are called (free) isotopic if for an open interval $I$ containing $[0,1]$
there is a smooth $(\textsf{g},\textsf{m})$ fiber bundle
$({\mathcal{Y}},{\mathcal{P}},\mathbold{E},X\times I)$ over the base $X\times
I$ (called an isotopy) with the following property. For each $t\in[0,1]$ we
put ${\mathcal{Y}}^{t}={\mathcal{P}}^{-1}(X\times\\{t\\})$ and
$\mathbold{E}^{t}=\mathbold{E}\cap{\mathcal{P}}^{-1}(X\times\\{t\\})$. The