# Trace Systoles and Sink Constants

Frédéric Palesi  Aix Marseille Université, CNRS, Centrale Marseille, I2M, UMR
7373, 13453 Marseille, France<EMAIL_ADDRESS>www.i2m.univ-
amu.fr/perso/frederic.palesi/

###### Abstract.

Let $S$ be a surface with $\chi(S)<0$, and a representation
$\rho:\pi_{1}(S)\rightarrow\rm{SL}(2,\mathbb{C})$. We define the _trace
systole_ $\mathrm{tys}(\rho)$ as the infimum of
$|\operatorname{\mathrm{tr}}(\rho(\gamma))|$ for $\gamma$ a simple closed
curve. We know that this map is bounded on relative character varieties, and
in this article we compute explicitly the optimal bounds for the one-holed
torus, the four-holed sphere and the non-orientable surface of genus $3$,
using results on generalized Markoff maps. From this, we infer various
consequences on the optimal systolic inequalities of some hyperbolic manifolds
and also informations on non-Fuchsian representations for certain surface
groups.

##  1\. Introduction

Let $S$ be a surface of finite type with $\chi(S)<0$, and denote by
$\widehat{\Omega}$ the set of homotopy classes of unoriented essential simple
closed curves, which can be seen as a subset of $\pi_{1}(S)/(x\sim x^{-1})$.
We define the _trace systole_ $\rm{tys}(\rho)$ of a representation
$\rho\in\mathrm{Hom}(\pi_{1}(S),\mathrm{SL}(2,\mathbb{C}))$ as follows :

$\rm{tys}(\rho)=\inf\left\\{|\operatorname{\mathrm{tr}}(\rho(\gamma))|\ ,\
\gamma\in\widehat{\Omega}\right\\}.$

By invariance of the trace by conjugation, this descends to a map on the
character variety
$\mathcal{X}(S)=\mathrm{Hom}(\pi_{1}(S),\mathrm{SL}(2,\mathbb{C}))/\mathrm{SL}(2,\mathbb{C})$.

When $S$ is a closed surface, one can show that $\rm{tys}$ is bounded on the
character variety and attains its maximum, that we denote by $\rm{Tys}(S)$.
When $S$ has boundary components, the fundamental group $\pi_{1}(S)$ is a free
group, and $\rm{tys}$ is an unbounded function on the whole character variety.
However, one can consider its restriction on relative character varieties,
with prescribed traces on the boundary. More precisely, if $S$ has $p$
boundary components, with $c_{1},\dots,c_{p}\in\pi_{1}(S)$ representing
boundary loops, and we let $\mathfrak{B}=(b_{1},\dots,b_{p})\in\mathbb{C}^{p}$
be arbitrary boundary data, we can consider

$\mathcal{X}_{\mathfrak{B}}(S)=\left\\{[\rho]\in\mathcal{X}(S)\ ,\ \forall
i\in\\{1,\dots,p\\},\operatorname{\mathrm{tr}}(\rho(c_{i}))=b_{i}\right\\}$

In that case we define $\rm{Tys}_{\mathfrak{B}}(S)$ as the maximum of
$\rm{tys}$ on $\mathcal{X}_{\mathfrak{B}}(S)$. The purpose of this article is
the computation of $\rm{Tys}(S)$ and $\rm{Tys}_{\mathfrak{B}}(S)$ for simple
surfaces such as the one-holed torus, the four-holed sphere and the non-
orientable surface of genus $3$.

To study the trace systole in these cases, we use the fact that the curve
complex has a simple description. Indeed, for each surface studied here, the
curve complex will correspond to the Farey graph on
$\mathbb{Q}\cup\\{\infty\\}$, or is constructed naturally from it. One can
then consider the dual graph $\Sigma$ to this complex which is an infinite
trivalent tree embedded in the disk where the set $\Omega$ of complimentary
regions corresponds to the simple closed curves on $S$. Then, for any
representation $\rho\in\mathcal{X}_{\mathfrak{B}}(S)$, one can associate a so-
called Markoff map.

These are maps $\phi:\Omega\rightarrow\mathbb{C}$, satisfying compatibility
conditions around vertices and edges, depending on a parameter
$\boldsymbol{\mu}=(\lambda_{1},\lambda_{2},\lambda_{3},s)\in\mathbb{C}^{4}$.
They were introduced by Bowditch [1] and then later studied by Tan-Wong-Zhang
[13] and Maloni-Palesi-Tan [9]. For such a map $\phi$, one can define a
natural orientation of the edges of the tree $\Sigma$ from the values of
$\phi$. A _sink_ for a Markoff map is a vertex of $\Sigma$ such that all three
incident edges point towards that vertex. It was already known that the
minimal value of the Markoff map on the regions adjacent to a sink is bounded
above by a constant depending only on $\boldsymbol{\mu}$. However, the optimal
value of this constant, that we call the _sink constant_ and denote by
$M(\boldsymbol{\mu})$ was unknown. This question was already asked by Tan-
Wong-Zhang ([13]) but remained open since then, even in the simplest cases.
The first main result of this article is the answer to that question in the
case where $\lambda_{1}=\lambda_{2}=\lambda_{3}=0$.

######  Theorem A.

Let $\mu\in\mathbb{C}$ and $\boldsymbol{\mu}=(0,0,0,\mu)$. Then
$M(\boldsymbol{\mu})=|t_{\mu}|$ where $t_{\mu}$ is the dominant root of the
polynomial equation $X^{3}-3X^{2}+\mu=0$.

One can then directly rely the sink constant $M(\boldsymbol{\mu})$ with the
trace systole constant $\rm{Tys}_{\mathfrak{B}}(S)$ for some surface $S$ and
boundary data $\mathfrak{B}$. Identifying Markoff maps with representations of
certain surface groups, we obtain :

######  Theorem B.

  1. (1)

Let $\Sigma_{1,1}$ be a one-holed torus and $k\in\mathbb{C}\setminus\\{2\\}$.
We have

$\mathrm{Tys}_{k}(\Sigma_{1,1})=|t_{k+2}|$

  2. (2)

Let $N_{3}$ be the closed non-orientable surface of genus $3$.

$\mathrm{Tys}(N_{3})=\sqrt{3+\sqrt{17}}$

We also obtain a similar result when $\boldsymbol{\mu}$ belongs to a
particular subset of $\mathbb{R}^{4}$. For these parameters, the Markoff maps
correspond to representations of the four-holed sphere $\Sigma_{0,4}$ where
all four boundary traces are real and positive. Hence we obtain the following
result on the trace systole for $\Sigma_{0,4}$ :

######  Theorem C.

Let $\mathfrak{B}=(a,b,c,d)\in\mathbb{R}_{+}^{4}$. We denote by
$T_{\mathfrak{B}}$ the largest positive real root of

$X^{3}-3X^{2}-(ab+bc+cd+ac+bd+ad)X+(4-a^{2}-b^{2}-c^{2}-d^{2}-abcd)=0$

Then we have

$\rm{Tys}_{\mathfrak{B}}(\Sigma_{0,4})=T_{\mathfrak{B}}$

When $X$ is an hyperbolic surface which is homeomorphic to $S$ then it gives
rise to an holonomy representation
$\rho_{X}:\pi_{1}(S)\rightarrow\mathrm{PGL}(2,\mathbb{R})$ which is well-
defined up to conjugacy. The length $l_{X}(\gamma)$ of a closed geodesic
$\gamma$ on the hyperbolic surface $X$ is related to the trace of its image by
the holonomy representation using the formula

$l_{X}(\gamma)=\left\\{\begin{array}[]{ll}2\cosh^{-1}\left(\frac{\operatorname{\mathrm{tr}}(\rho_{X}(\gamma))}{2}\right)&\mbox{
if }\gamma\mbox{ is 2-sided}\\\
2\sinh^{-1}\left(\frac{\operatorname{\mathrm{tr}}(\rho_{X}(\gamma))}{2}\right)&\mbox{
if }\gamma\mbox{ is 1-sided}\end{array}\right.$

This gives a natural relation between the value of the trace systole of a
Fuchsian representation and the usual systole $\mathrm{sys}(X)$ of the
corresponding hyperbolic surface. This relation can be generalised for
hyperbolic $3$-manifolds coming from quasi-Fuchsian representation of these
surfaces, and also for incomplete hyperbolic structure on singular surfaces
with conical singularities. Using this, we obtain the following results :

######  Theorem D.

  1. (1)

If $X$ is an hyperbolic one-holed torus, with geodesic boundary of length $l$,
then

$\cosh\left(\frac{sys(X)}{2}\right)\leq\cosh\left(\frac{l}{6}\right)+\frac{1}{2}$

  2. (2)

If $X$ is a quasi-Fuchsian structure of a once-punctured torus (with a cusp),
then

$\cosh\left(\frac{sys(X)}{2}\right)\leq\frac{3}{2}$

  3. (3)

If $X$ is a singular hyperbolic structure on a torus with a conical
singularity of angle $\theta$, then

$\cosh\left(\frac{sys(X)}{2}\right)\leq\cos\left(\frac{\theta}{6}\right)+\frac{1}{2}$

  4. (4)

If $X$ is a quasi-Fuchsian hyperbolic structure of a four-punctured sphere,
then :

$\cosh\left(\frac{sys(X)}{2}\right)\leq\frac{7}{2}$

  5. (5)

If $X$ is a quasi-Fuchsian structure on $N_{3}$, then

$\cosh(sys(X))=\frac{5+\sqrt{17}}{2}$

The first two results were already known see [3, 12], but the others appear to
be new, at least in this form. In any case, the proofs that we give here uses
a completely different approach, as previous proofs of similar results usually
rely on hyperbolic geometry. One can hope to generalize these results to
compute optimal systolic inequalities for other hyperbolic surfaces, which is
still an open problem in most cases.

We can also use the methods developped here to obtain properties of non-
Fuchsian representations of surface groups. In particular, the notion of trace
systole is related to a question of Bowditch [1] : for a given type-preserving
representation of a surface that is not discrete, does there exists a simple
closed curve such that $\rho(\gamma)$ is not an hyperbolic element ? Note that
this would imply that the trace systole of such a representation is less than
$2$. The answer to this question is already known for $\Sigma_{2}$ the closed
surface of genus $2$ and was proven by Marche and Wolff [10], using results on
domination of non-Fuchsian representations by Fuchsian ones, and the explicit
value of the Bers constant.

We give here an independant and alternative proof using our results on trace
systoles of representations of the one-holed torus and the four-holed sphere,
for the case of representations with Euler class $\pm 1$. We hope that this
approach could be used for more general surfaces, where the answer to Bowditch
question is still unknown.

Plan of the paper.

We first recall the necessary background on Markoff maps in Section 2. The
Section 3 will be devoted to the definition of the the sink constant of
Markoff maps, and the proofs of Theorems A and C providing explicit values of
these constant in various cases. In Section 4 we will give the precise
relation between character varieties of surface groups and Markoff maps, so
that in Section 5 we can give the main results in terms of trace systoles with
Theorems B and D. Finally, in Section 6 we will use these results to give an
alternaive proof of Bowditch’s question for the surface of genus $2$.

##  2\. Generalized Markoff maps

In this section we recall the main definitions and properties of generalized
Markoff maps, and refer to previous works [1, 13, 9] for more details.

###  2.1. Farey triangulation and binary tree.

Let $\mathcal{F}$ be the Farey triangulation of the hyperbolic plane
${\mathbb{H}}^{2}$. Recall that the ideal vertices of $\mathcal{F}$ correspond
to $\mathbb{Q}\cup\\{\infty\\}\subset\partial{\mathbb{H}}^{2}$ and that two
vertices $\frac{p}{q}$, $\frac{r}{s}$ of $\mathcal{F}$ are joined by an edge
if $|pq-rs|=1$ (where we assume that $p,q,r,s\in\mathbb{Z}$ and $p\wedge
q=r\wedge s=1$). Let $\Sigma$ be the dual graph to $\mathcal{F}$, where
vertices correspond to the triangles of $\mathcal{F}$ and edges come from
adjacency of triangles; see Figure 2. We know that $\Sigma$ is a countably
infinite simplicial tree properly embedded in the plane all of whose vertices
have degree $3$. We note $V(\Sigma)$ and $E(\Sigma)$ the set of vertices and
edges of $\Sigma$ respectively.

A _complementary region_ of $\Sigma$ is the closure of a connected component
of the complement. We denote by $\Omega=\Omega(\Sigma)$ the set of
complementary regions of $\Sigma$. The regions are in correspondance with
vertices of $\mathcal{F}$ and hence they are indexed by elements of
$\mathbb{Q}\cup\\{\infty\\}$.

Figure 1. The Farey triangulation (in black) and its dual graph $\Sigma$ (in
blue).

Figure 2. A tricoloring of the regions and edge with edge
$e\leftrightarrow(X,Y;Z,W)$.

We will use capital letters $X,Y,Z,W,\ldots$ to denote elements of $\Omega$.
For $e\in E(\Sigma)$, we will note $e\leftrightarrow(X,Y;Z,W)$ to indicate
that $e=X\cap Y$ and $e\cap Z$ and $e\cap W$ are the endpoints of $e$; see
Figure 2.

We choose a tri-coloring of the regions and edges, namely a map
$\mathcal{C}\colon\thinspace\Omega(\Sigma)\cup
E(\Sigma)\longrightarrow\\{1,2,3\\}$ such that for any edge
$e\leftrightarrow(X,Y;Z,W)$ we have
$\mathcal{C}(e)=\mathcal{C}(Z)=\mathcal{C}(W)$ and such that $\mathcal{C}(e)$,
$\mathcal{C}(X)$ , $\mathcal{C}(Y)$ are all different. This implies that the
colors of three regions meeting at a vertex are all different, and the same
also holds for the three edges meeting at a vertex. In fact, the coloring is
completely determined by a coloring of the three regions (or three edges)
around any specific vertex, and hence is unique up to a permutation of the set
$\\{1,2,3\\}$. We denote by $\Omega_{i}=\Omega_{i}(\Sigma)$ the set of
complementary regions with color $i$, and by $E_{i}=E_{i}(\Sigma)$ the set of
edges with color $i$; see Figure 2.

As a notational convention in the following, when $X,Y,Z$ are complementary
regions around a vertex, we will consider that $X\in\Omega_{1}$,
$Y\in\Omega_{2}$ and $Z\in\Omega_{3}$, or in general that
$X_{i}\in\Omega_{i}(\Sigma)$ .

###  2.2. $\boldsymbol{\mu}$–Markoff triples

For a complex quadruple
$\boldsymbol{\mu}=(\lambda_{1},\lambda_{2},\lambda_{3},s)\in\mathbb{C}^{4}$, a
$\boldsymbol{\mu}$-Markoff triple is an ordered triple $(x_{1},x_{2},x_{3})$
of complex numbers satisfying the $\boldsymbol{\mu}$–Markoff equation, also
called the vertex equation:

$\displaystyle
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{1}x_{2}x_{3}+\lambda_{1}x_{1}+\lambda_{2}x+\lambda_{3}x_{3}=s$
(1)

######  Remark 2.1.

Note that we slightly changed the convention used in our previous paper [9]
for consistence with the notation of Tan-Wong-Zhang [13]. To pass from one
convention to the other, one simply has to replace $(x_{1},x_{2},x_{3})$ by
$(-x_{1},-x_{2},-x_{3})$.

Note that, with this convention, if $(x_{1},x_{2},x_{3})$ is a $\mu$–Markoff
triple in the sense of Tan-Wong-Zhang, with $\mu\in\mathbb{C}$, then
$(x_{1},x_{2},x_{3})$ is a $\boldsymbol{\mu}$–Markoff triple in our sense,
with $\boldsymbol{\mu}=(0,0,0,\mu)$.

It is easily verified that, if $(x_{1},x_{2},x_{3})$ is a
$\boldsymbol{\mu}$–Markoff triple, so are the triples

$(x_{1},x_{2},x_{1}x_{2}-x_{3}-\lambda_{3}),\hskip
8.5359pt(x_{1},x_{1}x_{3}-x_{2}-\lambda_{2},x_{3})\mbox{ and
}(x_{2}x_{3}-x_{1}-\lambda_{1},x_{2},x_{3}).$ (2)

It is important to note that in general, permutations triples are not
$\boldsymbol{\mu}$–Markoff triples, unlike the $\mu$–Markoff triples
considered by [13]. Namely, if $(x_{1},x_{2},x_{3})$ is a
$\boldsymbol{\mu}$-Markoff triple, then the triples
$(x_{2},x_{3},x_{1}),(x_{3},x_{1},x_{2}),\dots$ have no reason to be
$\boldsymbol{\mu}$–Markoff triples.

###  2.3. $\boldsymbol{\mu}$–Markoff map

######  Definition 2.2.

A $\boldsymbol{\mu}$-Markoff map is a function
$\phi\colon\thinspace\Omega\longrightarrow\mathbb{C}$ such that

  * (i)

for every vertex $v\in V(\Sigma)$, the triple
$(\phi(X_{1}),\phi(X_{2}),\phi(X_{3}))$ is a $\boldsymbol{\mu}$–Markoff
triple, where $X_{1},X_{2},X_{3}\in\Omega$ are the three regions meeting $v$
such that $X_{i}\in\Omega_{i}$;

  * (ii)

For any $i\in\\{1,2,3\\}$ and for every edge $e\in E_{i}(\Sigma)$ such that
$e\leftrightarrow(X_{j},X_{k};X_{i},X_{i}^{\prime})$ we have:

$\phi(X_{i})+\phi(X_{i}^{\prime})=\phi(X_{j})\phi(X_{k})-\lambda_{i}.$ (3)

We denote by ${\bf\Phi}_{\boldsymbol{\mu}}$ the set of all
$\boldsymbol{\mu}$–Markoff maps.

One may establish a bijective correspondence between
$\boldsymbol{\mu}$–Markoff maps and $\boldsymbol{\mu}$–Markoff triples, by
fixing three regions $X_{1},X_{2},X_{3}$ which meet at some vertex $v_{0}$,
and considering a map $\phi\mapsto(\phi(X_{1}),\phi(X_{2}),\phi(X_{3}))$.

This process may be inverted by constructing a tree of
$\boldsymbol{\mu}$–Markoff triples as Bowditch did in [1] for Markoff triples
and as Tan, Wong and Zhang did in [13] for the $\mu$–Markoff triples: given a
$\boldsymbol{\mu}$-Markoff triple $(x_{1},x_{2},x_{3})$, set
$\phi(X_{i})=x_{i}$, and extend over $\Omega$ as dictated by the edge
relations. That’s because if the edge relation (3) is satisfied along all
edges, then it suffices that the vertex relation (1) is satisfied at a single
vertex.

In this way one obtains an identification of ${\bf\Phi}_{\boldsymbol{\mu}}$
with the algebraic variety in $\mathbb{C}^{3}$ given by the
$\boldsymbol{\mu}$–Markoff equation. In particular,
${\bf\Phi}_{\boldsymbol{\mu}}$ gets an induced topology as a subset of
$\mathbb{C}^{3}$.

###  2.4. Arrows assigned by a $\boldsymbol{\mu}$–Markoff map

Let $\overrightarrow{E}(\Sigma)$ be the set of oriented edges. Let
$\phi\in{\bf\Phi}_{\boldsymbol{\mu}}$. We can assign to each undirected edge,
$e\in E(\Sigma)$, a particular directed edge,
$\overrightarrow{e_{\phi}}\in\overrightarrow{E}(\Sigma)$, with underlying edge
$e$, in the following way.

Suppose $e\leftrightarrow(X,Y;Z,W)$ and $\phi\in{\bf\Phi}_{\boldsymbol{\mu}}$.
If $|z|\geq|w|$, then we associate the element $\overrightarrow{e_{\phi}}$ in
$\overrightarrow{E}(\Sigma)$, such that the arrow on $e$ points towards $W$;
in other words, $\overrightarrow{e_{\phi}}=(X,Y;Z\rightarrow W)$.
Reciprocally, if $|z|\leq|w|$, we put an arrow on $e$ pointing towards $Z$,
that is, $\overrightarrow{e_{\phi}}=(X,Y;W\rightarrow Z)$. If it happens that
$|z|=|w|$ then we consider that there is an arrow in both directions, as this
will not affect the arguments in the latter part of this paper.

For a given map $\phi$, a vertex with all three arrows pointing towards it is
called a sink, one where two arrows point towards it and one away is called a
merge, and vertex with two (respectively three) arrows pointing away from it
is called a fork (respectively _source_).

##  3\. Sink Constant

The following result is from Maloni-Palesi-Tan ([9], Lemma 3.5):

######  Lemma 3.1.

There exists a constant $m(\boldsymbol{\mu})\in\mathbb{R}_{>0}$ such that for
all $\phi\in{\bf\Phi}_{\boldsymbol{\mu}}$, if three regions
$X_{1},X_{2},X_{3}$ meet at a sink, then

$\min_{i}\left\\{|\phi(X_{i})|\right\\}\leq m(\boldsymbol{\mu}).$

A natural question is to understand the lowest possible value of this constant
$m(\boldsymbol{\mu})$, which we call the _sink constant_. We denote this
optimal constant by $M(\boldsymbol{\mu})$. This question was first asked by
Tan-Wong-Zhang in [13], where the authors only study the $(0,0,0,\mu)$-case
and remark that finding the optimal value seems to be a difficult problem.
Previously, Bowditch [1] considered the simplest possible case
$\boldsymbol{\mu}=(0,0,0,0)$ and found the sink constant
$M(\boldsymbol{\mu})=3$.

In this section, we answer the question of Tan-Wong-Zhang in the $(0,0,0,\mu)$
case, and also compute the Sink constant in certain particular cases.

###  3.1. Case $\lambda_{1}=\lambda_{2}=\lambda_{3}=0$

####  3.1.1. Complex case

The exact value of the sink constant will be expressed using the following
function:

######  Definition 3.2.

Let $a\in\mathbb{C}$. We denote by $t_{a}\in\mathbb{C}$ the dominant root of
the polynomial equation $X^{3}-3X^{2}+a=0$.

We first make an observation on the value $t_{a}$.

######  Lemma 3.3.

For all $a\in\mathbb{C}$, we have $\Re(t_{a})\geq 2$ with equality if and only
if $a=4$.

###### Proof.

Let $x_{1}\leq x_{2}\leq x_{3}\in[-2,2]$ such that $x_{1}+x_{2}+x_{3}=3$. Then
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leq 9$ with equality when
$(x_{1},x_{2},x_{3})=(-1,2,2)$.

Indeed, we get $x_{1}\in[-1,1]$ directly from $x_{1}=3-x_{2}-x_{3}$. And
$x_{3}\geq\frac{3-x_{1}}{2}$. If we fix $x_{1}$, it is clear that the maximum
of the function $f(x)=x_{1}^{2}+(3-x_{1}-x)^{2}+x^{2}$ on the interval
$[\frac{3-x_{1}}{2},2]$ is attained when $x=2$ and is equal to
$2x_{1}^{2}-2x_{1}+5$. And the maximum of this function on $[-1,1]$ occurs
when $x_{1}=-1$.

Now, Let $t_{1},t_{2},t_{3}$ be the three roots with multiplicity of
$x^{3}-3x^{2}+\mu=0$. From Vieta’s formula, we have that $t_{1}+t_{2}+t_{3}=3$
and $t_{1}^{2}+t_{2}^{2}+t_{3}^{2}=9$. If we denote $x_{i}=\Re(t_{i})$, and
assume that $|x_{i}|<2$, then we can apply the previous result and get
$x_{1}^{2}+x_{2}^{2}+x_{3}^{2}<9$. But As $\Re(z^{2})\leq(\Re(z)))^{2}$, we
see that if $\Re(t_{j})<2$, then $\Re(t_{1}^{2}+t_{2}^{2}+t_{3}^{2})\leq
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}<9$, which is a contradiction. So at least one of
the $t_{j}$ is such that $\Re(t_{j})\geq 2$. ∎

The rest of this section will be devoted to the proof of the following
Theorem.

######  Theorem 3.4.

Let $\mu\in\mathbb{C}$ and $\boldsymbol{\mu}=(0,0,0,\mu)$, then

$M(\boldsymbol{\mu})=|t_{\mu}|.$

The Theorem can then be seen as a consequence of the following technical lemma
:

######  Lemma 3.5.

The minimum of the function $f(p,q,r)=|pq|$ on the domain :

$\mathcal{D}_{\mu}=\left\\{(p,q,r)\in\mathbb{C}^{3}\ ,\
|p|\geq|q|\geq|r|,\Re(p),\Re(q),\Re(r)\leq\frac{1}{2},p+q+r-1=\mu
pqr\right\\}$

occurs for $p=q=r$.

###### Proof of Theorem 3.4 using Lemma 3.5.

Let $\phi\in{\bf\Phi}_{\boldsymbol{\mu}}$, and assume that
$X_{1},X_{2},X_{3}\in\Omega$ are three regions meeting at a sink for $\phi$.
We denote $x_{i}=\phi(X_{i})$. Then, the directions of the three arrows give
the three inequalities, called the _sink conditions_ :

$|x_{1}|\leq|x_{2}x_{3}-x_{1}|,\quad|x_{2}|\leq|x_{1}x_{3}-x_{2}|,\quad|x_{3}|<|x_{1}x_{2}-x_{3}|$

Without loss of generality, we can assume that $x_{i}\neq 0$, for otherwise
$\min_{i}\\{|x_{i}|\\}=0$, and also that $|x_{1}|\geq|x_{2}|\geq|x_{3}|$. We
want to prove that $|x_{3}|<|t_{\mu}|$.

We set $p=\frac{x_{1}}{x_{2}x_{3}}$, $q=\frac{x_{2}}{x_{3}x_{1}}$ and
$r=\frac{x_{3}}{x_{1}x_{2}}$. So we have $|p|\geq|q|\geq|r|$ and
$|pq|=\frac{1}{|z|^{2}}$.

We have that $|x|\leq|yz-x|\Leftrightarrow\Re(p)\leq\frac{1}{2}$. So, the sink
conditions are equivalent to $\Re(p),\Re(q),\Re(r)\leq\frac{1}{2}$. And the
vertex equation becomes:

$p+q+r-1=\mu pqr$

The Lemma shows that the minimal value of $|pq|$ is obtained when $p=q=r$,
which means that this minimum is larger than the smallest root $\tau_{\mu}$ of
the polynomial equation $3X-1-\mu X^{3}=0$. It’s easy to see that
$\tau_{\mu}=\frac{1}{t_{\mu}}$.

So we have $\frac{1}{|x_{3}|^{2}}=|pq|\geq\frac{1}{|t_{\mu}|^{2}}$, and hence
$|x_{3}|\leq|t_{\mu}|$ as wanted.

Moreover, the triple $(t_{\mu},t_{\mu},t_{\mu})$ defines a
$\boldsymbol{\mu}$-Markoff map such that the three regions meet at a sink.
Indeed, as $|t_{\mu}|\geq 2$ from Lemma 3.3, we get that
$|t_{\mu}^{2}-t_{\mu}|\geq||t_{\mu}|^{2}-|t_{\mu}||\geq|t_{\mu}|$. So the
constant $|t_{\mu}|$ is optimal. ∎

###### Proof of Lemma 3.5.

Let $\mu\neq 4$. Recall that $\tau_{\mu}$ is the smallest root of the
polynomial equation $3X-1-\mu X^{3}=0$. As $\tau_{\mu}=\frac{1}{t_{\mu}}$ and
$\Re(t_{\mu})>2$, we have that $\tau_{\mu}$ is included in the disk of
diameter $[0,\frac{1}{2}]$. As the triple $(\tau_{\mu},\tau_{\mu},\tau_{\mu})$
satisfies the sink conditions, we know that the minimum of the function $f$ on
the domain $\mathcal{D}$ is less or equal to $|\tau_{\mu}|^{2}$.

Let $(p,q,r)\in\mathcal{D}$ realizing the minimum of the function $f$ on the
domain $\mathcal{D}$.

  1. (1)

As $|pq|<|\tau_{\mu}|^{2}$ and $|p|\geq|q|$, we have directly that
$|q|<|\tau_{\mu}|<\frac{1}{2}$. Note that for any $z\in\mathbb{C}$ such that
$|z|<\frac{1}{2}$, we have $\Re(z)<\frac{1}{2}$

Assume by contradiction that $|q|>|r|$. Then for $\epsilon>0$, we consider
$p^{\prime}=p^{\prime}(\varepsilon)=(1-\varepsilon)p$ and
$q^{\prime}=q^{\prime}(\varepsilon)=(1-\varepsilon)q$. We let
$r^{\prime}=r^{\prime}(\varepsilon)$ be the unique complex number such that
$p^{\prime}+q^{\prime}+r^{\prime}-1=\mu p^{\prime}q^{\prime}r^{\prime}$. By
continuity of $r^{\prime}$ with respect to $\varepsilon$, we get that for
$\varepsilon$ small enough, we have
$(p^{\prime},q^{\prime},r^{\prime})\in\mathcal{D}$ and
$|p^{\prime}q^{\prime}|<|pq|$, which contradicts the minimality.

So we have $|q|=|r|$.

  2. (2)

Let $R>0$ such that $|q|=|r|=R$.

Let $f$ be the function $f(x,y)=\dfrac{1-x-y}{1-\mu xy}$ defined on the set
$K=\\{(x,y)\in\mathbb{C},|x|=|y|=R\\}$. The image $f(K)$ is a compact set in
$\mathbb{C}$. As $|x|,|y|<\frac{1}{2}$ we see that this image avoids $0$,
hence the minimum of $|f|$ occurs on the boundary of its image. The partial
functions $f(x,\cdot)$ and $f(\cdot,y)$ are Möbius maps. This means that for
$x$ fixed, the image of $f(x,\cdot)$ of the circle of radius $R$, is also a
circle denoted $\mathcal{C}$. And a direct computation shows that
$\mathcal{C}$ is disjoint from the circle of radius $|\tau_{\mu}|$ centered at
$0$, so the condition $|f(x,y)|\geq|\tau_{\mu}|\geq R$ is always satisfied.

Finally, by symmetry of the two partial functions, the image of $f(\cdot,r)$
is also the same circle $\mathcal{C}$. This implies that the boundary of the
image of the function $f$ is exactly the set $\\{f(x,x),|x|=R\\}$. So, the
minimum of $|f(x,y)|$ on $K$ is attained when $x=y$. In other words, we have
$q=r$.

  3. (3)

Now consider the function $F(y)=y\dfrac{1-2y}{1-\mu y^{2}}=f(y,y)$ on the
domain

$K^{\prime}=\left\\{y\in\mathbb{C}\ ,\
|y|\leq|\tau_{\mu}|,\Re\left(\frac{1-2y}{1-\mu
y^{2}}\right)\leq\frac{1}{2}\right\\}$

Again, this function is bounded away from $0$, so by the maximum principle,
the minimum of $|F|$ is attained on one of the boundaries of the domain
$K^{\prime}$. From the previous discussion, the minimum of this function is
precisely the minimum of $\phi$ on $\mathcal{D}$.

On the boundary corresponding to $|y|=|\tau_{\mu}|$, we easily see that the
minimum of $|F|$ occurs precisely when $y=\tau_{\mu}$. Indeed, we already
proved that if $|y|=|\tau_{\mu}|$, then $|f(y,y)|\geq|\tau_{\mu}|$, and on the
other hand we have $F(\tau_{\mu})=\tau_{\mu}^{2}$

  4. (4)

All is left to prove is that the minimum on the boundary given by the equation
$\Re\left(\frac{1-2y}{1-\mu y^{2}}\right)\leq\frac{1}{2}$ is greater than
$|\tau_{\mu}|^{2}$. We see that the equation $\frac{1-2y}{1-\mu y^{2}}=\alpha$
has two solutions which are given by :

$y_{\pm}(\alpha)=\dfrac{1\pm\sqrt{1-\mu\alpha+\mu\alpha^{2}}}{\mu\alpha}$

If we denote by $\alpha=\frac{1}{2}+i\beta$ for $\beta\in\mathbb{R}$, we only
have to consider a function of a single real variable
$\psi(\beta)=\left|\frac{1}{2}+i\beta\right||q_{\pm}(\frac{1}{2}+i\beta)|$. A
direct computation proves that the minimum of this function occurs at $b=0$
and is equal to :

$\left|\dfrac{1\pm\sqrt{1-\frac{\mu}{4}}}{\mu}\right|<|\tau_{\mu}|^{2}$

Note that this inequality is an equality if and only if $\mu=4$.

So this means that the minimum of $|F|$ occurs when $q=r=\tau_{\mu}$, which
implies that $p=\tau_{\mu}$, so this conlcudes the proof of the Lemma.

∎

####  3.1.2. Upper bound for the minimum of a Markoff map

There are Markoff maps with no sinks. In that case, we can use the following
proposition ([13], Lemma 3.11),

######  Proposition 3.6.

Let $\phi$ be a $(0,0,0,\mu)$-Markoff map with $\mu\neq 4$, and assume that
$\beta$ is an infinite ray in $\Sigma$ consisting of a sequence
$(e_{n})_{n\in\mathbb{N}}$ of edges of $\Sigma$ such that the arrow on each
$e_{n}$ assigned by $\phi$ is directed towards $e_{n+1}$. Then there exists at
least one region $X\in\Omega$ with $|\phi(X)|<2$.

Combining this proposition with Theorem 3.4 we obtain :

######  Theorem 3.7.

Let $\mu\neq 4$ and $\phi$ be a $(0,0,0,\mu)$-Markoff map. Then there exists a
region $X$ such that $|\phi(X)|\leq|t_{\mu}|$.

###### Proof.

We consider a trace reduction algorithm. Starting from three regions $X,Y,Z$
meeting at a vertex $v_{0}$, we follow a flow defined by $\phi$, which means
that we consider a maximal sequence of vertices $(v_{n})$ such that $v_{n+1}$
is a vertex adjacent to $v_{n}$ and the arrow from $v_{n}$ to $v_{n+1}$ is
pointing towards the latter. There are only two behaviors that can occur :

  * •

The sequence is finite. In that case, the last vertex of the sequenc is a
sink. In that case, one of the regions around that sink such that
$|\phi(X)|\leq|t_{\mu}|$

  * •

The sequence is infinite, in which case we have an infinite descending ray,
and hence there exists a region such that $|\phi(X)|<2\leq|t_{\mu}|$.

∎

Note that when $\mu=4$, we have $t_{\mu}=2$, and yet there are Markoff maps
such that for all regions $X$ we have $|\phi(X)|>2$. However, in that case
$\inf\\{|\phi(X)|,X\in\Omega\\}=2$.

####  3.1.3. Real Case

When $\mu\in\mathbb{R}^{4}$, one can consider real Markoff maps
$\phi:\Omega\longrightarrow\mathbb{R}$, and we denote by
${\bf\Phi}_{\boldsymbol{\mu}}^{\mathbb{R}}$ the set of such maps. In that
case, one can refine the previous theorem, as only the real roots of the
polynomial $X^{3}-3X^{2}+\mu=0$ will play a role.

######  Theorem 3.8.

Let $\mu\in\mathbb{R}\setminus\\{4\\}$. Let $t^{\prime}_{\mu}$ be the largest
(in absolute value) real root of the polynomial $X^{3}-3X^{2}+\mu=0$. Then for
all $\phi\in{\bf\Phi}_{\boldsymbol{\mu}}^{\mathbb{R}}$, if three regions
$(X,Y,Z)$ meet at a sink, then :

$\min\\{|x|,|y|,|z|\\}\leq\max(|t^{\prime}_{\mu}|,2)$

Note that this theorem gives more information than the previous one in the
case of real Markoff maps. For example, consider the case $\mu=54$. Then the
only real root of the equation $X^{3}-3X^{2}+54=0$ is $t^{\prime}_{\mu}=-3$,
however, the dominant complex roots is $t_{\mu}=3+3i$, and hence
$|t^{\prime}_{\mu}|<|t_{\mu}|$.

In the real case we have precise information on the real roots of the
polynomial. The discriminant of such an equation is

$\Delta=108\mu-27\mu^{2}=27\mu(4-\mu)$

So we can distinguish 5 possible cases depending on the value of $\mu$.

######  Proposition 3.9.

The real roots of $X^{3}-3X^{2}+\mu=0$ satisfy the following properties :

  1. (1)

If $\mu<0$, then the equation has a unique real solution. This solution is
positive and greater than $3$.

  2. (2)

If $\mu=0$, then there are two real solutions : $0$ and $3$.

  3. (3)

If $\mu\in]0,4[$, then $(E_{\mu})$ has three real solutions. Exactly one of
them is in $]2,3[$ and the other two are in $]-2,2[$.

  4. (4)

If $\mu=4$, then there are two real solutions : $-1$ and $2$.

  5. (5)

If $\mu>4$, then this equation has a unique real solution, which is negative.
In that case $|t^{\prime}_{\mu}|<2$ if and only if $\mu<20$.

We can now prove the Theorem 3.8, using a different strategy.

###### Proof.

We consider the set (that we will call the sink domain)

$\mathscr{S}=\left\\{(x,y,z)\in(\mathbb{R}\setminus\\{0\\})^{3}\ |\
\frac{x}{yz}\leq\frac{1}{2},\frac{y}{xz}\leq\frac{1}{2},\frac{z}{xy}\leq\frac{1}{2}\right\\}$

First, we can see that for all $\mu\in\mathbb{R}$, the triple
$(t_{\mu},t_{\mu},t_{\mu})\in\mathscr{S}$. Indeed, if $\mu>4$, then
$t_{\mu}<0$ is negative and all the equations are trivially satisfied because
$\frac{1}{t_{\mu}}<0<\frac{1}{2}$. On the other hand, if $\mu\leq 4$, then
$t_{\mu}\geq 2$ and hence $\frac{1}{t_{\mu}}\leq\frac{1}{2}$. Note that
$(2,2,2)$ is also in $\mathscr{S}$.

Let $f(x,y,z)=x^{2}+y^{2}+z^{2}-xyz$. We compute the gradient of the function
$f$, so we get

$\nabla f=\begin{pmatrix}2x-yz\\\ 2y-xz\\\ 2z-xy\end{pmatrix}$

We can see easily that if $(x,y,z)\in\mathscr{S}$ and $x,y,z$ are al of the
same sign, then each coordinates of $\nabla f$ is negative at $(x,y,z)$. We
have $\nabla f=(0,0,0)$ if and only if $x=y=z=2$.

Now, let $(x,y,z)\in\mathbb{R}^{3}$ such that $f(x,y,z)=\mu$ and
$(x,y,z)\in\mathscr{S}$. Without loss of generality, as
$f(x,y,z)=f(-x,-y,z)=f(-x,y,-z)=f(x,-y,-z)$, we can assume that $x,y,z$ are
all of the same sign, and non-zero.

Assume by contradiction that $|x|,|y|,|z|>\max(|t_{\mu}|,2)$.

  * •

If $xyz<0$, then we have $\mu=f(x,y,z)=x^{2}+y^{2}+z^{2}-xyz\geq 20>4$. This
implies that $t_{\mu}<0$, and hence $(t_{\mu},t_{\mu},t_{\mu})$ and $(x,y,z)$
are both in $]-\infty,0[^{3}$. Consider the straight path
$\nu:[0,1]\rightarrow\mathbb{R}^{3}$ from such that
$\nu(0)=(t_{\mu},t_{\mu},t_{\mu})$ and $\nu(1)=(x,y,z)$. This path is entirely
contained in $\mathscr{S}$ and hence $f\circ\nu$ is a strictly increasing
function. This means that $f(x,y,z)>f(t_{\mu},t_{\mu},t_{\mu})=\mu$, which
gives a contradiction.

  * •

If $xyz>0$, assume without loss of generality that $2<x\leq y\leq z$ then
consider the jagged path $\nu:[0,1]\rightarrow\mathbb{R}^{3}$ defined by three
straight path :

$(2,2,2)\rightarrow(x,x,x)\rightarrow(x,y,y)\rightarrow(x,y,z)$

This path is strictly increasing in each variable and stays in $\mathscr{S}$.
Hence $f\circ\nu$ is a strictly decreasing function. So $f(x,y,z)<f(2,2,2)=4$,
which implies that $\mu<4$ and $t_{\mu}>2$.

Now, that same path $\nu$ also joins $(t_{\mu},t_{\mu},t_{\mu})$ to
$\rightarrow(x,y,z)$ . By the same argument, we have
$f(x,y,z)<f(t_{\mu},t_{\mu},t_{\mu})=\mu$, which gives a contradiction.

∎

###  3.2. General case

We now consider the general case of $\boldsymbol{\mu}\in\mathbb{C}^{4}$, and
start with the following remark

######  Remark 3.10.

Let $\boldsymbol{\mu}=(-50,30,50,0)$. In that case, the triple $(-10,-10,10)$
is a $\boldsymbol{\mu}$–Markoff triple, and it’s easy to check that this
triple corresponds to a sink. However the largest root of

$X^{3}-3X^{2}-(-50+30+50)X=0$

is $\frac{1}{2}(3+\sqrt{129})=7.17...$ which is smaller than $10$.

In fact, the triple $(t_{\mu},t_{\mu},t_{\mu})$ is not a sink, which indicates
that the previous arguments cannot work.

This suggests that the optimal constant in the general case is much more
difficult to compute, and the natural generalization of previous results is
false. But adapting the previous proof, we can see that there are certain
cases of geometrical significance that can be studied with similar methods. In
particular, we will consider the following set of parameters :

$U=[0,+\infty[^{3}\times]-\infty,4]\subset\mathbb{R}^{4}$

######  Definition 3.11.

Let $\boldsymbol{\mu}\in U$. A Markoff map
$\phi\in{\bf\Phi}_{\boldsymbol{\mu}}$ is said to be _positive_ if
$\rm{Im}(\phi)\subset\mathbb{R}_{+}$. We denote by
${\bf\Phi}_{\boldsymbol{\mu}}^{\geq 0}$ the set of positive Markoff maps.

######  Theorem 3.12.

Let $\boldsymbol{\mu}=(\lambda_{1},\lambda_{2},\lambda_{3},s)$ in $U$. For all
$\phi\in{\bf\Phi}_{\boldsymbol{\mu}}^{\geq 0}$, if three regions
$X_{1},X_{2},X_{3}$ meet at a sink, then

$\min\\{x_{1},x_{2},x_{3}\\}\leq T_{\boldsymbol{\mu}}$

where $T_{\boldsymbol{\mu}}$ is the largest positive real root of
$X^{3}-3X^{2}-(\lambda_{1}+\lambda_{2}+\lambda_{3})X+s=0$.

###### Proof.

Let $\phi\in{\bf\Phi}_{\boldsymbol{\mu}}^{\geq 0}$ and let $X_{1},X_{2},X_{3}$
be three regions meeting at a sink. The sink condition can be written :

$|x_{i}|\leq|x_{j}x_{k}-x_{i}-\lambda_{i}|\
,\mbox{with}\\{i,j,k\\}=\\{1,2,3\\}$

As the Markoff map is positive, we have that both $x_{i}$ and
$x_{j}x_{k}-x_{i}-\lambda_{i}$ are positive and hence the condition is
equivalent to $x_{j}x_{k}-2x_{i}-\lambda_{i}\geq 0$. This leads to the
following definition of the _Sink set_

$\mathscr{S}=\left\\{(x_{i},x_{j},x_{k})\in[2,+\infty[\,|\,x_{j}x_{k}-2x_{i}-\lambda_{i}\geq
0,\forall\\{i,j,k\\}=\\{1,2,3\\}\right\\}$

Let $t=T_{\mu}$ be the largest real root of
$P(X)=X^{3}-3X^{2}-(\lambda_{1}+\lambda_{2}+\lambda_{3})X+s$. As
$P(2)=s-4-2(\lambda_{1}+\lambda_{2}+\lambda_{3})<0$, we know that $t>2$.
Moreover, we see that $(t,t,t)\in\mathscr{S}$ as
$t(t^{2}-2t-p)=t^{2}+(q+r)t-s>0$.

Now, consider the function
$f(x,y,z)=x^{2}+y^{2}+z^{2}-xyz+\lambda_{1}x+\lambda_{2}y+\lambda_{3}z$. The
gradient of this map is

$\nabla f=\begin{pmatrix}2x-yz+\lambda_{1}\\\ 2y-xz+\lambda_{2}\\\
2z-xy+\lambda_{3}\end{pmatrix}$

So on $\mathscr{S}$, the coordinates of the gradient are negative. So if
$\min(x_{1},x_{2},x_{3})>t$, then as before we can consider a path $\nu$ from
$(t,t,t)$ to $(x_{1},x_{2},x_{3})$ that stays in $\mathscr{S}$ and is
increasing in each variable. This implies that $f\circ\nu$ is striclty
decreasing and hence $f(x_{1},x_{2},x_{3})<f(t,t,t)=s$ which gives a
contradiction. ∎

##  4\. Character Varieties of surface groups

In this section, we recall the precise relationship between Markoff maps and
representations of certain surface group. For more details, we refer to [13,
9].

###  4.1. One-holed Torus

Let $T$ be a topological one-holed torus, and $\Gamma$ be its fundamental
group. The group $\Gamma=\langle\alpha,\beta\rangle$ is the free group of rank
two, and $\alpha$ and $\beta$ correspond to simple closed curves with
geometric intersection number one.

The set $\widehat{\Omega}$ of free homotopy classes of essential unoriented
simple closed curves on $T$ can be naturally identified with
$\mathbb{Q}\cup\\{\infty\\}$ by considering the “slope” of the curve (see
[13]). It is a well-known fact that the curve complex of the one-holed torus
is isomorphic to the Farey triangulation $\mathcal{F}$ described before. This
means that we can naturally identify $\widehat{\Omega}$ with $\Omega$, so that
each region in $\Omega$ correspond to a simple closed curve. As a consequence,
two regions share an edge if and only if the corresponding curves intersect
exactly once on $T$. Similarly, if three regions meet at a vertex, then there
exists a generating set $(\alpha,\beta)$ of $\Gamma$ such that the
corresponding curves have representatves $\alpha,\beta,\alpha\beta\in\Gamma$.

For $k\in\mathbb{C}$, a representation
$\rho:\Gamma\rightarrow\mathrm{SL}(2,\mathbb{C})$ is said to be a
$k$-representation, if $\operatorname{\mathrm{tr}}([\alpha,\beta])=k$, where
$[\alpha,\beta]=\alpha\beta\alpha^{-1}\beta^{-1}$ is the element corresponding
to the boundary curve of $T$. Note that this element is independent of the
chosen basis for $\Gamma$. The space of equivalence classes of
$k$-representations is denoted $\mathcal{X}_{k}$ and is called the
$k$-relative character variety.

There is a natural one-to-one correspondance between $\mathcal{X}_{k}$ and
${\bf\Phi}_{\boldsymbol{\mu}}$ with $\boldsymbol{\mu}=(0,0,0,k+2)$ obtained by
fixing a generating set $\alpha,\beta$ for $\Gamma$. Indeed, from this
generating set, one can identify $\Omega$ and $\widehat{\Omega}$, and hence a
character $\rho:\Gamma\rightarrow\mathrm{SL}(2,\mathbb{C})$ gives rise to a
map $\phi:\Omega\rightarrow\mathbb{C}$ defined by
$\phi(X)=\operatorname{\mathrm{tr}}(\rho(g))$ where $g\in\Gamma$ is the
representative of the element in$\widehat{\Omega}$ corresponding to $X$.

The edge and vertex relations then follow from the classical trace identities
in $\mathrm{SL}(2,\mathbb{C})$ :

$\displaystyle\operatorname{\mathrm{tr}}A\operatorname{\mathrm{tr}}B=$
$\displaystyle\operatorname{\mathrm{tr}}AB+\operatorname{\mathrm{tr}}AB^{-1},$
(4)
$\displaystyle(\operatorname{\mathrm{tr}}A)^{2}+(\operatorname{\mathrm{tr}}B)^{2}+(\operatorname{\mathrm{tr}}C)^{2}-\operatorname{\mathrm{tr}}A\operatorname{\mathrm{tr}}B\operatorname{\mathrm{tr}}C=$
$\displaystyle\operatorname{\mathrm{tr}}[A,B]+2=k+2$ (5)

Conversely, any $\boldsymbol{\mu}$–Markoff map $\phi$ gives rise to an
equivalence class of representation in $\mathcal{X}_{\mu-2}$, once given a
choice of three adjacent regions $(X,Y,Z)$. Indeed, if we consider the $\mu$
Markoff triple $(x,y,z)=(\phi(X),\phi(Y),\phi(Z))$. We know that
$\mathcal{X}_{\mu-2}$ is identified with

$\\{(x,y,z)\in\mathbb{C}^{3}\ |\ x^{2}+y^{2}+z^{2}-xyz-2=\mu-2\\},$

and hence the triple $(x,y,z)$ defines a unique $k$-character in
$\mathcal{X}_{k}$.

###  4.2. Four-holed Sphere

Let $S$ be a topological four-holed sphere, and $\Gamma$ be its fundamental
group. The group $\Gamma$ admits the following standard presentation

$\Gamma=\langle\alpha,\beta,\gamma,\delta\mid\alpha\beta\gamma\delta\rangle$

where $\alpha,\beta,\gamma,\delta$ correspond to homotopy classes of the four
boundary components. Note that $\Gamma$ is isomorphic to the free group on
three generators $\langle\alpha,\beta,\gamma\rangle$.

As in the one-holed torus case, the set $\widehat{\Omega}$ of free homotopy
classes of essential unoriented simple closed curves on $T$ can be naturally
identified with $\mathbb{Q}\cup\\{\infty\\}$ by considering the “slope” of the
curve, see [9].

For $\boldsymbol{\tau}=(a,b,c,d)\in\mathbb{C}^{4}$, a representation
$\rho:\Gamma\rightarrow\mathrm{SL}(2,\mathbb{C})$ is said to be a
$\boldsymbol{\tau}$ representation, when
$\operatorname{\mathrm{tr}}(\rho(\alpha))=a$,
$\operatorname{\mathrm{tr}}(\rho(\beta))=b$,
$\operatorname{\mathrm{tr}}(\rho(\gamma))=c$ and
$\operatorname{\mathrm{tr}}(\rho(\delta))=d$.

The space of equivalence classes of $\boldsymbol{\tau}$-representation is
denoted $\mathfrak{X}_{\boldsymbol{\tau}}$ and is called the
$\boldsymbol{\tau}$-relative character variety. We can consider the map

$\displaystyle\chi:\mathfrak{X}_{\boldsymbol{\tau}}$
$\displaystyle\longrightarrow\mathbb{C}^{3}$ $\displaystyle[\rho]$
$\displaystyle\longmapsto\left(\operatorname{\mathrm{tr}}(\rho(\alpha\beta)),\operatorname{\mathrm{tr}}(\rho(\beta\gamma)),\operatorname{\mathrm{tr}}(\rho(\gamma\alpha))\right)$

A classical result (see for example [4]) on the character variety of the free
group in three generators states that this map is injective and its image is
given by :

$\mathfrak{X}_{\boldsymbol{\tau}}=\\{(x,y,z)\in\mathbb{C}^{3}\ |\
x^{2}+y^{2}+z^{2}+xyz=px+qy+rz+s\\}$

with $(p,q,r,s)=GT(a,b,c,d)$ where
$GT:\mathbb{C}^{4}\rightarrow\mathbb{C}^{4}$ is the map defined by :

$\begin{pmatrix}a\\\ b\\\ c\\\
d\end{pmatrix}\longmapsto\begin{pmatrix}ab+cd\\\ ad+bc\\\ ac+bd\\\
4-a^{2}-b^{2}-c^{2}-d^{2}-abcd\end{pmatrix}=\begin{pmatrix}p\\\ q\\\ r\\\
s\end{pmatrix}$

This map is onto and proper (see [5, 2]).

This allows us to identify the relative character variety with the space of
Markoff maps.

######  Proposition 4.1.

Let $\boldsymbol{\tau}\in\mathbb{C}^{4}$. A representation $\rho$ is in
$\mathfrak{X}_{\boldsymbol{\tau}}$ if and only if the triple $-\chi(\rho)$ is
a $\boldsymbol{\mu}$-Markoff triple with
$\boldsymbol{\mu}=GT(\boldsymbol{\tau})$.

###  4.3. Closed surface of characteristic $-1$

Let $N_{3}$ be a topological closed surface of characteristic $-1$. It is the
non-orientable surface of genus $3$, namely the connected sum of three
projective plane.

####  4.3.1. Curves on $N_{3}$

Recall that a closed curve on a non-orientable surface is said to be two-sided
if it admits a regular neighborhood which is orientable, else it is said to be
one-sided. We also say that a simple closed curve is orientable (resp. non-
orientable) if the surface cut along that curve is orientable (resp. non-
orientable). Note that on a non-orientable surface, a separating curve is
necessarily $2$-sided and non-orientable. And if the genus of the non-
orientable surface is odd, there are no orientable $2$-sided curves.

So, on the surface $N_{3}$, there are exactly four types of simple closed
curves :

  1. (1)

Separating curves. These are necessarily $2$-sided and non-essential, as they
bound a Mobius band.

  2. (2)

Non-separating $2$-sided curves. The surface obtained by cutting along such a
curve is a 2-holed projective plane.

  3. (3)

Orientable $1$-sided curve. There is a unique such curve

  4. (4)

Non-orientable $1$-sided curves. A curve such that $S\setminus\gamma$ is non-
orientable.

The curve complex of $N_{3}$ has the following structure :

  * •

The unique orientable 1-sided curve is disjoint from all non-separating
$2$-sided curves, but intersect all non-orientable 1-sided curve.

  * •

Two $2$-sided non-separating curves intersect at least once. And for each
$2$-sided non-separating curve, there is a unique non-orientable curve that is
disjoint from it.

  * •

Finally, the subcomplex formed by non-orientable $1$-sided curves is
equivalent to the curve complex of the one-holed torus.

####  4.3.2. Character variety of $N_{3}$

The fundamental group of $N_{3}$ is given by the following presentation :

$\Gamma_{3}=\pi_{1}(N_{3})=\langle\alpha,\beta,\gamma\mid\alpha^{2}\beta^{2}\gamma^{2}\rangle$

where $\alpha,\beta,\gamma$ are homotopy classes of disjoint simple non-
orientable $1$-sided curves. The unique orientable $1$-sided curve is given by
$\delta=\alpha\beta\gamma$.

The character variety of $\Gamma_{3}$ in $\mathrm{SL}(2,\mathbb{C})$ is given
by the following Theorem, which already appeared in the author’s thesis ([11])
:

######  Theorem 4.2.

The map

$\displaystyle\mathcal{X}(\Gamma_{3})$
$\displaystyle\longrightarrow\mathbb{C}^{4}$ $\displaystyle[\rho]$
$\displaystyle\longmapsto(\operatorname{\mathrm{tr}}(\rho(\alpha)),\operatorname{\mathrm{tr}}(\rho(\beta)),\operatorname{\mathrm{tr}}(\rho(\gamma)),\operatorname{\mathrm{tr}}(\rho(\delta)))$

is injective. Its image is the set

$\mathfrak{N}=\left\\{(a,b,c,d)\in\mathbb{C}^{4}\,\mid|\,a^{2}+b^{2}+c^{2}-abc\frac{d}{2}=4\right\\}$

###### Proof.

We know that $\mathcal{X}(\Gamma_{3})$ is an algebraic subset of the character
variety of the free group in three generators. Which means that we have a
first injection $\mathcal{X}(\Gamma_{3})\rightarrow\mathbb{C}^{7}$ where the
image is the set

$\left\\{(a,b,c,d,x,y,z)\in\mathbb{C}^{7},\left|\begin{array}[]{l}a^{2}+b^{2}+c^{2}+d^{2}+x^{2}+y^{2}+z^{2}-abcd-4\\\
-(ab+cd)x-(ad+bc)y-(ac+bd)z+xyz=0\end{array}\right.\right\\}$

The relation in $\Gamma_{3}$ implies that
$(\alpha\beta)^{-1}=\beta\gamma\gamma\alpha$ and hence any
$(a,b,c,d,x,y,z)\in\mathcal{X}(N_{3})$ satisfies $x=cd-x$. Similarly, we have
$y=ad-y$ and $z=bd-z$. Equivalently, we can write
$x=\frac{cd}{2},y=\frac{ad}{2},z=\frac{bd}{2}$. When we substitute the
expression of $x,y,z$ in terms of $a,b,c,d$ in the equation defining
$\mathcal{X}(F_{3})$ we get :

$\displaystyle
a^{2}+b^{2}+c^{2}+d^{2}+\left(\frac{cd}{2}\right)^{2}+\left(\frac{ad}{2}\right)^{2}+\left(\frac{bd}{2}\right)^{2}-abcd-4$
$\displaystyle-(ab+cd)\left(\frac{cd}{2}\right)-(ad+bc)\left(\frac{ad}{2}\right)-(ac+bd)\left(\frac{bd}{2}\right)+\left(\frac{cd}{2}\right)\left(\frac{ad}{2}\right)\left(\frac{bd}{2}\right)=0$

which is equivalent to $a^{2}+b^{2}+c^{2}-abc\frac{d}{2}=4$

This proves the injectivity of the map described in the theorem, and that its
image is included in $\mathfrak{N}$. To get surjectivity, we can use Theorem
3.2 in [6] which describe $X(G)$ as an explicit closed algebraic set (see the
author thesis for more details). ∎

This allows us to parametrize directly $\mathcal{X}(\Gamma_{3})$ as a set of
Markoff maps, except in the $d=0$ case.

######  Proposition 4.3.

An element $(a,b,c,d)\in\mathbb{C}^{4}$ with $d\neq 0$, is the character of a
representation $\rho$ in $\mathcal{X}(\Gamma_{3})$ if and only if
$\left(\frac{cd}{2},\frac{ad}{2},\frac{bd}{2}\right)$ is a
$(0,0,0,d^{2})$-Markoff map.

An element $(a,b,c,0)$ is the character of a representation $\rho$ in
$\mathcal{X}(\Gamma_{3})$ if and only if $a^{2}+b^{2}+c^{2}=4$.

###### Proof.

When $d\neq 0$, the equation defining $\mathfrak{N}$ is equivalent to :

$x^{2}+y^{2}+z^{2}-xyz=d^{2}$

with the change of variable $x=\frac{cd}{2},y=\frac{ad}{2},z=\frac{bd}{2}$. ∎

######  Remark 4.4.

A natural generalisation of Markoff maps can be related to representations of
a three-holed projective plane. The interested reader can refer to [8, 7],
where the theory of Markoff quads is developped.

##  5\. Trace Systoles

###  5.1. From hyperbolic surfaces to representations

Let $X$ be an orientable hyperbolic surface of finite type that can be closed
or with geodesic boundaries and cusps, and with conical singularities, so that
$\chi(X)<0$ (Recall that $\chi(X)=2-2g-b-s+\sum(\frac{\alpha_{i}}{2\pi}-1)$,
where $g$ is the genus, $b$ the number of geodesic boundaries, $s$ the number
of cusps and $\alpha_{i}$ the angles of the conical singularities). We denote
by $\mathfrak{B}$ the boundary data, namely the number of cusps, the lengths
of boundary components and the angles of conical singularities, if any.

The _systole_ of $X$ is the minimal length of an essential simple closed curve
on $X$ and is denoted $\mathrm{sys}(X)$. This defines a function
$\mathrm{sys}:\mathcal{T}(\Sigma,\mathfrak{B})\rightarrow\mathbb{R}_{+}$ where
$\mathcal{T}(\Sigma,\mathfrak{B})$ is the Teichmuller space of equivalence
classes of hyperbolic structures on the topological surface $\Sigma$ with
prescribed boundary data $\mathfrak{B}$.

The holonomy representation of such a structure gives rise to an homomorphism
from $\pi_{1}(\Sigma)$ into $\mathrm{PSL}(2,\mathbb{R})$, where $\Sigma$ is
the topological surface corresponding to $X$. For consistence, we consider
that conical singularities do not belong to $X$, so that a closed curve around
such a singularity is a non-trivial element of the fundamental group. There is
a relation between the length of a closed geodesic on $\Sigma$ and the trace
of its image by the holonomy representation given by

$l_{X}(\gamma)=2\mathrm{arccosh}\left(\frac{|\operatorname{\mathrm{tr}}(\rho_{X}(\gamma))|}{2}\right)$
(6)

From this relation, we see that we can generalize the notion of systole to the
entire space of representations
$\mathrm{Hom}(\pi_{1}(\Sigma),\rm{SL}(2,\mathbb{C}))$.

######  Definition 5.1.

Given a representation $\rho:\pi_{1}(\Sigma)\rightarrow\rm{SL}(2,\mathbb{C})$,
we define the trace systole of $\rho$ as :

$\rm{tys}(\rho)=\inf\left\\{|\operatorname{\mathrm{tr}}(\rho(\gamma))|\ ,\
\gamma\in\pi_{1}(\Sigma),\mbox{essential simple closed curve}\right\\}$

The definition of $\rm{tys}$ can be used for any representation into
$G=\rm{PSL}(2,\mathbb{R})$ or $\rm{PSL}(2,\mathbb{C})$, because the modulus
$|\operatorname{\mathrm{tr}}(\rho(\gamma))|$ is still well-defined in each of
these groups. So we will use the same notation for such representations. We
also note that as the trace is conjugation invariant, the function $\rm{tys}$
can be defined on the character variety
$\mathfrak{X}(\pi_{1}(\Sigma),G)=\rm{Hom}(\pi_{1}(\Sigma),G)/G$ which is the
space of orbit closures for the action of $G$ by conjugation on
representations.

It is important that we restrict ourselves to simple curves in the definition
of the trace systole to have an interesting function for representations that
are not discrete. Indeed, if
$\rho:\pi_{1}(\Sigma)\rightarrow\rm{PSL}(2,\mathbb{R})$ is a representation
that is not discrete, then its image is dense in $\rm{PSL}(2,\mathbb{R})$ and
hence
$\inf\left\\{|\operatorname{\mathrm{tr}}(\rho(\gamma))|,\gamma\in\pi_{1}(\Sigma)\right\\}=0$.

###  5.2. Maximum of the trace systole

When $S$ is a closed surface, one can show that the function $\rm{tys}$ is
bounded on the character variety and attains its maximum. We denote by
$\rm{Tys}(\Sigma)$ this maximum. When $\Sigma$ has boundary components, the
fundamental group $\pi_{1}(\Sigma)$ is a free group and the function
$\rm{tys}$ is unbounded on the whole character variety. However, one can
consider its restriction on relative character variety with prescribed traces
on the boundary.

If $\Sigma$ has $p$ boundary components, with
$c_{1},\dots,c_{p}\in\pi_{1}(\Sigma)$ representing loops around each boundary,
and we let $\mathfrak{B}=(b_{1},\dots,b_{p})\in\mathbb{C}^{b}$, we can
consider

$\mathfrak{X}_{\mathfrak{B}}(\Sigma)=\left\\{[\rho]\in\mathfrak{X}(\Sigma),\forall
i\in\\{1,\dots,p\\},\operatorname{\mathrm{tr}}(\rho(c_{i})=b_{i}\right\\}$

So we can define $\mathrm{Tys}_{\mathfrak{B}}(\Sigma)$ as the maximum of
$\rm{tys}$ on $\mathfrak{X}_{\mathfrak{B}}(\Sigma)$.

###  5.3. One holed torus

A reformulation of Theorem 3.7 in terms of trace systole gives the following

######  Theorem 5.2.

Let $T$ be a one-holed torus, and $k\in\mathbb{C}\setminus\\{2\\}$. We have

$\mathrm{Tys}_{k}(T)=|t_{k+2}|$

where $t_{a}$ is the dominant root of $X^{3}-3X^{2}+a=0$.

####  5.3.1. Hyperbolic one-holed torus

Ine the case of representations coming from holonomy representations of
hyperbolic structures, we can use the results on the trace systole to infer
optimal systolic inequalities.

######  Theorem 5.3.

Let $T$ be a surface with an hyperbolic metric and $\rm sys$ the length of its
systole.

  1. (1)

If $T$ is an hyperbolic one-holed torus, with geodesic boundary of lenth $l$,
then

$\cosh\left(\frac{{\rm
sys}}{2}\right)\leq\cosh\left(\frac{l}{6}\right)+\frac{1}{2}$

  2. (2)

If $T$ is a once-punctured torus, then

$\cosh\left(\frac{{\rm sys}}{2}\right)\leq\frac{3}{2}$

  3. (3)

If $T$ is a singular hyperbolic structure on the torus with a conical
singularity of angle $\theta$, then

$\cosh\left(\frac{{\rm
sys}}{2}\right)\leq\cos\left(\frac{\theta}{6}\right)+\frac{1}{2}$

The first two systolic inequalities were well-known before (see [12, 3]). The
last inequality seems to be new, at least in this form, but might have been
known to the specialists. In any case, the proof that we propose is quite
different than the one in the litterature.

###### Proof.

Let $\rho:\pi_{1}(T)\rightarrow\rm{PSL}(2,\mathbb{R})$ be the holonomy
representation of the hyperbolic structure on $T$. . Using Theorem 3.4 we have
that $\rm{tys}(\rho)\leq|t_{\mu}|$ with $t_{\mu}^{3}-3t_{\mu}^{2}=\mu$.

On the other hand, if we denote $\pi_{1}(T)=\langle A,B\rangle$, then the
element $[A,B]$ corresponds to the boundary component of $T$ (or the conical
singularity), and we have $\operatorname{\mathrm{tr}}([A,B])=\mu-2$. Depending
on the case we have

$\operatorname{\mathrm{tr}}([A,B])=\mu-2=\left\\{\begin{array}[]{ll}-2\cosh\left(\frac{l}{2}\right)&\mbox{
if }T\mbox{ is an hyperbolic one-holed torus}\\\ -2&\mbox{ if }T\mbox{ is once
punctured torus}\\\ -2\cos\left(\frac{\theta}{2}\right)&\mbox{ if }T\mbox{ is
a singular torus}\end{array}\right.$

In the first case, using the trigonometric identity
$\cosh(3x)=4\cosh^{3}(x)-3\cosh(x)$, and setting $x=\frac{l}{6}$ we see that
the equation can be rewritten

$\displaystyle t_{\mu}^{3}-3t_{\mu}^{2}$ $\displaystyle=2+2\cosh(3x)$
$\displaystyle\Leftrightarrow$ $\displaystyle
4\left(\frac{t_{\mu}}{2}-\frac{1}{2}\right)^{3}-3\left(t_{\mu}-\frac{1}{2}\right)$
$\displaystyle=\cosh(3x)$ $\displaystyle\Leftrightarrow$
$\displaystyle\frac{t_{\mu}}{2}-\frac{1}{2}$ $\displaystyle=\cosh(x)$

We know that $\rm{tys}(\rho)=2\cosh\left(\frac{\rm{sys}(T)}{2}\right)$. Hence
we have that
$\cosh\left(\frac{\rm{sys}(T)}{2}\right)\leq\cosh(x)+\frac{1}{2}$.

The exact same computations apply in the third case, replacing $\cosh(3x)$ by
$\cos(3x)$.

∎

####  5.3.2. Non-Fuchsian component for the one-holed torus

We also get original results for representations of the fundamental group of
the one-holed torus that do not correspond to hyperbolic structure (singular
or not) on a one-holed torus. These corresponds to representations with
relative Euler class $0$.

######  Theorem 5.4.

Let $\rho\in\mathcal{X}_{k}(T)$, with $k>2$.

  1. (1)

If $k\in]2,18[$, there exists a simple closed curve that is sent to an
elliptic element.

  2. (2)

If $k\geq 18$, there exists a simple closed curve $\gamma\in\pi_{1}(T)$ with
$|\operatorname{\mathrm{tr}}(\rho(\gamma)|\leq
2\cosh\left(\frac{l}{6}\right)-1$, with
$l=2\cosh^{-1}\left(\frac{k}{2}\right)$.

###### Proof.

A representation in $\mathcal{X}_{k}(T)$ with $k>2$, corresponds to a
$\mu$-Markoff map with $\mu>4$. Using the same reasoning as in Theorem 3.7, we
deduce that this Markoff map either has a sink or an infinite descending ray.
If there is an infinite descending ray, then there exists a curve $\gamma$
such that $|\operatorname{\mathrm{tr}}(\rho(\gamma))|<2$.

Otherwise, we use Theorem 3.8 and Proposition 3.9 to get the following :

  * •

If $k\in]2,18[$, then $\mu=k+2\in]4,20[$, and hence there exists a simple
closed curve $\gamma$ such that
$|\operatorname{\mathrm{tr}}(\rho(\gamma))|\leq|t_{\mu}|<2$. So this curve
$\gamma$ is sent to an elliptic element.

  * •

If $k\geq 18$, then there exists a curve $\gamma$ such that
$|\operatorname{\mathrm{tr}}(\rho(\gamma))|\leq|t_{\mu}|$. A computation
similar to the one in the proof of Theorem 5.3 gives the desired inequality.

∎

###  5.4. Four-holed sphere

####  5.4.1. Markoff maps coming from hyperbolic structures

Let $S$ be a four-holed sphere and let
$\boldsymbol{\tau}=(a_{1},a_{2},a_{3},a_{4})\in[0,+\infty[^{4}$.

When $a_{i}\geq 2$ for all $i\in\\{1,2,3,4\\}$, then we can endow $S$ with an
hyperbolic structure with geodesic boundaries or cusps such that the length of
the boundaries are given by $l_{i}=l(a_{i})$ for $i=1,2,3,4$ (if $l_{i}=0$,
then it’s a cusp). The equivalence class of the holonomy representation of
such a structure is an element of $\mathfrak{X}_{\boldsymbol{\tau}}$.

From the previous chapter, this representation corresponds to a
$\boldsymbol{\mu}$-Markoff map with $\boldsymbol{\mu}=GT(\boldsymbol{\tau})$.
As $a_{i}\geq 2$, we have naturally that $GT(a_{1},a_{2},a_{3},a_{4})\in U$.
Such a Markoff map takes values in $[2,+\infty[$, as all simple closed curves
are sent to hyperbolic elements. So, we have that any Markoff map constructed
from the holonomy representation of an hyperbolic structure is a positive
Markoff map

A similar reasoning can be made if we replace one or several boundaries of the
sphere by conical singularities of angle $\theta_{i}<\pi$. This corresponds to
the case where $a_{i}\in[0,2[$ and in that case we have
$\theta_{i}=2\cos^{-1}\left(\frac{a_{i}}{2}\right)$. One can still endow $S$
with an incomplete hyperbolic structure with geodesic boundaries, cusps and
conical singularities. The holonomy representation of such a structure is not
discrete and faithful, but all the simple closed curves are sent to hyperbolic
elements, and hence the corresponding Markoff map remains positive.

So a direct consequence of Theorem 3.12 is the following.

######  Theorem 5.5.

Let $S$ be an hyperbolic structure on sphere with $b$ geodesic boundaries, $c$
cusps and $d$ conical singularities of angles such that $b+c+d=4$, and let
$\gamma_{1},\dots,\gamma_{4}$ be the curves around them.

For $i\in\\{1,\dots,4\\}$ we set $a_{i}=2\cosh(\frac{l_{i}}{2})$ if
$\gamma_{i}$ corresponds to a geodesic boundary of length, $a_{i}=2$ if
$\gamma_{i}$ corresponds to a cusp, and $a_{i}=2\cos(\frac{\theta_{i}}{2})$ if
$\gamma_{i}$ is a conical singularity of angle $\theta_{i}$. Finally, let
$(\lambda_{1},\lambda_{2},\lambda_{3},s)=GT(_{1},a_{2},a_{3},a_{4})$. Then

$\mathrm{sys}(S)\leq 2\cosh^{-1}(t/2)$

where $t$ is the solution of
$t^{3}-3t^{2}-(\lambda_{1}+\lambda_{2}+\lambda_{3})t-s=0$.

####  5.4.2. Quasi-Fuchsian representations

We can consider Quasi-Fuchsian representations of a four-punctured sphere,
where each boundary is sent to a parabolic element. In that case we can assume
without loss of generalities that $a=b=c=d=2$. The Markoff equation becomes :

$x^{2}+y^{2}+z^{2}-xyz+8x+8y+8z=-28$

We define an auxillary map $\hat{\phi}:\Omega\rightarrow\mathbb{C}$ by
$\hat{\phi}(X)=\phi(X)+2$. This gives a map with properties that are slightly
different from the initial Markoff map. If we denote $\hat{x}=\hat{\phi}(X)$
and so on, the vertex relation and the edge relation become :

$(\hat{x}+\hat{y}+\hat{z})^{2}=\hat{x}\hat{y}\hat{z}$ (7)
$\dfrac{\hat{x}+\hat{y}+\hat{z}}{\hat{x}\hat{y}}+\dfrac{\hat{x}+\hat{y}+\hat{z}^{\prime}}{\hat{x}\hat{y}}=1$
(8)

This new map still gives rise to an orientation of the edges $E(\Sigma)$, and
hence we can still consider sinks in this context.

######  Lemma 5.6.

Let $X,Y,Z$ be three regions meeting at a vertex and assume that is is a sink
for the map $\hat{\phi}$. Then

$\min\\{|\hat{x}|,|\hat{y}|,|\hat{z}|\\}\leq 9$

###### Proof.

Let
$p=\Re\left(\dfrac{\hat{x}+\hat{y}+\hat{z}}{\hat{z}\hat{y}}\right),q=\Re\left(\dfrac{\hat{x}+\hat{y}+\hat{z}}{\hat{x}\hat{z}}\right),r=\Re\left(\dfrac{\hat{x}+\hat{y}+\hat{z}}{\hat{x}\hat{y}}\right)$,
and assume without loss of generalities that $p\geq q\geq r$. At a sink, we
have $p,q,r\leq\frac{1}{2}$. The vertex equation states that $p+q+r=1$ and
hence we know that $p\geq\frac{1}{3}$. Similarly, we get that
$q\geq\frac{1}{2}(1-p)$. So we have

$pq\geq\frac{1}{2}p(1-p)\geq\frac{1}{2}\frac{1}{3}\frac{2}{3}=\frac{1}{9}$

Now we have that

$\frac{1}{|\hat{z}|}=\left|\left(\dfrac{\hat{x}+\hat{y}+\hat{z}}{\hat{z}\hat{y}}\right)\left(\dfrac{\hat{x}+\hat{y}+\hat{z}}{\hat{x}\hat{z}}\right)\right|\geq
pq\geq\frac{1}{9}$

Which means that $|\hat{z}|\leq 9$. ∎

From this we can deduce the following systolic inequality

######  Proposition 5.7.

Let $\rho$ denote a Quasi-Fuchsian representation for a four-punctured sphere
and $X_{\rho}$ the corresponding hyperbolic $3$-manifold, then

$\rm{sys}(X_{\rho})\leq 2\rm{arccosh}\left(\frac{7}{2}\right)$ (9)

In particular the maximum of the systole function over the moduli space of all
hyperbolic four-punctured sphere is $2\rm{arccosh}\left(\frac{7}{2}\right)$.

###### Proof.

the previous Lemma implies that there exists a simple closed curve such that
$|2+2\cosh(\frac{L}{2})|<9$. As $x=2\cosh\left(\frac{L}{2}\right)$ with
$L=l+i\theta$ the complex translation length of $\rho(X)$, we can infer that

$\left|2+2\cosh\left(\frac{L}{2}\right)\right|=4\left|\cosh^{2}\left(\frac{L}{4}\right)\right|<4\left|\cosh^{2}\left(\frac{l}{4}\right)\right|.$

Finally, we obtain :

$|l|\leq
4\rm{arccosh}\left(\frac{3}{2}\right)=2\rm{arccosh}\left(\frac{7}{2}\right)$

To prove equality, we consider the Markoff triple $(7,7,7)$, which is a
$(8,8,8,-28)$-Markoff triple, which is a sink in the corresponding Markoff
map. ∎

###  5.5. Closed non-orientable surface of genus 3

We end this section with results concerning representations of
$\pi_{1}(N_{3})$.

####  5.5.1. Trace systole

######  Theorem 5.8.

We have :

$\mathrm{Tys}(N_{3})=\sqrt{3+\sqrt{17}}$

In addition, for any $\rho:\pi_{1}(N_{3})\rightarrow\rm{SL}(2,\mathbb{C})$,
there exists a $1$-sided simple closed curve such that
$|\operatorname{\mathrm{tr}}(\rho(\gamma))|\leq\mathrm{Tys}(N_{3})$.

###### Proof.

Let $\rho$ be such a representation. Recall that if $d\neq 0$, then
$(\frac{cd}{2},\frac{ad}{2},\frac{bd}{2})$ is a $(0,0,0,d^{2})$-Markoff triple
and hence corresponds to a $d^{2}$-Markoff map, denoted $\phi$.

Assume that $|d|>\sqrt{3+\sqrt{17}}$. From Theorem 3.4, there exists an
element $X\in\Omega$ such that $|\phi(X)|\leq|t_{d^{2}}|$.

The maximal value of $t_{\mu}$ for $\mu\in\mathbb{C}$ with $|\mu|=r$ occurs
for $\mu=-r$. This means that $|t_{d^{2}}|\leq|t_{-|d|^{2}}|$. Moreover, as
the function $\mu\rightarrow t_{\mu}$ is decreasing on $\mathbb{R}_{<0}$, we
get that $|t_{d^{2}}|<t_{-3-\sqrt{17}}$.

A simple computation gives that $t_{-3-\sqrt{17}}=\dfrac{3+\sqrt{17}}{2}$. And
hence we have

$|\phi(X)|<\dfrac{3+\sqrt{17}}{2}.$

So there exists a $1$-sided simple closed curve $\gamma$ such that
$\phi(X)=\operatorname{\mathrm{tr}}(\rho(\gamma))\frac{d}{2}$. Which means
that

$\operatorname{\mathrm{tr}}(\rho(\gamma))\leq\dfrac{3+\sqrt{17}}{2}\dfrac{1}{\sqrt{3+\sqrt{17}}}<\sqrt{3+\sqrt{17}}$

and this ends the proof of the inequality.

The representation given by the character $(it,it,it,it)\in\mathfrak{N}$ with
$t=\sqrt{3+\sqrt{17}}$ satisfies the equality $\rm{tys}(\rho)=t$ ∎

####  5.5.2. Twisted I-bundles

For non-orientable surface, we have to distinguish whether the curve is
$2$-sided or $1$-sided. As the holonomy representation of an hyperbolic
structure sends every $1$-sided curve to an orientation-reversing isometry, we
have :

$l_{X}(\gamma)=\left\\{\begin{array}[]{ll}2\mathrm{arccosh}\left(\frac{|\operatorname{\mathrm{tr}}(\rho_{X}(\gamma))|}{2}\right)&\mbox{
if }\gamma\mbox{ is 2-sided}\\\
2\mathrm{arcsinh}\left(\frac{|\operatorname{\mathrm{tr}}(\rho_{X}(\gamma))|}{2}\right)&\mbox{
if }\gamma\mbox{ is 1-sided}\end{array}\right.$

This allows us to get the following theorem when one restricts to quasi-
fuchsian representation of $N_{3}$.

######  Proposition 5.9.

Let $\rho$ be a Quasi-Fuchsian representation for the surface $N_{3}$, and
$X_{\rho}$ the corresponding hyperbolic $3$-manifold. Then

$\rm{sys}(X_{\rho})\leq\cosh^{-1}\left(\dfrac{5+\sqrt{17}}{2}\right)$

###### Proof.

From the previous theorem, there exists a $1$-sided simple closed cuve such
that $|\operatorname{\mathrm{tr}}(\rho(\gamma))|^{2}\leq 3+\sqrt{17}$. As we
have
$|\operatorname{\mathrm{tr}}(\rho(\gamma))|^{2}=\left|2\sinh\left(\frac{l_{\gamma}(X)}{2}\right)\right|^{2}=|2\cosh(l_{\gamma}(X))-2|$
we get that

$\cosh(l_{\gamma}(X))\leq\dfrac{5+\sqrt{17}}{2}$

∎

Note that this inequality was already determined by Gendulphe in the case of
hyperbolic surfaces [3]

##  6\. Trace systole in non-Fuchsian components

As a final section, we apply similar methods to show that a representation of
the genus 2 surface with euler class $\pm 1$ sends a simple closed curve on a
non-hyperbolic element. This result was already proven by Marche and Wolff,
but their proof is using hyperbolic geometry, fine estimations of the Bers
constant and domination of non-Fuchsian representation by Fuchsian ones. We
give a completely independent proof.

######  Theorem 6.1.

Let $\rho:\pi_{1}(\Sigma_{2})\rightarrow\mathrm{PSL}(2,\mathbb{R})$ be a
representation with Euler class $\pm 1$. Then there exists a simple closed
curve $\gamma\in\pi_{1}(\Sigma_{2})$ such that
$|\operatorname{\mathrm{tr}}(\rho(\gamma))|\leq 2$.

###### Proof.

Assume that there is a representation such that each simple closed curve is
sent to an hyperbolic element in $\mathrm{PSL}(2,\mathbb{R})$. We consider a
naive trace reduction algorithm.

Start from any pants decomposition $\mathcal{P}_{0}$ of the surface and choose
the largest curve. Consider the surface obtained by gluing the one or two pair
of pants that share this curve (which is a one-holed torus or a pair-of-
pants), and find the systole of that subsurface. If the chosen curve is a
systole, then choose the next curve in the pants decomposition, if not then
this gives a new pants decomposition, and one can continue the same process.
Each new pants decomposition arising in this way has one of its curve that is
strictly shorter than the previous one.

As all the simple closed curves are sent to hyperbolic elements, only two
situations can arise :

  1. (1)

After a finite number of steps, we arrive at a pants decomposition where every
curve is a systole of the subsurface (one holed torus or four-holed sphere)
defined by it.

  2. (2)

The process continues indefinitely.

Assume that we are in case $(1)$, and consider the final pants decomposition
$\mathcal{P}$. Let $\alpha$ and $\delta$ be the two curves of minimal length
of this pants decomposition. We first show that $\alpha$ and $\delta$ are both
non-separating. Indeed, if $\mathcal{P}$ contains a separating curve $\gamma$,
then this curves separates $\Sigma_{2}$ into two one-holed torus $T_{1}$ and
$T_{2}$. The restriction of the representation to each of these torus give two
representations $\rho_{1}$ and $\rho_{2}$. By additivity of the Euler class,
we have that one of these representations has Euler class $0$, for example
$\rho_{1}$. By hypothesis, $\rm{tys}(\rho_{1})\geq 2$, which means that
$\operatorname{\mathrm{tr}}(\rho(\gamma))\geq 18$ by Theorem 5.4. But for
$|\operatorname{\mathrm{tr}}(\rho(\gamma))|\geq 18$, we know for both
representations $\rho_{1}$ and $\rho_{2}$, there exists a curve whose trace is
smaller than $|\operatorname{\mathrm{tr}}(\rho(\gamma))|$ and hence the two
smallest curves of the pants decomposition are non-separating.

We consider the four-holed sphere, obtained by cutting $\Sigma_{2}$ along
$\alpha$ and $\delta$. As the Euler class of the representation
$\rho^{\prime}$ restricted to $S$ is $+1$, without loss of generality, we can
assume that the boundary traces of this representation are given by
$(a,a,d,-d)$. As the third curve of the pants decomposition is sent to an
hyperbolic element and is the systole of the representation $\rho^{\prime}$ we
can consider a triple of simple closed curve $(X,Y,Z)$ in $S$, such that the
corresponding vertex is a sink of the associated Markoff map, and where $X$ is
the separating curve. Without loss of generalities, up to changing signs of
the generators, we can assume that $2<a<d<z<|y|$.

The curve $X$ separates $\Sigma_{2}$ into two one-holed torus and the induced
representations have Euler class $-1$ and $0$. Which means that
$|x|>\max(a^{3}-3a^{2}+2,d^{3}+3d^{2}-2)=\lambda$. And $(x,y,z)$ is a sink so
that $|y|<|xz+y|$ and $|x|<|yz+x-(a^{2}-d^{2})|$.

Consider $z$ as a constant and consider the function :

$f(x,y)=x^{2}+y^{2}+z^{2}+xyz-(a^{2}-d^{2})x-(a^{2}-2)(d^{2}-2)$

We want to prove that in this situation $f(x,y)\neq 0$, which will give a
contradiction.

We first consider the case $xy>0$. In that case, $x>a^{2}-4$, and $d>2$, so we
have $x-(a^{2}-d^{2})>0$. And similarly $x>(d^{2}-2)$ and $yz>a^{2}>a^{2}-2$
so $xyz>(a^{2}-2)(d^{2}-2))$ . Combining these arguments we have

$f(x,y)=(x^{2}-(a^{2}-d^{2})x)+(xyz-(a^{2}-2)(d^{2}-2)))+y^{2}+z^{2}>8$

Now assume $xy<0$. In that case, the sink inequalities become $|2y|<|xz|$ and
$|2x|<|yz-(a^{2}-d^{2})|$. Without loss of generality, up to changing signs of
the generators, we restrict our study of the function $f(x,y)$ on the domain
defined by $y>0$, and the inequalities given by $2y<-xz$ and
$-2x<yz-(a^{2}-d^{2})$. This is a convex domain whose boundary is a union of
lines, and the partial derivatives of $f$ are given by :

$\frac{\partial f}{\partial x}=2x+yz-(a^{2}-d^{2})>0$ $\frac{\partial
f}{\partial y}=2y+xz<0$

So it suffices to prove that $f(x,y)<0$ on the lower right corner of the
domain.This corner has coordinates

$(x_{0},y_{0})=\left(\lambda,\max\left(z,\frac{2\lambda+(a^{2}-d^{2})}{z}\right)\right)$

Case 1 : If $z^{2}>2\lambda+(a^{2}-d^{2})$.

$\displaystyle f(\lambda,z)$ $\displaystyle=\lambda^{2}+2z^{2}-\lambda
z^{2}+(a^{2}-d^{2})\lambda-(a^{2}-2)(d^{2}-2)$
$\displaystyle=-(\lambda-2)(z^{2}-\lambda)+2\lambda+(a^{2}-d^{2})\lambda-(a^{2}-2)(d^{2}-2)$
$\displaystyle<-(\lambda-2)(\lambda+(a^{2}-d^{2}))+2\lambda+(a^{2}-d^{2})\lambda-(a^{2}-2)(d^{2}-2)$
$\displaystyle<-(\lambda-4)+2(a^{2}-d^{2})-(a^{2}-2)(d^{2}-2)<0$

Case 2 : If $z^{2}<2\lambda+(a^{2}-d^{2})$

$\displaystyle f\left(\lambda,\frac{2\lambda+(a^{2}-d^{2})}{z}\right)=$
$\displaystyle-\lambda^{2}+z^{2}+\left(\frac{2\lambda+(a^{2}-d^{2})}{z}\right)^{2}-(a^{2}-2)(d^{2}-2)$
$\displaystyle<$
$\displaystyle-\lambda^{2}+2\lambda+(a^{2}-d^{2})+(2\lambda+(a^{2}-d^{2}))\frac{2\lambda+(a^{2}-d^{2})}{z^{2}}-(a^{2}-2)(d^{2}-2)$
$\displaystyle<$
$\displaystyle-\lambda^{2}+2(2\lambda+(a^{2}-d^{2}))-(a^{2}-2)(d^{2}-2)$
$\displaystyle<$
$\displaystyle-\lambda^{2}+4\lambda+2(a^{2}-d^{2})-(a^{2}-2)(d^{2}-2)$
$\displaystyle<$ $\displaystyle\ 0$

Now we just need to treat the case $(2)$. In that case, the sequence of
absolute values of traces of the curves in the pants decomposition are
decreasing and bounded, so they converge. So we can consider that for any
$\varepsilon>0$ there is a step $n$ such that the pants decomposition
satisfies that each pants curve is within $\varepsilon$ of the trace systole
of the corresponding subsurface.

In that case one can infer that

  * •

The two shortest curves $\alpha$ and $\delta$ of that pants decomposition are
non-separating.

  * •

In the four-holed sphere obtained by cutting along $\alpha$ and $\delta$, we
have a sink $(X,Y,Z)$ such that $X$ corresponds to a separating curve, and
$2<a<d<z+\varepsilon<|y|+\varepsilon$.

So we can reproduce the same computations with the additional $\varepsilon$,
and we can see that when $\varepsilon$ is small enough, the final inequalities
still hold.

∎

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#  $\mathcal{Q}$-groups satisfy the equation $T_{G}(r,s)=0$

Francesco G. Russo  Department of Mathematics and Applied Mathematics
University of Cape Town Private Bag X1, Rondebosch 7701, Cape Town, South
Africa<EMAIL_ADDRESS>
###### Abstract.

The present note shows that $\mathcal{Q}$-groups in [H. Heineken and F.G.
Russo, Groups described by element numbers, Forum Math. 27 (2015), 1961–1977]
are solvable groups (not necessarily nilpotent) for which the equation
$T_{G}(r,s)=0$ is satisfied.

###### Key words and phrases:

Breadth ; $\mathcal{Q}$-groups ; Euler’s function; Inequalities ; Inverse
Problem to Frobenius’ Theorem

###### 2010 Mathematics Subject Classification:

20D10, 20D15, 20D60.

##  1\. Two different approaches of investigation for the local breadth

The present note deals with finite groups only. It is shown that the
$\mathcal{Q}$-groups in [7, Page 1962] and the groups of Meng and Shi in [1,
10, 11] satisfy the equation $T_{G}(r,s)=0$, mentioned in [5, Questions 1.4
and 1.5].

Chronologically [5] appears two years after [7], but even [2, 6, 12, 13] do
not mention the so called “Inverse Problem to Frobenius’ Theorem” in the study
of structural properties of groups by restrictions of numerical nature. A
brief historical remark is then appropriate. Frobenius [3, 4] and Yamaki [9,
14, 15, 16, 17] worked on the following problem long time ago:

######  Problem 1.1.

Given a group $G$ and an integer $m\geq 1$ dividing the exponent $\exp(G)$,
define

(1.1) $L_{m}(G)=\\{x\in G\ |\ x^{m}=1\\}$

and study relations between $m$, $|L_{m}(G)|$ and the structure of $G$.

Frobenius showed that a divisor $m$ of $|G|$ divides also $|L_{m}(G)|$, that
is, $|L_{m}(G)|=mk$ for some $k\geq 1$, on the other hand, he conjectured that
for $k=1$, then the $m$ elements of $L_{m}(G)$ form a characteristic subgroup
of $G$. This was indeed shown by Iiyori and Yamaki in their aforementioned
contributions with the help of the classification of simple groups. The
elegance of the proof of Frobenius’ Conjecture cannot be summarized here, but
its importance is clear since no restriction on $G$ is given (of any nature
!). A more recent problem, studied in [1, 7, 8, 10, 11], is to look at the
size of $k$, and look for structural conditions on $G$. More precisely:

######  Problem 1.2 (Inverse Problem to Frobenius’ Theorem).

Classify all groups $G$ such that $m^{-1}\ |L_{m}(G)|\leq\psi(m)$ for all $m$
dividing $\exp(G)$, where $\psi:m\in\mathbb{N}\mapsto\psi(m)\in\mathbb{N}$ is
a prescribed function depending only on $m$.

Now we change completely perspective and look numerically at the size of
$L_{m}(G)$. Consider the Euler function $\varphi(k)$ and define the number

$c_{k}=|\\{C\leq G\ |\ C\ \mbox{is a cyclic subgroup of order }\ k\\}|$

of cyclic subgroups of $G$ of order exactly $k$. One can see that

(1.2) $|L_{m}(G)|=\sum_{k\ |\ m}c_{k}\varphi(k)=m\cdot f(m),$

where $f(m)\geq 1$ is an integer depending only on $m$ and $G$.

Denoting by $r,s$ two real numbers and by $n=|G|$, Garonzi and Patassini [5,
Page 683, §2.4] worked on the function

(1.3) $T_{G}(r,s)=\sum_{k\ |\ n}g_{k,n/k}^{r,s}\ k\ (f(k)-1),$

where $g_{k,\ n/k}^{r,s}$ are the coefficient expressed by the expansion [5,
(1) of Lemma 7]. These coefficients turn out to be non-negative and the case
$g_{k,\ n/k}^{r,s}=0$ is completely described by [5, final part of (1) in
Lemma 7].

Restrictions on $T_{G}(r,s)$ allows us to detect nilpotency in $G$ by [5,
Theorem 5] and further properties were also noted by De Medts and Tǎrnǎuceanu
in [2, 12, 13]. In particular, [5, Questions 1.4] asks whether the equality
$T_{G}(r,s)=0$ detects solvability for $G$ for some $(r,s)$, while [5,
Questions 1.5] asks whether the equality $T_{G}(r,s)=0$ implies structural
properties for $G$ for some $(r,s)\in\mathbb{R}^{2}$.

##  2\. The main result and its proof

In order to attack Problem 1.2, the authors of [1, 7, 8, 10, 11] studied what
is called the local breadth, that is,

$\mathbf{b}_{m}(G)=|L_{m}(G)|\ \cdot\ m^{-1}$

of $G$ (under the general condition that $m$ is a divisor of the exponent of
$|G|$, so not necessarily of $|G|$). In particular, we get

$f(m)=\mathbf{b}_{m}(G)$

comparing (1.1) with (1.2).

A first body of results is the following, where only nilpotent groups appear.

######  Theorem 2.1 (See [10], Main Theorem and [8], list in §4).

Let $G$ be a group and $m\geq 1$ a divisor of $\exp(G)$. Then $f(m)\leq 2$ if
and only if

  * (i)

$G$ is cyclic;

  * (ii)

$G\simeq C_{h}\times C_{2^{k-1}}\times C_{2}$ with $h$ odd and $k\geq 2$;

  * (iii)

$G\simeq C_{h}\times Q_{8}=C_{h}\times\langle a,b\ |\
a^{4}=1,a^{2}=b^{2},b^{-1}ab=a^{-1}\rangle$ with $h$ odd;

  * (iv)

$G\simeq C_{h}\times\langle a,b\ |\
a^{2^{t-1}}=b^{2}=1,b^{-1}ab=a^{1+2^{t-2}}\rangle$ with $t\geq 4$ and $h$ odd;

  * (v)

$G\simeq C_{h}\times\langle a,b\ |\ a^{3}=b^{2^{s}}=1,b^{-1}ab=a^{-1}\rangle$
with $s\geq 1$ and $\gcd(h,6)=1$.

One can see without difficulties that the previous result remains true when
$m$ is a divisor of $|G|$. After Theorem 2.1, Chen, Meng and Shi [1, Theorems
1.1, 1.2] improved the classification based on the bound $f(m)\leq 2$ with
another one, based on the bound $f(m)\leq 3$. Successively it was introduced
the more comprehensive notion of $\mathcal{Q}$-$group$ in [7] (i.e.: a group
$G$ such that $f(m)\leq m$ with $m$ divisor of $\exp(G)$) and corresponding
classifications are presented in [7, Theorems 3.2, 3.5, 3.6, 3.8, 3.12, 3.14],
which we are going to summarize below.

######  Theorem 2.2 (Structure of $\mathcal{Q}$-groups, see [7]).

Let $G$ be a group and $m\geq 1$ a divisor of $\exp(G)$ such that $f(m)\leq
m$.

  * (i)

$G$ is solvable;

  * (ii)

$G$ has a $2$-nilpotent normal subgroup $M$ of index $|G:M|\in\\{1,3\\}$;

  * (iii)

if $p>3$ is a prime and $G$ a $p$-group, then $G$ is the product of two cyclic
groups with trivial intersection;

  * (iv)

if $\gcd(|G|,6)=1$, then $G$ is metabelian and the quotient $G/F(G)$ through
the Fitting subgroup $F(G)$ is cyclic;

  * (v)

$G=\mathrm{SL}(2,3)$ is a solvable non-metabelian $\mathcal{Q}$-group with
$\gcd(|G|,6)\neq 1$.

The notation for the Fitting subgroup, the notion of $p$-nilpotent group and
the notation for the special linear group of dimension $2$ over the field with
$3$ elements are very common and recalled in most of the references in the
bibliography. Thanks to Theorem 2.2, we are in the position to show that a
large class of solvable groups, not necessarily nilpotent, satisfies the
equation $T_{G}(r,s)=0$.

######  Theorem 2.3.

If $(r,s)\in\mathbb{R}^{2}$ satisfy $1\leq g_{k,\ n/k}^{r,s}$ and
$f(k)\leq{\big{(}g_{k,\ n/k}^{r,s}\big{)}}^{-\frac{1}{2}}\leq k$ for all $k$
divisors of $n$, then $T_{G}(r,s)=0$ and $G$ is solvable.

###### Proof of Theorem 2.3.

It is useful to note that (1.3) becomes equal to zero when $f(k)=1$ and
$f(k)=1$ happens if and only if $G$ satisfies (i) of Theorem 2.1, that is, $G$
is cyclic. Therefore the real issue of studying $T_{G}(r,s)=0$ is when
$f(k)\geq 2$, so there is no loss of generality in assuming $f(k)\geq 2$.
Since all $g_{k,\ n/k}^{r,s}\geq 1$, we have $T_{G}(r,s)\geq 0$ and it is
enough to show $T_{G}(r,s)\leq 0$ in order to conclude that $T_{G}(r,s)=0$.
Now

$f(k)\leq{\big{(}g_{k,\ n/k}^{r,s}\big{)}}^{-\frac{1}{2}}\ \ \Rightarrow\ \
g_{k,n/k}^{r,s}\ f(k)\leq\sqrt{g_{k,n/k}^{r,s}}$

implies

$\sum_{k\ |\ n}g_{k,n/k}^{r,s}\ f(k)\ k\leq\sum_{k\ |\
n}\sqrt{g_{k,n/k}^{r,s}}\ k\leq\sum_{k\ |\ n}g_{k,\ n/k}^{r,s}\ k$

and so

$\sum_{k\ |\ n}g_{k,n/k}^{r,s}\ k\ (f(k)-1)\leq 0$

which gives what we claimed $T_{G}(r,s)=0$. On the othe hand, $G$ is a
$\mathcal{Q}$-group because $f(k)\leq k$ and so it is solvable by (i) of
Theorem 2.2. It follows that there exist $(r,s)\in\mathbb{R}^{2}$ such that
$T_{G}(r,s)=0$ and $G$ is solvable. ∎

While the existence of $(r,s)$ is shown under the conditions of Theorem 2.3,
the structural conditions of $G$ are elucidated in Theorem 2.2.

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