# Tits Alternative for the $3$-dimensional tame automorphism group

Stéphane Lamy∗ Institut de Mathématiques de Toulouse UMR 5219, Université de
Toulouse, UPS F-31062 Toulouse Cedex 9, France<EMAIL_ADDRESS>and  Piotr Przytycki†∗ Department of Mathematics and Statistics, McGill
University, Burnside Hall, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9,
Canada<EMAIL_ADDRESS>
###### Abstract.

We prove the strong Tits alternative for the tame automorphism group of
$\mathbbm{k}^{3}$, for $\mathbbm{k}$ of characteristic zero.

${\dagger}$ Partially supported by NSERC, AMS, CNRS and National Science
Centre, Poland UMO-2018/30/M/ST1/00668.

$*$ This work was partially supported by the grant 346300 for IMPAN from the
Simons Foundation and the matching 2015–2019 Polish MNiSW fund

##  1\. Introduction

A group $G$ satisfies the _strong Tits alternative_ (resp.  _weak Tits
alternative_) if each subgroup of $G$ (resp. each finitely generated subgroup
of $G$) is virtually solvable or contains a non-abelian free group. For each
$n\geq 1$ and each field $\mathbbm{k}$ of characteristic zero (resp. of
positive characteristic), Tits proved that the linear group
$\textbf{GL}_{n}(\mathbbm{k})$ satisfies the strong (resp. weak) Tits
alternative [T]. This result was then extended to many classes of groups. We
mention first some groups of polynomial or rational maps, also known as
Cremona transformations. The group $\mathrm{Aut}(\mathbb{C}^{2})$ of
polynomial automorphisms of the complex plane satisfies the strong Tits
alternative by [L]. For an arbitrary field $\mathbbm{k}$, one can adapt the
argument and obtain the (strong, or weak, according to the characteristic of
$\mathbbm{k}$) Tits alternative for $\mathrm{Aut}(\mathbbm{k}^{2})$
[LamyBook]. For birational maps, the weak and then strong Tits alternative for
the 2-dimensional Cremona group $\mathrm{Bir}(\mathbb{C}^{2})$ was established
in [Cantat] and [Urech]. Again, the argument can be extended to any field of
characteristic zero [LamyBook], while the case of positive characteristic
remains unclear. In higher dimension, very little is known. However, the
strong Tits alternative for $\mathrm{Tame}(\textbf{SL}_{2}(\mathbb{C}))$, a
particular subgroup of the Cremona group $\mathrm{Bir}(\mathbb{C}^{3})$, was
established in [BFL].

The aim of this article is to establish the Tits alternative for the following
subgroup of the 3-dimensional Cremona group. The _tame automorphism group_
$\mathrm{Tame}(\mathbbm{k}^{3})$ of the affine space $\mathbbm{k}^{3}$ is the
subgroup of the polynomial automorphism group generated by the affine
automorphisms and by the automorphisms of the form

$(x_{1},x_{2},x_{3})\mapsto(x_{1}+P(x_{2},x_{3}),x_{2},x_{3}),\
P\in\mathbbm{k}[x_{2},x_{3}].$

######  Main Theorem.

Let $\mathbbm{k}$ be a field of characteristic zero. Then
$\mathrm{Tame}(\mathbbm{k}^{3})$ satisfies the strong Tits alternative.

A common feature of the proof of the Tits alternative for
$\mathrm{Aut}(\mathbb{C}^{2})$, $\mathrm{Bir}(\mathbb{C}^{2})$ and
$\mathrm{Tame}(\textbf{SL}_{2}(\mathbb{C}))$ is the use of an action on a
CAT(0) metric space. Namely, $\mathrm{Aut}(\mathbb{C}^{2})$ acts on the
Bass–Serre tree associated to its structure of amalgamated product,
$\mathrm{Bir}(\mathbb{C}^{2})$ acts on an infinite dimensional hyperbolic
space $\mathbb{H}^{\infty}$ and $\mathrm{Tame}(\textbf{SL}_{2}(\mathbb{C}))$
acts on a CAT(0) square complex. None of these actions is proper.

In general, suppose that a group $G$ acts by isometries on a Gromov-hyperbolic
geodesic metric space $X$. By [G]*§3.1, the group $G$ (or its index $2$
subgroup) fixes a point of $X$ or the Gromov boundary of $X$, or contains a
non-abelian free group. In particular, if $G$ is word-hyperbolic, if we take
$X$ to be the Cayley graph of $G$, the stabilisers of points in the Gromov
boundary of $X$ are virtually cyclic and so $G$ satisfies the strong Tits
alternative. More generally, if $G$ acts acylindrically on a Gromov-hyperbolic
geodesic metric space $X$, and the subgroups of $G$ with bounded orbits in $X$
satisfy the weak (resp. strong) Tits alternative, then $G$ satisfies the weak
(resp. strong) Tits alternative [DGO]. Taking for $X$ the coned-off Deligne
complex, this gives the strong Tits alternative for $2$-dimensional Artin
groups of hyperbolic type [MP2].

For groups $G$ that are not word-hyperbolic, but possess some hyperbolic-like
features, various strategies were employed. Strong Tits alternative was proved
for mapping class groups of hyperbolic surfaces [Iv, McC] and outer
automorphism groups of free groups [BFH, BFH2]. This was generalised to outer
automorphism groups of right-angled Artin groups [CV, Ho].

It is conjectured that if $X$ is a CAT(0) metric space, and $G$ acts on $X$
properly and cocompactly, then it satisfies the strong Tits alternative. This
conjecture was verified for $2$-dimensional complexes $X$ in [OP]. The methods
can be extended to ‘recurrent’ spaces [OP0], implying that the Artin groups of
large type satisfy the strong Tits alternative. For CAT(0) cube complexes $X$
the conjecture was established in [SW]. More generally, if $G$ acts on a
finite-dimensional CAT(0) cube complex $X$ with the stabilisers of points in
$X\cup\partial_{\infty}X$ satisfying the weak (resp. strong) Tits alternative,
then $G$ also satisfies the weak (resp. strong) Tits alternative [CS].
Applying this, one verifies the strong Tits alternative for Artin groups $G$
of type FC [MP1], by computing the stabilisers of points in
$\partial_{\infty}X$, where $X$ is the (cubical) Deligne complex.

The group $\mathrm{Tame}(\mathbbm{k}^{3})$ is the amalgamated product of three
subgroups along their pairwise intersection [Wright, LamyToulouse]. This gives
a natural action of $\mathrm{Tame}(\mathbbm{k}^{3})$ on a $2$-dimensional
simplicial complex, which was proved in [LP1] to be contractible and Gromov-
hyperbolic in order to find normal subgroups in
$\mathrm{Tame}(\mathbbm{k}^{3})$. However, this complex cannot be endowed with
an equivariant CAT(0) metric, and so we do not know how to classify the
subgroups of $\mathrm{Tame}(\mathbbm{k}^{3})$ with bounded orbits in that
complex.

In [LP2], a new $2$-dimensional CAT(0) complex $\mathbf{X}$ with an action of
$\mathrm{Tame}(\mathbbm{k}^{3})$ was constructed, leading to the description
of the finite subgroups of $\mathrm{Tame}(\mathbbm{k}^{3})$. However, this
action is not proper and $\mathbf{X}$ is not Gromov-hyperbolic. Therefore, in
order to prove the Main Theorem, we need to classify and analyse all elements
of $\mathrm{Tame}(\mathbbm{k}^{3})$ that do not fix a point of $\mathbf{X}$
and are not loxodromic of ‘rank 1’. In order to do this, we classify locally
and globally nearly flat subcomplexes of $\mathbf{X}$.

We apply an original mixture of metric and combinatorial techniques, where we
introduce ‘relative disc diagrams’, which are, roughly speaking, disc diagrams
that are required to be combinatorial only away from the boundary. We often
study the limit of relative disc diagrams $\phi_{n}\colon D\to\mathbf{X}$,
whose domain $D$ might degenerate in the limit into a ‘frilling’ $D^{\prime}$.
We believe that these concepts will be useful for studying the actions of
other groups on non-proper CAT(0) complexes.

Organisation. We present now a simplified outline of the proof of the Main
Theorem; see Section 12 for the full proof.

The group $\mathrm{Tame}(\mathbbm{k}^{3})$ acts on the non-proper
2-dimensional $\mathrm{CAT}(0)$ complex $\mathbf{X}$, which we discuss in
Section 3. An important property of $\mathbf{X}$ is the existence of a proper
quotient $\rho_{+}\colon\mathbf{X}\to\nabla^{+}$, with $\nabla^{+}$ isometric
to a Weyl chamber in $\mathbb{R}^{2}$.

Let $G<\mathrm{Tame}(\mathbbm{k}^{3})$ and suppose that $G$ does not stabilise
a point in $\mathbf{X}$, or a ‘principal’ or an ‘antiprincipal’ point $\zeta$
in $\partial_{\infty}\mathbf{X}$. (Stabilisers of such points $\zeta$ are
conjugate to subgroups denoted by
$C,B^{\prime}<\mathrm{Tame}(\mathbbm{k}^{3})$, which satisfy the strong Tits
alternative.) In particular, by [NOP]*Cor 2.5, there is an element of $G$ that
is not elliptic, i.e. has no fixed point in $\mathbf{X}$. If all the elements
of $G$ are elliptic or loxodromic of ‘rank 1’, one can find a non-abelian free
group in $G$ appealing to the work of Bestvina and Fujiwara. However, one
needs to carry over the classical results of Ballmann and Ruane to the non-
proper setup, which we do in Section 2.

It remains to classify the elements of $\mathrm{Tame}(\mathbbm{k}^{3})$ that
are parabolic or loxodromic not of rank 1. We do so eventually in Sections 8
and 9, where we show that all such elements are conjugate into $C$ or
$B^{\prime}$. To do that, we first classify all cycles in the vertex links of
$\mathbf{X}$ of length $<2\pi+\varepsilon_{0}$, for a uniform
$\varepsilon_{0}$ (Sections 4 and 5). We then study nearly flat diagrams
$\phi\colon D\to\mathbf{X}$, and we describe their ‘folding locus’ in $D$
where the composition $\rho_{+}\circ\phi\colon D\to\nabla^{+}$ is not a local
isometry (Section 7). Note that the classification of $g$ parabolic, which are
all conjugate into $C$, is much easier and can be read without preparation.

In Section 10, we study diagrams for pairs $(\xi,\eta)$ of limit points of
distinct isometries $g,g^{\prime}\in G$ that are parabolic or loxodromic not
of rank 1. Since $\mathbf{X}$ is not proper, we have to study the limits of
disc diagrams $\phi_{n}$ by postcomposing them with $\rho_{+}$. This requires
introducing a procedure of passing to the limit with non-combinatorial
‘relative disc diagrams’ (see Section 6, portions of which could be ignored at
a first reading by pretending that all the diagrams are combinatorial).
Finally, in Section 11 we prove that since $G$ does not stabilise a principal
or an antiprincipal point in $\partial_{\infty}\mathbf{X}$, there exist
$g,g^{\prime}\in G$ with $\xi$ and $\eta$ ‘far’ enough so that some $g^{\prime
n}g^{n}$ has rank 1, giving rise to the non-abelian free group.

Acknowledgements. We thank Shaked Bader, Werner Ballmann, Pierre-Emmanuel
Caprace, Sami Douba, Bruno Duchesne, Koji Fujiwara, Camille Horbez, and Marcin
Sabok for helpful explanations.

##  2\. Constructing rank 1 isometries

In this section, we consider an arbitrary complete $\mathrm{CAT}(0)$ space
$(X,d_{X})$. We allow $X$ to be not proper. We generalise the methods of
Ballmann and Ruane of constructing rank 1 isometries to this setting.

###  2.1. Boundary

Two geodesic rays $r,r^{\prime}\colon[0,\infty)\to X$ are _asymptotic_ if
there is $d>0$ with $d_{X}\big{(}r(t),r^{\prime}(t)\big{)}\leq d$ for all $t$.
This defines an equivalence relation, and we denote by $[r]$ the equivalence
class of a geodesic ray $r$. The _visual boundary_ $\partial_{\infty}X$ is the
set of equivalence classes of geodesic rays. Note that for each geodesic ray
$r$ and each $y\in X$ there is a geodesic ray representing $[r]$ starting at
$y$ [BH]*II.8.2, which we call the _geodesic ray from $y$ to $[r]$_. We extend
the topology on $X$ to the _cone topology_ on $X\cup\partial_{\infty}X$
described in [BH]*II.8.6. Namely, given a geodesic ray $r$, and
$t,\epsilon>0$, a _basic neighbourhood of $[r]$, based at $r(0)$,_ is the set
of $x\in X\cup\partial_{\infty}X$ such that the geodesic (possibly a ray)
$r^{\prime}$ from $r(0)$ to $x$ has length $>t$ and satisfies
$d_{X}\big{(}r^{\prime}(t),r(t)\big{)}<\epsilon$. Note that since $X$ is not
assumed to be proper, $\partial_{\infty}X$ might not be compact.

For $\xi,\eta\in\partial_{\infty}X$, the _angle_ $\angle(\xi,\eta)$ is the
supremum of the Alexandrov angle $\angle_{x}(r,r^{\prime})$ (see [BH]*I.1.12)
over $x\in X$ and $r$ representing $\xi$ and $r^{\prime}$ representing $\eta$
starting at $x$. The angle equips $\partial_{\infty}X$ with a complete
$\mathrm{CAT}(1)$ metric [BH]*II.9.13. However, the induced topology on
$\partial_{\infty}X$ is stronger than the cone topology.

###  2.2. Isometries

Let $g$ be an isometry of $X$. The _translation length_ of $g$ is
$|g|=\inf_{x\in X}d_{X}\big{(}x,g(x)\big{)}$. If the infimum is attained, we
say that $g$ is _elliptic_ if $|g|=0$ and _loxodromic_ otherwise. If the
infimum is not attained, we say that $g$ is _parabolic_. If $g$ is loxodromic,
then by [BH]*II.6.8.(1) it has (at least one) _axis_ , which is a
$g$-invariant geodesic line in $X$.

Let $\xi\in\partial_{\infty}X$. We say that $\xi$ is the _positive_ (resp.
_negative_) _limit point of $g$_, if for some (hence any) point $x\in X$, we
have $g^{n}(x)\to\xi$ for $n\to\infty$ (resp. $n\to-\infty$). Note that a
limit point is fixed by $g$. If $g$ is loxodromic, it has the positive limit
point $\xi^{+}$ and the negative limit point $\xi^{-}$ that are the endpoints
of any of the axes for $g$. In particular $\angle(\xi^{+},\xi^{-})=\pi$. If
$g$ is parabolic with positive translation length, then by [D]*lem 23.2 $g$
also has the positive and negative limit points $\xi^{+},\xi^{-}$ with
$\angle(\xi^{+},\xi^{-})=\pi$. A parabolic isometry of translation length zero
without a point $\xi\in\partial_{\infty}X$ that is its positive and negative
limit point is called _vile_. (We do not know if vile isometries exist.)

###  2.3. Rank 1

Let $\xi,\eta\in\partial_{\infty}X$. In what follows we generalise the
condition $d_{T}(\xi,\eta)>\pi$ used by Ruane in the setting of proper $X$
(where $d_{T}$ is the _Tits metric_ , i.e. the path metric induced by the
angle metric). In [R]*Prop 2.4 she proved that for proper $X$ this condition
is equivalent to $\xi,\eta$ being the endpoints of a geodesic line that does
not bound an isometrically embedded Euclidean half-plane.

######  Definition 2.1.

Let $p\in X$ and let $\varepsilon,R>0$. A geodesic quadrilateral
$xyy^{\prime}x^{\prime}$ in $X$ is a _$(p,\varepsilon,R)$ -quadrilateral_ if
$p\in xy$, the concatenation $yy^{\prime}\cdot y^{\prime}x^{\prime}\cdot
x^{\prime}x$ is at distance $>R$ from $p$, and the sum of the four Alexandrov
angles of $xyy^{\prime}x^{\prime}$ is $>2\pi-\varepsilon$.

######  Lemma 2.2.

Let $\square=xyy^{\prime}x^{\prime}$ be a $(p,\varepsilon,R)$-quadrilateral in
$X$. Then there is an embedded $(p,\varepsilon,R)$-quadrilateral
$\overline{\square}=\bar{x}\bar{y}\bar{y}^{\prime}\bar{x}^{\prime}$ in $X$
with $\bar{x}\bar{y}\subset xy$ and all the other sides of
$\overline{\square}$ contained in some sides of $\square$.

###### Proof.

If for some vertex $v$ of $\square$ the consecutive sides containing $v$
intersect along a geodesic $v\bar{v}$, then we replace $v$ by $\bar{v}$. Note
that all the Alexandrov angles remain the same except that the angle $0$ at
$v$ is replaced by a nonnegative angle at $\bar{v}$.

If opposite sides of $\square$ intersect at a point $z$ of $X$, then $\square$
decomposes as a concatenation of two geodesic triangles with Alexandrov angle
sum $>2\pi-\varepsilon$. Let $\triangle$ be the triangle containing $p$. By
[BH]*II.1.7(4), the Alexandrov angle sum in the other triangle is $\leq\pi$,
so the sum of the Alexandrov angles of $\triangle$ is $>\pi-\varepsilon$.

As in the first paragraph, we can assume that $\triangle$ is embedded. We
introduce an extra vertex (of Alexandrov angle $\pi$) on a side of $\triangle$
not containing $p$ to treat $\triangle$ as a
$(p,\varepsilon,R)$-quadrilateral. ∎

######  Definition 2.3.

Let $\xi,\eta\in\partial_{\infty}X$. We say that $\xi,\eta$ are
_$(\varepsilon,R)$ -far_ (or, shortly, _far_), if there are neighbourhoods
$U,V$ of $\xi,\eta$ in $X\cup\partial_{\infty}X$ and a closed metric ball
$P\subset X$ such that for each $u\in U,v\in V,$ with at least one of $u,v$ in
$X$,

  * •

the geodesic (possibly a ray) $uv$ in $X$ intersects $P$, and, furthermore,

  * •

there is $p\in uv\cap P$ that does not lie in any segment of $uv$ that is a
side of a $(p,\varepsilon,R)$-quadrilateral.

######  Lemma 2.4.

Let $x\in X$, let $\xi,\eta\in\partial_{\infty}X$, and let $u,v\in X$ distinct
from $x$ lie on the geodesic rays from $x$ to $\xi,\eta$, respectively. Then
for any point $p$ on the geodesic $uv$, distinct from $u,v$, we have
$\angle_{u}(x,v)+\angle_{v}(x,u)\geq\pi-\angle_{p}(\xi,\eta)$.

###### Proof.

We have $\angle_{u}(x,v)\geq\pi-\angle_{u}(\xi,v)$, which is
$\geq\angle_{p}(\xi,u)$ by [BH]*II.9.3(1). See Figure 1. Analogously, we have
$\angle_{v}(x,u)\geq\angle_{p}(\eta,v)$ and the lemma follows. ∎

Figure 1.

######  Lemma 2.5.

Suppose that $\xi,\eta$ are far and $P$ is as in Definition 2.3. Then
$\angle(\xi,\eta)=\pi$ and for each $\delta>0$ there is $p\in P$ with
$\angle_{p}(\xi,\eta)>\pi-\delta$.

###### Proof.

Let $x\in X$ and let $r,r^{\prime}\colon[0,\infty)\to X$ be the geodesic rays
from $x$ to $\xi,\eta$. Then for $t$ sufficiently large, we have $r(t)\in
U,r^{\prime}(t)\in V$. If $\angle(\xi,\eta)<\pi$, then by [BH]*II.9.8(4) the
geodesic $r(t)r^{\prime}(t)$ is arbitrarily far from $x$ for $t$ sufficiently
large. Consequently, $r(t)r^{\prime}(t)$ is disjoint from $P$, which is a
contradiction.

Thus $\angle(\xi,\eta)=\pi$, and so we can assume
$\angle_{x}(\xi,\eta)>\pi-\delta$. Then
$\angle_{r(t)}\big{(}x,r^{\prime}(t)\big{)}+\angle_{r^{\prime}(t)}\big{(}r(t),x\big{)}<\delta$.
Consequently, for $p\in P\cap r(t)r^{\prime}(t)$, by Lemma 2.4 we have
$\angle_{p}(\xi,\eta)>\pi-\delta$. ∎

Following [BF]*Def 5.1, an isometry $g$ of $X$ has _rank 1_ if for some (hence
any) $x\in X$, there is $M>0$ such that, for each $n>0$,

  * •

the geodesic $\beta_{n}=g^{n}(x)g^{-n}(x)$ is at Hausdorff distance $\leq M$
from the set $\\{g^{n}(x),g^{n-1}(x),\ldots,g^{-n}(x)\\}$, and

  * •

$\beta_{n}$ is _$M$ -contracting_, in the sense that the projection to
$\beta_{n}$ of any closed metric ball $B$ disjoint from $\beta_{n}$ has length
$\leq M$.

It is not hard to prove (but we will not need it) that if $X$ is proper, then
$\xi,\eta$ are $(\varepsilon,R)$-far (for some $\varepsilon,R$) if and only if
$\xi,\eta$ are the endpoints of a geodesic line that does not bound an
isometrically embedded Euclidean half-plane. Thus the following theorem
generalises [BF]*Thm 5.4.

######  Theorem 2.6.

Let $g$ be an isometry of $X$ with far limit points. Then $g$ has rank 1.

Note that if $X$ has finite telescopic dimension, then such $g$ is loxodromic,
since by [CL]*Thm 1.1 (see also [D]*thm 6.1), for parabolic $g$ all of its
fixed points in $\partial_{\infty}X$ are at Tits distance $\leq\pi$.

Before giving the proof, we record the following immediate consequence of
Theorem 2.6 and [BF]*Prop 5.9.

######  Corollary 2.7.

Let $g$ (resp. $g^{\prime}$) be isometries of $X$ with far limit points
$\xi,\eta$ (resp. $\xi^{\prime},\eta^{\prime}$). If
$\\{\xi,\eta\\},\\{\xi^{\prime},\eta^{\prime}\\}$ are disjoint, then some
powers of $g,g^{\prime}$ are a basis of a free subgroup in the isometry group
of $X$.

To prove Theorem 2.6, we first establish the following.

######  Lemma 2.8.

Let $\varepsilon,R>0$ and let $\beta\subset X$ be a geodesic with a family of
points $\mathcal{P}$ dividing it into segments of length $\leq D$. Then
$\beta$ is
$2R+\big{\lceil}\frac{2\pi}{\varepsilon}\big{\rceil}(D+2R)$-contracting or
there is $p\in\mathcal{P}$ and a $(p,\varepsilon,R)$-quadrilateral with a side
on $\beta$.

###### Proof.

Suppose that $\beta$ is not
$2R+\big{\lceil}\frac{2\pi}{\varepsilon}\big{\rceil}(D+2R)$-contracting. Then
there is a closed ball $B\subset X$ of radius $R^{\prime}>R$ and centre $z$
disjoint from $\beta$, and points $x^{\prime\prime},y^{\prime\prime}\in B$
with projections to $\beta$ at distance
$>2R+\big{\lceil}\frac{2\pi}{\varepsilon}\big{\rceil}(D+2R)$. Let $x^{\prime}$
be the point on the geodesic $x^{\prime\prime}z$ at distance
$\min\\{R,d_{X}(x^{\prime\prime},z)\\}$ from $x^{\prime\prime}$. Define
$y^{\prime}$ analogously. Then the geodesic $x^{\prime}y^{\prime}$ is
contained in the closed ball of radius $R^{\prime}-R$ and centre $z$ and hence
it is at distance $>R$ from $\beta$. Let $x,y$ be the projections of
$x^{\prime},y^{\prime}$ to $\beta$. Note that $x$ and $y$ are at distance
$\leq R$ from the projections of $x^{\prime\prime},y^{\prime\prime}$. Thus the
geodesic $xy$ has length
$>\big{\lceil}\frac{2\pi}{\varepsilon}\big{\rceil}(D+2R)$. Consequently, $xy$
can be divided into $N=\big{\lceil}\frac{2\pi}{\varepsilon}\big{\rceil}$
segments $\gamma_{1}=xx_{1},\gamma_{2}=x_{1}x_{2},\ldots,\gamma_{N}=x_{N-1}y$
of length $>D+2R$.

Figure 2.

Each $\gamma_{i}$ contains some $p_{i}\in\mathcal{P}$ at distance $>R$ from
the endpoints of $\gamma_{i}$. Choose $x_{1}^{\prime},\ldots,x_{N-1}^{\prime}$
on $x^{\prime}y^{\prime}$ in that order so that the projection of each
$x^{\prime}_{i}$ to $\beta$ is $x_{i}$. See Figure 2. Denoting
$x_{0}^{\prime}=x^{\prime},x_{N}^{\prime}=y^{\prime},x_{0}=x,x_{N}=y$, for
$i=1,\ldots,N$ the geodesics $x^{\prime}_{i-1}x_{i-1},x^{\prime}_{i}x_{i}$ are
at distance $>R$ from $p_{i}$. Thus each
$x_{i-1}x_{i}x_{i}^{\prime}x_{i-1}^{\prime}$ is either a
$(p_{i},\varepsilon,R)$-quadrilateral, or its sum of the Alexandrov angles is
$\leq 2\pi-\varepsilon$. Since
$\angle_{x_{i}}(x^{\prime}_{i},x_{i-1})+\angle_{x_{i}}(x^{\prime}_{i},x_{i+1})\geq\pi$
and
$\angle_{x^{\prime}_{i}}(x_{i},x^{\prime}_{i-1})+\angle_{x^{\prime}_{i}}(x_{i},x^{\prime}_{i+1})\geq\pi$,
the latter possibly would imply that the sum of the Alexandrov angles of
$xyy^{\prime}x^{\prime}$ is $\leq 2\pi-N\varepsilon\leq 0$, which is a
contradiction. ∎

###### Proof of Theorem 2.6.

Suppose that the limit points of $g$ are $(\varepsilon,R)$-far. Let $P,U,V$ be
provided by Definition 2.3. Let $p\in P$. By possibly passing to a power of
$g$, we can assume $g^{n}(p)\in U,g^{-n}(p)\in V$ for all $n\geq 1$.
Consequently, given $n\geq 1$, for $-(n-1)\leq k\leq n-1$, we have
$g^{n-k}(p)\in U,g^{-n-k}(p)\in V$, and so there is a point $p_{k}\in P$ on
the geodesic $g^{n-k}(p)g^{-n-k}(p)$ that does not lie in a segment of
$g^{n-k}(p)g^{-n-k}(p)$ that is a side of a
$(p_{k},\varepsilon,R)$-quadrilateral. Denote $D=\mathrm{diam}(P\cup g(P))$.
The points $g^{n-1}(p_{n-1}),g^{n-2}(p_{n-2}),\ldots,g^{-(n-1)}(p_{-(n-1)})$
divide the geodesic $\beta_{n}=g^{n}(p)g^{-n}(p)$ into segments of length
$\leq D$. In particular, $\beta_{n}$ is at Hausdorff distance $\leq D$ from
$\\{g^{n}(p),g^{n-1}(p),\ldots,g^{-n}(p)\\}$.

Let $\beta=\beta_{n}$ and $\mathcal{P}=\\{g^{k}(p_{k})\\}$. By Lemma 2.8,
$\beta_{n}$ is
$2R+\big{\lceil}\frac{2\pi}{\varepsilon}\big{\rceil}(D+2R)$-contracting or
there is $-(n-1)\leq k\leq n-1$ and a
$(g^{k}(p_{k}),\varepsilon,R)$-quadrilateral with a side on $\beta_{n}$.
However, the latter possibility would imply that there is a
$(p_{k},\varepsilon,R)$-quadrilateral with a side on
$g^{-k}(\beta_{n})=g^{n-k}(p)g^{-n-k}(p)$, which would be a contradiction.

Thus each $\beta_{n}$ is
$2R+\big{\lceil}\frac{2\pi}{\varepsilon}\big{\rceil}(D+2R)$-contracting, as
desired. ∎

###  2.4. Construction

The main result of this section is the following construction of rank 1
isometries, which is a generalisation of [R]*Main Thm.

######  Proposition 2.9.

Let $f,g$ be isometries of $X$ that have positive limit points
$\xi_{f}^{+},\xi_{g}^{+}$ that are far and negative limit points
$\xi_{f}^{-},\xi_{g}^{-}$ that are far. Then for any neighbourhoods
$U_{0},V_{0}$ of $\xi_{f}^{+},\xi_{g}^{+}$, for sufficiently large $n$, the
isometry $f^{n}g^{-n}$ is vile or has positive translation length and far
limit points in $U_{0},V_{0}$.

To prove Proposition 2.9, we first need the following variant of [Ball]*Lem
III.3.2.

######  Lemma 2.10.

Suppose that $\xi,\eta\in\partial_{\infty}X$ are far. Let $h_{n}$ be a
sequence of isometries of $X$ such that for some (hence every) $p\in X$ we
have $h_{n}(p)\to\xi,h^{-1}_{n}(p)\to\eta$. Then for any neighbourhoods
$U_{0},V_{0}$ of $\xi,\eta$, for sufficiently large $n$, the isometry $h_{n}$
is vile or has positive translation length and far limit points in
$U_{0},V_{0}$.

###### Proof.

Suppose that $\xi,\eta$ are $(\varepsilon,R)$-far, and let $P,U,V$ be provided
by Definition 2.3. We can assume $U\subset U_{0},V\subset V_{0}$. By Lemma
2.5, we can assume that for some $p\in P$ and any $u\in U,v\in V$ we have
$\angle_{p}(u,v)>\pi-\frac{\varepsilon}{2}$. Furthermore, we can assume that
$U,V$ are basic neighbourhoods based at $p$ and that for all $n>0$ we have
$h_{n}(B_{R}(P))\subset U,h^{-1}_{n}(B_{R}(P))\subset V$, where $B_{R}$
denotes the closed $R$-neighbourhood.

We first justify that for sufficiently large $n$ the isometry $h_{n}$ has no
fixed point $x_{n}\in X\cup\partial_{\infty}X$ outside $U\cup V$. Indeed,
otherwise let $p^{\prime}\in P$ be a point on the geodesic
$h_{n}(p)h_{n}^{-1}(p)$ as in Definition 2.3. Since $V$ is based at $p$, and
$h^{-1}_{n}(B_{R}(P))\subset V$ whereas $h_{n}^{-1}(x_{n})=x_{n}\notin V$, we
have that the geodesic $ph_{n}^{-1}(x_{n})$ (which is possibly a ray) is
disjoint from $h^{-1}_{n}(B_{R}(p^{\prime}))$. Consequently, the geodesic
$h_{n}(p)x_{n}$ is disjoint from $B_{R}(p^{\prime})$. Analogously, the
geodesic $h^{-1}_{n}(p)x_{n}$ is disjoint from $B_{R}(p^{\prime})$.

Figure 3.

If $x_{n}\in X$, then let $x=h_{n}(p),y=h^{-1}_{n}(p),x^{\prime}=x_{n}$, and
let $y^{\prime}$ be any point on the geodesic $x^{\prime}y$ distinct from
$x^{\prime},y$. See Figure 3. We now compute the Alexandrov angles of the
geodesic quadrilateral $xyy^{\prime}x^{\prime}$. The triangles $x^{\prime}xp$
and $x^{\prime}py$ are isometric and so
$\angle_{y}(p,x^{\prime})+\angle_{x}(x^{\prime},p)=\angle_{p}(x,x^{\prime})+\angle_{p}(x^{\prime},y)>\pi-\frac{\varepsilon}{2}$.
Consequently,
$\angle_{y}(x,x^{\prime})+\angle_{x}(x^{\prime},y)>\pi-\frac{\varepsilon}{2}-\angle_{y}(p,x)-\angle_{x}(y,p)\geq\pi-\frac{\varepsilon}{2}-\frac{\varepsilon}{2}$.
Thus the sum of the Alexandrov angles of $xyy^{\prime}x^{\prime}$ at $x$ and
$y$ exceeds $\pi-\varepsilon$, whereas at $y^{\prime}$ it equals $\pi$.
Consequently, $xyy^{\prime}x^{\prime}$ is a
$(p^{\prime},\varepsilon,R)$-quadrilateral, which is a contradiction.

If $x_{n}\in\partial_{\infty}X$, then we take $x^{\prime}$ sufficiently far on
the geodesic ray $h_{n}(p)x_{n}$, and we set
$y^{\prime}=h_{n}^{-2}(x^{\prime})$. Then, since horoballs are limits of
balls, the Alexandrov angles of $xyy^{\prime}x^{\prime}$ at $x^{\prime}$ and
$y^{\prime}$ are $\geq\frac{\pi}{2}$. Thus an analogous calculation to the one
in the case of $x_{n}\in X$ leads to a contradiction.

Suppose now that $h_{n}$ has a fixed point $x_{n}\in X\cup\partial_{\infty}X$
in $V$. Then observe that the geodesic $h_{n}(p)x_{n}$ passes through $P$,
which contains $p$. If $x_{n}\in X$, then for large $n$ this contradicts
$d_{X}\big{(}h_{n}(p),x_{n}\big{)}=d_{X}(p,x_{n})$. If
$x_{n}\in\partial_{\infty}X$ and $x_{n}$ is the positive fixed point of
$h_{n}$, then for large $n$ this observation contradicts the fact that the
Busemann function [BH]*II.8.17 satisfies $b_{x_{n}}(h_{n}(p))\leq
b_{x_{n}}(p)$. The only remaining possibility is that for large $n$ the
isometry $h_{n}$ is vile or has positive translation length, with a positive
limit point in $U$ and a negative limit point in $V$. Choose neighbourhoods
$U_{n},V_{n}\subset X\cup\partial_{\infty}X$ of these limits points inside
$U,V$. Then $P,U_{n},V_{n}$ satisfy Definition 2.3 for the limit points of
$h_{n}$. ∎

The following is essentially [R]*Lem 6.3.

######  Lemma 2.11.

Let $f$ be an isometry of $X$ with limit points
$\xi^{+},\xi^{-}\in\partial_{\infty}X$. Let $\eta\in\partial_{\infty}X$ with
$\xi^{-},\eta$ far. Then there is a neighbourhood
$V\subset\partial_{\infty}X\cup X$ of $\eta$ such that for every neighbourhood
$U^{+}\subset\partial_{\infty}X\cup X$ of $\xi^{+}$, for sufficiently large
$n$ we have $f^{n}(V)\subset U^{+}$.

###### Proof.

Let $P,U^{-},V$ be provided by Definition 2.3 for $\xi^{-},\eta$. Choose a
basepoint $x\in X$ such that $f^{-n}(x)$ lie in $U^{-}$ for all $n\geq 0$.
Then for each $v\in V$ and all $n\geq 0$ there is a point of the ball $P$ on
the geodesic $f^{-n}(x)v$ (which is possibly a ray). Consequently, each
geodesic $xf^{n}(v)$ contains a point at a uniform distance from $f^{n}(P)$.
See Figure 4. This proves that for every neighbourhood
$U^{+}\subset\partial_{\infty}X\cup X$ of $\xi^{+}$ we have $f^{n}(V)\subset
U^{+}$ for $n$ sufficiently large. ∎

Figure 4.

###### Proof of Proposition 2.9.

Let $U^{+}$ be a neighbourhood of $\xi_{f}^{+}$. Let $V$ be a neighbourhood of
$\xi_{g}^{-}$ from Lemma 2.11. Choose $x\in X$. Then for $n$ sufficiently
large we have $g^{-n}(x)\in V$ and hence by Lemma 2.11 for $n$ sufficiently
large we have $f^{n}g^{-n}(x)\in U^{+}$. Consequently for $h_{n}=f^{n}g^{-n}$
we have $h_{n}(x)\to\xi_{f}^{+}$. Analogously $h^{-1}_{n}(x)\to\xi_{g}^{+}$.
By Lemma 2.10, for sufficiently large $n$ the isometry $h_{n}$ is vile or has
positive translation length and far limit points in arbitrary neighbourhoods
of $\xi_{f}^{+},\xi_{g}^{+}$. ∎

##  3\. Building-like space $\mathbf{X}$

We recall the construction from [LP2] of a complete $\mathrm{CAT}(0)$ space
$\mathbf{X}$ with a natural action of $\mathrm{Tame}(\mathbbm{k}^{3})$.

###  3.1. Metric

Let $\mathbb{R}_{>0}$ be the positive reals and let $\Pi=\mathbb{R}_{>0}^{3}$.
The _weight space_ $\nabla$ is the projectivisation of $\Pi$. Let
$\Pi^{+}\subset\Pi$ consist of $(\alpha_{1},\alpha_{2},\alpha_{3})$ satisfying
$\alpha_{1}\geq\alpha_{2}\geq\alpha_{3}$, and let $\nabla^{+}\subset\nabla$ be
the projectivisation of $\Pi^{+}$. We equip $\nabla$ and $\nabla^{+}$ with the
topology induced from the projective plane. In particular, $\nabla^{+}$ is a
topological surface with boundary $\partial\nabla^{+}$ where
$\alpha_{1}=\alpha_{2}$ or $\alpha_{2}=\alpha_{3}$. Then
$\mathrm{int}\,\nabla^{+}=\nabla^{+}\setminus\partial\nabla^{+}$ consists of
$[\alpha]$ represented by $\alpha$ with $\alpha_{1}>\alpha_{2}>\alpha_{3}$.

For $\alpha\in\Pi$, we define the following map
$\nu_{\mathrm{id},\alpha}\colon\mathbbm{k}[x_{1},x_{2},x_{3}]\to\mathbb{R}$.
Let $I$ stand for $(m_{1},m_{2},m_{3})\in\mathbb{N}^{3}$. Namely, for
$P=\sum_{I}c_{I}x_{1}^{m_{1}}x_{2}^{m_{2}}x_{3}^{m_{3}}\in\mathbbm{k}[x_{1},x_{2},x_{3}]$,
we set

$\nu_{\mathrm{id},\alpha}\left(P\right)=\min_{c_{I}\neq
0}\left(-\sum_{k=1}^{3}\alpha_{k}m_{k}\right).$

For $f\in\mathrm{Tame}(\mathbbm{k}^{3})$, we define the _monomial valuation_
$\nu_{f,\alpha}\colon\mathbbm{k}[x_{1},x_{2},x_{3}]\to\mathbb{R}$ by
$\nu_{f,\alpha}(P)=\nu_{\mathrm{id},\alpha}(P\circ f)$. The group
$\mathrm{Tame}(\mathbbm{k}^{3})$ acts on monomial valuations by
$g\cdot\nu(P)=\nu(P\circ g)$. The projective class of $\nu_{f,\alpha}$ depends
only on $[\alpha]$ and is denoted by $\nu_{f,[\alpha]}$. Instead of
$[(\alpha_{1},\alpha_{2},\alpha_{3})]$ we shortly write
$[\alpha_{1},\alpha_{2},\alpha_{3}]$.

We equip the space $\mathbf{X}$ of projective classes of all monomial
valuations with the following metric. First, consider the map
$\nabla\to\mathbb{R}^{3}$ sending each $[\alpha]\in\nabla$ represented by
$\alpha$ with $\Pi_{k=1}^{3}\alpha_{k}=1$ to
$(\log\alpha_{1},\log\alpha_{2},\log\alpha_{3})$, with image in the Euclidean
plane $\mathbb{R}^{2}\subset\mathbb{R}^{3}$ of equation
$\log\alpha_{1}+\log\alpha_{2}+\log\alpha_{3}=0$. We equip $\nabla$ (and
consequently $\nabla^{+}$) with the Euclidean metric $|\cdot,\cdot|$ pulled
back from $\mathbb{R}^{2}$.

We view $\mathbf{X}$ as the quotient (by an equivalence relation $\sim$) of
the disjoint union of _chambers_
$\mathbf{E}^{+}_{f}=\\{\nu_{f,[\alpha]}\colon[\alpha]\in\nabla^{+}\\}$. We
also consider _apartments_
$\mathbf{E}_{f}=\\{\nu_{f,[\alpha]}\colon[\alpha]\in\nabla\\}$. The following
proves in particular that the projective classes of monomial valuations
$\nu_{\mathrm{id},[\alpha]}$ and $\nu_{\mathrm{id},[\alpha^{\prime}]}$ are
distinct for $[\alpha]\neq[\alpha^{\prime}]$.

######  Corollary 3.1 ([LP2]*cor 2.5(i)).

The map $\bigsqcup\mathbf{E}^{+}_{f}\to\nabla^{+}$ sending each
$\nu_{f,[\alpha]}$ to $[\alpha]$ is well-defined and descends to a map
$\rho_{+}\colon\mathbf{X}\to\nabla^{+}$.

We equip each chamber with the Euclidean metric $|\cdot,\cdot|$ coming from
$\nabla^{+}$. A _chain_ is a sequence $(x^{\prime}_{0},x_{1}\sim
x_{1}^{\prime},x_{2}\sim x_{2}^{\prime},\ldots,x_{k-1}\sim
x^{\prime}_{k-1},x_{k})$ in $\bigsqcup\mathbf{E}^{+}_{f}$ such that there are
$f_{0},\ldots,f_{k-1}\in\mathrm{Tame}(\mathbbm{k}^{3})$ with
$x_{i}^{\prime},x_{i+1}\in\mathbf{E}_{f_{i}}^{+}$. For
$\overline{x},\overline{y}\in\mathbf{X}$ we define the _quotient pseudo-
metric_ $d_{\mathbf{X}}(\overline{x},\overline{y})$ to be the infimum of the
_lengths_ $\sum_{i=0}^{k-1}|x^{\prime}_{i},x_{i+1}|$ of chains with
$x_{0}^{\prime}\in\overline{x}$ and $x_{k}\in\overline{y}$.

######  Theorem 3.2 ([LP2]*Thm A).

$(\mathbf{X},d_{\mathbf{X}})$ is complete and $\mathrm{CAT}(0)$.

Here are some other basic properties of $(\mathbf{X},d_{\mathbf{X}})$ that we
will be using. By $B_{\mathbf{X}}(\overline{x},\epsilon)$ we denote the radius
$\epsilon$ open ball in $\mathbf{X}$ around $\overline{x}\in\mathbf{X}$.

######  Lemma 3.3.

  1. $(i)$

For each $\overline{x}\in X$ there is $\epsilon=\epsilon(\overline{x})>0$ such
that $B_{\mathbf{X}}(\overline{x},\epsilon)$ coincides as a set with the union
of radius $\epsilon$ open balls around $x$ in $\bigsqcup\mathbf{E}^{+}_{f}$
for $x\in\overline{x}$.

  2. $(ii)$

Each $\mathbf{E}^{+}_{f}$ is isometrically embedded in $\mathbf{X}$.

  3. $(iii)$

For each pair $\overline{x},\overline{y}\in\mathbf{X}$, the infimum in the
definition of the distance $d_{\mathbf{X}}(\overline{x},\overline{y})$ is
realised.

###### Proof.

Item (i) is [LP2]*lem 5.5(a), and item (ii) is [LP2]*lem 5.6(i). For item
(iii), let $\gamma$ be a geodesic from $\overline{x}$ to $\overline{y}$
guaranteed by Theorem 3.2. By the compactness of $\gamma$, we can find points
$\overline{x}_{0}=\overline{x},\overline{x}_{1},\ldots,\overline{x}_{k}=\overline{y}$
such that the consecutive balls
$B_{\mathbf{X}}\Big{(}\overline{x}_{i},\frac{\epsilon(\overline{x}_{i})}{2}\Big{)}$
intersect. Consequently, for each $0\leq i<k$, we have $\overline{x}_{i}\in
B_{\mathbf{X}}\big{(}\overline{x}_{i+1},\epsilon(\overline{x}_{i+1})\big{)}$
or $\overline{x}_{i+1}\in
B_{\mathbf{X}}\big{(}\overline{x}_{i},\epsilon(\overline{x}_{i})\big{)}$. By
item (i), this shows that $\overline{x}_{i},\overline{x}_{i+1}$ have
representatives in a common $\mathbf{E}^{+}_{f_{i}}$. By item (ii), the length
of $\gamma$ coincides with the length of the chain determined by the
$\overline{x}_{i}$. ∎

###  3.2. Point stabilisers

Let $A<\mathrm{Tame}(\mathbbm{k}^{3})$ be the subgroup consisting of the
affine transformations of $\mathbbm{k}^{3}$. We will also consider the
following subgroups of $\mathrm{Tame}(\mathbbm{k}^{3})$:

$B=\\{(ax_{1}+P(x_{2},x_{3}),bx_{2}+cx_{3}+d,b^{\prime}x_{2}+c^{\prime}x_{3}+d^{\prime});\\\
P\in\mathbbm{k}[x_{2},x_{3}],a\neq 0,bc^{\prime}-b^{\prime}c\neq 0\\},$
$H=\\{(ax_{1}+P_{1}(x_{2},x_{3}),bx_{2}+P_{2}(x_{3}),cx_{3}+d);\\\
P_{1}\in\mathbbm{k}[x_{2},x_{3}],P_{2}\in\mathbbm{k}[x_{3}]\ a,b,c\neq 0\\},$
$K=\\{(ax_{1}+bx_{2}+P_{1}(x_{3}),a^{\prime}x_{1}+b^{\prime}x_{2}+P_{2}(x_{3}),cx_{3}+d);\\\
P_{1},P_{2}\in\mathbbm{k}[x_{3}],ab^{\prime}-a^{\prime}b\neq 0,c\neq 0\\}.$

Let $C=\langle K,H\rangle,$ where $C=K\ast_{K\cap H}H$ by [LP1]*Prop 3.5.
Finally, let $B^{\prime}=\langle B,H\rangle$, where $B^{\prime}=B\ast_{B\cap
H}H$, which corresponds to the splitting of $\mathrm{Aut}(\mathbbm{k}^{2})$ in
[vdk].

Let $\alpha\in\Pi^{+}$. We consider the following subgroups
$M_{\alpha}<H,L_{\alpha}<\textbf{GL}_{3}(\mathbbm{k})<\mathrm{Tame}(\mathbbm{k}^{3})$.

  1. (1)

If $\alpha_{1}=\alpha_{2}=\alpha_{3}$, then
$L_{\alpha}=\textbf{GL}_{3}(\mathbbm{k})$ and $M_{\alpha}$ is the group of
translations, so that $M_{\alpha}\rtimes L_{\alpha}=A$.

  2. (2)

If $\alpha_{1}>\alpha_{2}>\alpha_{3}$, then $L_{\alpha}$ is the group of the
diagonal matrices and $M_{\alpha}$ is the group of the automorphisms

$(x_{1},x_{2},x_{3})\to(x_{1}+P_{1}(x_{2},x_{3}),x_{2}+P_{2}(x_{3}),x_{3}+d)$

where $P_{i}$ satisfy $\alpha_{i}\geq-\nu_{\mathrm{id},\alpha}(P_{i})$. In
particular, $M_{\alpha}\rtimes L_{\alpha}<H$.

  3. (3)

If $\alpha_{1}>\alpha_{2}=\alpha_{3}$, then

$\displaystyle L_{\alpha}$
$\displaystyle=\\{(ax_{1},bx_{2}+cx_{3},b^{\prime}x_{2}+c^{\prime}x_{3})\\},$
$\displaystyle M_{\alpha}$
$\displaystyle=\\{(x_{1}+P(x_{2},x_{3}),x_{2}+d,x_{3}+d^{\prime});\,\alpha_{1}\geq\alpha_{2}\deg
P\\}.$

In particular, $M_{\alpha}\rtimes L_{\alpha}<B$.

  4. (4)

If $\alpha_{1}=\alpha_{2}>\alpha_{3}$, then

$\displaystyle L_{\alpha}$
$\displaystyle=\\{(ax_{1}+bx_{2},a^{\prime}x_{1}+b^{\prime}x_{2},cx_{3})\\},$
$\displaystyle M_{\alpha}$
$\displaystyle=\\{(x_{1}+P_{1}(x_{3}),x_{2}+P_{2}(x_{3}),x_{3}+d);\,\alpha_{1}\geq\alpha_{3}\deg
P_{1},\alpha_{3}\deg P_{2}\\}.$

In particular, $M_{\alpha}\rtimes L_{\alpha}<K$.

Note that $L_{\alpha},M_{\alpha}$ depend only on the class $[\alpha]$.

######  Proposition 3.4 ([LP2]*prop 3.2).

Let $\alpha\in\Pi^{+}$. We have

$\mathrm{Stab}\big{(}\nu_{\mathrm{id},[\alpha]}\big{)}=M_{\alpha}\rtimes
L_{\alpha}.$

In particular, each element of the groups $H,B,$ and $K$, has a fixed point in
$\mathbf{E}^{+}_{\mathrm{id}}$.

The following result was proved in [L] for $\mathbbm{k}=\mathbb{C}$. In fact,
the proof extends to any field of characteristic zero, see [LamyBook].

######  Theorem 3.5.

Let $\mathbbm{k}$ be a field of characteristic zero. Then the polynomial
automorphism group $\mathrm{Aut}(\mathbbm{k}^{2})$ satisfies the strong Tits
alternative.

Below, the _step_ of a solvable group is its derived length, that is, the
length of its derived series. In particular, non-trivial abelian groups have
step 1.

######  Remark 3.6.

  1. (i)

The groups $M_{\alpha}$ are solvable and the groups $L_{\alpha}$ are linear.
Thus, since the strong Tits alternative is preserved under semidirect
products, by Proposition 3.4 all
$\mathrm{Stab}\big{(}\nu_{\mathrm{id},[\alpha]}\big{)}$ satisfy the strong
Tits alternative.

  2. (ii)

Since $M_{\alpha}$ are solvable of step $\leq 3$, and all virtually solvable
subgroups of all $L_{\alpha}$ contain a triangularisable subgroup of uniformly
bounded index (see [W] and [KM]*Thm 21.1.5), which is solvable of step $\leq
3$, we have that all virtually solvable subgroups of all
$\mathrm{Stab}\big{(}\nu_{\mathrm{id},[\alpha]}\big{)}$ contain a solvable
subgroup of uniformly bounded index and step $\leq 6$.

  3. (iii)

The group $B^{\prime}$ has a decomposition into a semidirect product of
$\mathrm{Aut}(\mathbbm{k}^{2})$ (with variables $x_{2},x_{3}$) and the
solvable normal subgroup consisting of automorphisms of the form
$(ax_{1}+P(x_{2},x_{3}),x_{2},x_{3})$. Thus by Theorem 3.5, the group
$B^{\prime}$ also satisfies the strong Tits alternative. Similarly, the group
$C$ is a semidirect product of the solvable subgroup consisting of
$(x_{1},x_{2},cx_{3}+d)$ and a normal subgroup that can be embedded in the
automorphism group $\mathrm{Aut}\big{(}\mathbbm{k}(x_{3})^{2}\big{)}$ with
variables $x_{1},x_{2}$. Consequently, $C$ also satisfies the strong Tits
alternative.

Proposition 3.4 can be rephrased as follows.

######  Corollary 3.7 ([LP2]*prop 4.4).

Let $f\in\mathrm{Tame}(\mathbbm{k}^{3})$ and let $\alpha\in\Pi^{+}$. Then
$\nu_{\mathrm{id},[\alpha]}$ is fixed by $f$ if and only if $\alpha$ satisfies
all the inequalities of the form

$\alpha_{i}\geq\sum_{k}m_{k}\alpha_{k},$

where $i=1,2,3$ and $\prod_{k}x_{k}^{m_{k}}$ are the monomials appearing in
$f_{i}$.

Figure 5.

The subsets of $\nabla^{+}$ and $\nabla$ given by the equations of the form
$\alpha_{1}=m_{2}\alpha_{2}+m_{3}\alpha_{3}$, up to interchanging
$\alpha_{i}$, are _admissible lines_. If additionally $m_{2}=0$ (or
$m_{3}=0$), then such a line is _principal_. See Figure 5 for the set of the
admissible lines in $\nabla$ with the projective structure (on the left), and
with the Euclidean metric $|\cdot,\cdot|$ (on the right). Note that the only
admissible lines intersecting the interior of $\nabla^{+}$ have the precise
form $\alpha_{1}=m_{2}\alpha_{2}+m_{3}\alpha_{3}$, where $m_{2}+m_{3}\geq 2$,
or $\alpha_{2}=m_{3}\alpha_{3}$, where $m_{3}\geq 2$.

The following Figure 6 illustrates some possible intersections between
$\mathbf{E}_{\mathrm{id}}$ and $\mathbf{E}_{f}$ (and
$\mathbf{E}^{+}_{\mathrm{id}}$ and $\mathbf{E}^{+}_{f}$ in bold), where $f$
fixes the point $\nu_{\mathrm{id},[3,2,1]}$.

######  Remark 3.8.

Suppose in Corollary 3.7 that $\nu_{\mathrm{id},[\alpha]}$ is fixed by $f$. If
$\alpha_{2}>\alpha_{3}$, then for each
$\alpha^{\prime}=(\alpha_{1},\alpha_{2},\alpha_{3}^{\prime})$ with
$\alpha_{3}^{\prime}<\alpha_{3}$, the point
$\nu_{\mathrm{id},[\alpha^{\prime}]}$ is fixed by $f$. If
$\alpha_{1}>\alpha_{2}$, then for each
$\alpha^{\prime}=(\alpha^{\prime}_{1},\alpha_{2},\alpha_{3})$ with
$\alpha_{1}^{\prime}>\alpha_{1}$, the point
$\nu_{\mathrm{id},[\alpha^{\prime}]}$ is fixed by $f$.

Figure 6. Intersections of apartments
$\mathbf{E}_{\mathrm{id}},\mathbf{E}_{f}$ and chambers
$\mathbf{E}^{+}_{\mathrm{id}},\mathbf{E}^{+}_{f}$ for particular $f$.

###  3.3. Combinatorial structure on $\mathbf{X}$

A map from a CW-complex $X$ to a CW-complex $Y$ is _combinatorial_ , if its
restriction to any open cell of $X$ is a homeomorphism onto an open cell of
$Y$. A CW-complex $X$ is _combinatorial_ , if the attaching map of each open
cell of $X$ is combinatorial for some subdivision of the sphere. The _star_ of
a cell $\sigma$ in a CW-complex $X$ is the union of all the open cells of $X$
containing $\sigma$ in their closure.

The space $\mathbf{X}$ is equipped with the following structure of a
combinatorial CW-complex (which induces a weaker topology than the metric
$d_{\mathbf{X}}$ due to the lack of local finiteness). Namely, we first equip
$\nabla^{+}$ with the following structure of a combinatorial CW-complex. Its
vertices are the intersection points of distinct admissible lines. Its open
edges are the connected components of the complement of the vertices in the
admissible lines. Finally, its open $2$-cells are the connected components of
the complement of the admissible lines in $\nabla^{+}$. In (the proof of)
[LP2]*prop 6.3, we identify $\mathbf{X}$ with the universal development of a
certain complex of groups with underlying complex $\nabla^{+}$ equipped with
that CW-structure. This equips $\mathbf{X}$ with a CW-complex structure.
Equivalently, each open cell of $\mathbf{X}$ is contained in a chamber
$\mathbf{E}_{f}^{+}$ and it is the preimage in $\mathbf{E}_{f}^{+}$ under
$\rho_{+}$ of a cell of $\nabla^{+}$.

##  4\. Triples of polynomials in two variables

In this section we collect several estimates on particular exponents appearing
in identities involving triples of polynomials in two variables. This will
enable us to study the links of the vertices of $\mathbf{X}$ in Section 5.

######  Lemma 4.1.

Let $Q\in\mathbbm{k}[y,z]$ be a nonzero homogeneous polynomial of degree
$d\geq 0$, let $c,c^{\prime}\in\mathbbm{k}^{*}$, and let $q\geq 1$ be an
integer. Then the partial derivatives $\partial_{y}[(cy+c^{\prime}z)^{q}Q]$
and $\partial_{z}[(cy+c^{\prime}z)^{q}Q]$ are both nonzero homogeneous
polynomials of degree $d+q-1$ that are multiples of $(cy+c^{\prime}z)^{q-1}$.

###### Proof.

Given a homogeneous polynomial $P\in\mathbbm{k}[y,z]$, if $\deg_{y}P\geq 1$,
then we have $\deg\partial_{y}P=\deg(P)-1$. This gives the assertion about the
degree. The assertion that $(cy+c^{\prime}z)^{q-1}$ divides the partial
derivatives is a straightforward computation. For instance, for the first one:

$\partial_{y}[(cy+c^{\prime}z)^{q}Q]=(cy+c^{\prime}z)^{q-1}[cqQ+(cy+c^{\prime}z)\partial_{y}Q].\qed$

######  Lemma 4.2.

Let $R,T,U\in\mathbbm{k}[x_{2},x_{3}]$ be nonzero homogeneous polynomials, and
let $\ell_{1},\ell_{2},\ell_{3}\in\mathbbm{k}[x_{2},x_{3}]$ be pairwise
independent linear forms such that we have a relation

$\ell_{1}^{r}R+\ell_{2}^{t}T+\ell_{3}^{u}U=0,$

for some $r,t,u\in\mathbb{N}$. In particular, the three terms of the sum have
the same degree $d=r+\deg R=t+\deg T=u+\deg U$. Then
$\min\\{r,t,u\\}\leq\left\lfloor\frac{2d+1}{3}\right\rfloor$.

###### Proof.

By a linear change of coordinates we can assume $\ell_{1}=x_{2}$,
$\ell_{2}=cx_{2}+c^{\prime}x_{3}$ and $\ell_{3}=x_{3}$, for some
$c,c^{\prime}\in\mathbbm{k}^{*}$, so that the relation becomes

$x_{2}^{r}R+(cx_{2}+c^{\prime}x_{3})^{t}T+x_{3}^{u}U=0.$

Observe that for an integer $a$, the condition
$a\leq\left\lfloor\frac{2d+1}{3}\right\rfloor$ is equivalent to
$a<\frac{2}{3}(d+1)$. For contradiction, assume $r,t,u\geq\frac{2}{3}(d+1)$.
Let $n$ be the unique integer such that
$\tfrac{1}{3}(d-2)<n\leq\tfrac{1}{3}(d+1)$. Then $2n\leq\frac{2}{3}(d+1)\leq
t$. We have

$\deg R=d-r\leq d-\tfrac{2}{3}(d+1)=\tfrac{1}{3}(d-2)<n,$

so $\partial_{x_{3}}^{n}[x_{2}^{r}R]=x_{2}^{r}\partial_{x_{3}}^{n}R=0$.
Similarly, $\deg U<n$ and $\partial_{x_{2}}^{n}[x_{3}^{u}U]=0$. Taking the
partial derivatives $\partial_{x_{2}}^{n}\partial_{x_{3}}^{n}$ of the
expression $E=x_{2}^{r}R+(cx_{2}+c^{\prime}x_{3})^{t}T+x_{3}^{u}U$, we obtain
$\partial_{x_{2}}^{n}\partial_{x_{3}}^{n}E=\partial_{x_{2}}^{n}\partial_{x_{3}}^{n}[(cx_{2}+c^{\prime}x_{3})^{t}T]$.
Since $2n\leq t$, we can apply $2n$ times Lemma 4.1 to obtain
$\partial_{x_{2}}^{n}\partial_{x_{3}}^{n}[(cx_{2}+c^{\prime}x_{3})^{t}T]\neq
0$, hence $E\neq 0$, contradiction. ∎

We will need the following weighted versions of Lemma 4.2.

######  Lemma 4.3.

Let $p\geq 1$ be an integer, and let $R,T,U\in\mathbbm{k}[x_{2},x_{3}]$ be
nonzero weighted homogeneous polynomials with $x_{2},x_{3}$ of respective
weights $p,1$. Assume that there exist $r,t,u,m\in\mathbb{N}$ such that

$\deg R+rp=\deg T+tp=\deg U+up=m.$

Suppose that for some distinct $c,c^{\prime}\in\mathbbm{k}^{*}$ we have

$x_{2}^{r}R+(x_{2}+cx_{3}^{p})^{t}T+(x_{2}+c^{\prime}x_{3}^{p})^{u}U=0.$

Let $m=\alpha p+\beta$ be the Euclidean division of $m$ by $p$, and assume
$\alpha\geq 2$. Then $\min\\{rp,tp,up\\}\leq\frac{3}{4}m$.

######  Remark 4.4.

In the case $\alpha=r=t=u=1$, up to scaling the only possibility for $R,T,U$
satisfying the identity in Lemma 4.3 is

$\displaystyle R=(c-c^{\prime})x_{3}^{\beta},$ $\displaystyle
T=c^{\prime}x_{3}^{\beta},$ $\displaystyle U=-cx_{3}^{\beta}.$

###### Proof of Lemma 4.3.

There exist (ordinary) homogeneous polynomials
$\widetilde{R},\widetilde{T},\widetilde{U}$ satisfying

$\displaystyle R(x_{2},x_{3})$
$\displaystyle=x_{3}^{\beta}\widetilde{R}(x_{2},x_{3}^{p}),$ $\displaystyle
T(x_{2},x_{3})$ $\displaystyle=x_{3}^{\beta}\widetilde{T}(x_{2},x_{3}^{p}),$
$\displaystyle U(x_{2},x_{3})$
$\displaystyle=x_{3}^{\beta}\widetilde{U}(x_{2},x_{3}^{p}).$

By assumption, we have

$x_{2}^{r}x_{3}^{\beta}\widetilde{R}(x_{2},x_{3}^{p})+(x_{2}+cx_{3}^{p})^{t}x_{3}^{\beta}\widetilde{T}(x_{2},x_{3}^{p})+(x_{2}+c^{\prime}x_{3}^{p})^{u}x_{3}^{\beta}\widetilde{U}(x_{2},x_{3}^{p})=0.$

Dividing by $x_{3}^{\beta}$ and changing variables we get

$x_{2}^{r}\widetilde{R}(x_{2},x_{3})+(x_{2}+cx_{3})^{t}\widetilde{T}(x_{2},x_{3})+(x_{2}+c^{\prime}x_{3})^{u}\widetilde{U}(x_{2},x_{3})=0,$

where each term of the sum is homogeneous of degree
$\frac{m-\beta}{p}=\alpha$. By Lemma 4.2, we have
$\min\\{r,t,u\\}\leq\left\lfloor\frac{2\alpha+1}{3}\right\rfloor$. Observe
that

($\spadesuit$) $\max_{\alpha\geq
2}\frac{\left\lfloor\frac{2\alpha+1}{3}\right\rfloor}{\alpha}=\frac{3}{4},\text{
realised for }\alpha=4.$

We conclude that

$\min\\{rp,tp,up\\}\leq\left\lfloor\frac{2\alpha+1}{3}\right\rfloor
p\leq\frac{\left\lfloor\frac{2\alpha+1}{3}\right\rfloor}{\alpha}\alpha
p\leq\frac{3}{4}m.\qed$

######  Lemma 4.5.

Let $p\geq 1$ be an integer, and let $R,T,U\in\mathbbm{k}[x_{2},x_{3}]$ be
nonzero weighted homogeneous polynomials with $x_{2},x_{3}$ of respective
weights $p,1$. Assume that there exist $r,t,u,m\in\mathbb{N}$ such that

$\deg R+rp=\deg T+tp=\deg U+u=m.$

Suppose that for some $c\in\mathbbm{k}^{*}$ we have

$x_{2}^{r}R+(x_{2}+cx_{3}^{p})^{t}T+x_{3}^{u}U=0.$

Let $m=\alpha p+\beta$ be the Euclidean division of $m$ by $p$, and assume
$\alpha\geq 2$. Then $\min\\{rp,tp,u\\}<\frac{4}{5}m$.

######  Remark 4.6.

In the case $\alpha=r=t=1$, and $u=m$, up to scaling the only possibility for
$R,T,U$ satisfying the identity in Lemma 4.5 is

$\displaystyle R=-x_{3}^{\beta},$ $\displaystyle T=x_{3}^{\beta},$
$\displaystyle U=-c.$

###### Proof of Lemma 4.5.

We can assume that $x_{3}$ does not divide $U$. Write the Euclidean division
$m-u=\alpha^{\prime}p+\beta^{\prime}$. Then there exist (ordinary) homogeneous
polynomials $\widetilde{R},\widetilde{T},\widetilde{U}$ satisfying

$\displaystyle R(x_{2},x_{3})$
$\displaystyle=x_{3}^{\beta}\widetilde{R}(x_{2},x_{3}^{p}),$ $\displaystyle
T(x_{2},x_{3})$ $\displaystyle=x_{3}^{\beta}\widetilde{T}(x_{2},x_{3}^{p}),$
$\displaystyle U(x_{2},x_{3})$
$\displaystyle=x_{3}^{\beta^{\prime}}\widetilde{U}(x_{2},x_{3}^{p}).$

By assumption, we have

$x_{2}^{r}x_{3}^{\beta}\widetilde{R}(x_{2},x_{3}^{p})+(x_{2}+cx_{3}^{p})^{t}x_{3}^{\beta}\widetilde{T}(x_{2},x_{3}^{p})+x_{3}^{u}x_{3}^{\beta^{\prime}}\widetilde{U}(x_{2},x_{3}^{p})=0.$

We see that $u+\beta^{\prime}-\beta$ must be a multiple of $p$, and writing
$u+\beta^{\prime}-\beta=pu^{\prime}$ with $u^{\prime}\geq 0$, we get

$x_{2}^{r}\widetilde{R}(x_{2},x_{3})+(x_{2}+cx_{3})^{t}\widetilde{T}(x_{2},x_{3})+x_{3}^{u^{\prime}}\widetilde{U}(x_{2},x_{3})=0.$

By Lemma 4.2, we have
$\min\\{r,t,u^{\prime}\\}\leq\left\lfloor\frac{2\alpha+1}{3}\right\rfloor$. By
definition, we have $u\leq pu^{\prime}+\beta,$ and so multiplying by $p$ we
obtain

$\min\\{rp,tp,u\\}\leq\left\lfloor\frac{2\alpha+1}{3}\right\rfloor p+\beta.$

Observe that for any real numbers $0<c_{1}<c_{2}$, we have
$c_{1}p+\beta<\frac{c_{1}+1}{c_{2}+1}(c_{2}p+\beta)$. Applying this
observation with $c_{1}=\left\lfloor\frac{2\alpha+1}{3}\right\rfloor$ and
$c_{2}=\alpha$, we obtain

$\min\\{rp,tp,u\\}\leq\left\lfloor\frac{2\alpha+1}{3}\right\rfloor
p+\beta<C(\alpha p+\beta)=Cm$

for

$C=\max_{\alpha\geq
2}\frac{\left\lfloor\frac{2\alpha+1}{3}\right\rfloor+1}{\alpha+1}=\frac{4}{5}.$

(The maximum is realised for $\alpha=4.$) ∎

##  5\. Links

In this section we will study the link of each vertex $\nu$ in $\mathbf{X}$,
which is the following metric graph [BB]*page 176. Namely, the vertices of the
link correspond to the edges of $\mathbf{X}$ containing $\nu$ and the edges of
the link correspond to the corners of the $2$-cells of $\mathbf{X}$ containing
$\nu$. The length of each edge of the link is the angle in $\mathbf{X}$ at the
corresponding corner. Note that the link is not finite but has finitely many
possible edge lengths.

Since $\mathrm{Tame}(\mathbbm{k}^{3})$ maps any vertex into
$\mathbf{E}^{+}_{\mathrm{id}}$, we will assume
$\nu\in\mathbf{E}^{+}_{\mathrm{id}}$. For example, by [LP2]*lem 3.4 the link
of $\nu_{\mathrm{id},[1,1,1]}$ in $\mathbf{X}$ can be identified with the
Bruhat–Tits building of $\textbf{GL}_{3}(\mathbbm{k})$. Consequently, each
cycle in the link of length $<2\pi+\frac{2\pi}{3}$ has length exactly $2\pi$
and is the link of $\nu$ in $\mathbf{E}_{f}$ for $f\in A$.

We will now perform a similar classification for other weights. In [LP2]*prop
7.9, 7.10 and 7.12 we proved that the link of $\nu$ is $\mathrm{CAT}(1)$, i.e.
all of its cycles have length $\geq 2\pi$. Our aim is to classify all of its
cycles of length $<2\pi+\varepsilon_{0}$, for some small $\varepsilon_{0}>0$
independent of $\nu$.

###  5.1. Links of vertices with weight $[m,m,1]$

Let $\nu=\nu_{\mathrm{id},[m,m,1]}$ for an integer $m\geq 2$. Let $\Gamma$ be
the link of $\nu$ in $\mathbf{X}$. Let $I$ be the link of $[m,m,1]$ in
$\nabla^{+}$, which is a simplicial path consisting of $3$ edges of length
$\frac{\pi}{3}$ [LP2]*rem 5.2, see Figure 7. We keep the notation $\rho_{+}$
for the induced map from $\Gamma$ to $I$. Let $s,q\in I$ be the directions
towards $[1,1,1]$ and towards $[1,1,0]$, respectively. For
$f\in\mathrm{Stab}(\nu)$, let $\Gamma_{f},\Gamma^{+}_{f}\subset\Gamma$ be the
cycle and the path, respectively, that are the links of $\nu$ in
$\mathbf{E}_{f},\mathbf{E}_{f}^{+}$. Let $s_{f},q_{f}\in\Gamma^{+}_{f}$ be the
preimages under $\rho_{+}$ of $s,q,$ respectively.

Figure 7.

######  Proposition 5.1.

Each cycle in $\Gamma$ of length $<2\pi+\frac{2\pi}{3}$ has length $2\pi$ and

  * •

equals $\Gamma_{f}$, or

  * •

is the closure of
$(\Gamma_{f}\cup\Gamma_{g})\setminus(\Gamma_{f}\cap\Gamma_{g})$, or

  * •

intersects $\Gamma_{f},\Gamma_{g},\Gamma_{h}$ along length $\frac{2\pi}{3}$
paths with midpoints $s_{f},s_{g},s_{h}$.

In the proof we will need the following.

######  Lemma 5.2 ([LP2]*lem 7.7(iii)).

Each vertex $u\in\Gamma\setminus\rho_{+}^{-1}\\{q,s\\}$ is contained in
exactly one edge $e$ such that $\rho_{+}(e)$ separates $\rho_{+}(u)$ from $q$
in $I$. Moreover, if $u\in\Gamma^{+}_{f}$ for some $f\in\mathrm{Stab}(\nu)$,
then $e\subset\Gamma^{+}_{f}$.

######  Lemma 5.3 ([LP2]*lem 7.8).

Suppose that $s_{f}=s_{f^{\prime}}$ for some
$f,f^{\prime}\in\mathrm{Stab}(\nu)$. Then there exists
$f^{\prime\prime}\in\mathrm{Stab}(\nu)$ with
$\Gamma^{+}_{f},\Gamma^{+}_{f^{\prime}}\subset\Gamma_{f^{\prime\prime}}$.

###### Proof of Proposition 5.1.

Since $\Gamma$ is bipartite and all of its edges have length $\frac{\pi}{3}$,
we only need to classify cycles $\gamma$ of length $2\pi$, each of which
consists of six edges. We orient each edge of $\gamma$, so that its image in
$I$ under $\rho_{+}$ is oriented from $q$ to $s$. By Lemma 5.2, $\gamma$
decomposes into $j=2,4,$ or $6$ oriented paths each of which lies in an
$\Gamma^{+}_{f}$ and ends with $s_{f}$. By Lemma 5.3, the cases $j=2,4,6,$
correspond to the respective bullet points of the proposition. ∎

###  5.2. Links of vertices with weight $[m,1,1]$

Let $m\geq 2$ be an integer.

######  Remark 5.4.

The weight $[\alpha]=[m,1,1]$ lies on exactly $m+2$ admissible lines, namely
$\alpha_{2}=\alpha_{3}$ and $\alpha_{1}=a\alpha_{2}+(m-a)\alpha_{3}$, where
$a=0,\dots,m$. The latter lines contain points $[a,1,0]$, and are principal
exactly for $a=0$ and $a=m$. See Figure 8.

Figure 8.

Let $\nu=\nu_{\mathrm{id},[\alpha]}$ with $[\alpha]=[m,1,1]$. Let $\Gamma_{0}$
be the link of $\nu$ in $\mathbf{X}$. We keep the notation $\rho_{+}$ for the
induced map from $\Gamma_{0}$ to the link $I$ of $[\alpha]$ in $\nabla^{+}$,
which is a simplicial path. Let $s,q\in I$ be the directions towards $[1,1,1]$
and towards $[1,0,0]$, respectively. Let $e^{s}\subset I$ be the edge
containing $s$. For $f\in\mathrm{Stab}(\nu)$, let
$\Gamma_{f},\Gamma^{+}_{f}\subset\Gamma_{0}$ be the cycle and the path,
respectively, that are the links of $\nu$ in
$\mathbf{E}_{f},\mathbf{E}_{f}^{+}$. Let
$s_{f},q_{f},e^{s}_{f}\subset\Gamma^{+}_{f}$ be the preimages under $\rho_{+}$
of $s,q,e^{s},$ respectively. Lemmas 5.2 and 5.3 remain true in this setting,
with $\Gamma$ replaced by $\Gamma_{0}$ (see the first paragraph of the proof
of [LP2]*prop 7.10).

######  Proposition 5.5.

There is $\varepsilon_{0}>0$ independent of $m$ such that each cycle in
$\Gamma_{0}$ of length $<2\pi+\varepsilon_{0}$

  * •

equals $\Gamma_{f}$, or

  * •

is the closure of
$(\Gamma_{f}\cup\Gamma_{g})\setminus(\Gamma_{f}\cap\Gamma_{g})$.

Note that unlike in Proposition 5.1, here in the second bullet point we can
have cycles of length $>2\pi$ arbitrary close to $2\pi$.

In Example 5.9 we will see that setting $\varepsilon_{0}=\frac{2\pi}{3}$ does
not suffice.

In the proof, we will need the following. For $f,g\in\mathrm{Stab}(\nu)$, we
write $f\sim_{\nu}g$ if $\Gamma^{+}_{f}=\Gamma^{+}_{g}$. Let
$\mathrm{Stab}^{\prime}(\nu)<\mathrm{Stab}(\nu)$ be the subgroup of elements
of form

$(x_{1}+P(x_{2},x_{3}),bx_{2}+cx_{3}+d,b^{\prime}x_{2}+c^{\prime}x_{3}+d^{\prime}),$

which differs from $\mathrm{Stab}(\nu)$ only by requiring the coefficient at
$x_{1}$ to be equal $1$. Let $N_{\alpha}<\mathrm{Stab}^{\prime}(\nu)$ be the
subgroup of elements of form

$(x_{1}+P(x_{2},x_{3}),x_{2}+d,x_{3}+d^{\prime})$

with $\deg P<m$.

######  Lemma 5.6.

  1. (i)

For each $f\in\mathrm{Stab}(\nu)$, there is $g\in\mathrm{Stab}^{\prime}(\nu)$
with $f\sim_{\nu}g$.

  2. (ii)

Let $h\in N_{\alpha}$. Then $h\sim_{\nu}\mathrm{id}$.

  3. (iii)

$N_{\alpha}$ is normal in $\mathrm{Stab}^{\prime}(\nu)$.

###### Proof.

Part (i) follows from the description of $\mathrm{Stab}(\nu)$ in Proposition
3.4 and the observation [LP2]*cor 3.3 that for $l=(ax_{1},x_{2},x_{3})$ we
have $l\sim_{\nu}\mathrm{id}$. Part (ii) follows from Corollary 3.7. Part
(iii) is an easy computation. ∎

######  Lemma 5.7.

Let $f,g\in\mathrm{Stab}^{\prime}(\nu)$. If $\Gamma^{+}_{f}\neq\Gamma^{+}_{g}$
and $\Gamma^{+}_{f}\cap\Gamma^{+}_{g}\neq\\{q_{f}\\}$, then either

  1. (i)

$\Gamma^{+}_{f}\cap\Gamma^{+}_{g}=\\{s_{f},q_{f}\\}$ and $f^{-1}g\in
hN_{\alpha}$ for some
$h=(x_{1},bx_{2}+cx_{3},b^{\prime}x_{2}+c^{\prime}x_{3})$ with $b^{\prime}\neq
0$, or

  2. (ii)

$\Gamma^{+}_{f}\cap\Gamma^{+}_{g}$ is the path from $q_{f}$ to the direction
mapping under $\rho_{+}$ to the direction towards $[a,1,0]$, where $0\leq
a\leq m$ and $f^{-1}g\in hN_{\alpha}$ for some

$h=(x_{1}+\sum_{i=0}^{a}c_{i}x_{2}^{i}x_{3}^{m-i},bx_{2}+cx_{3},c^{\prime}x_{3})$

with $c_{a}\neq 0$.

###### Proof.

After composing $f$ and $g$ with $f^{-1}$, we can assume $f=\mathrm{id}$. Let
$g=(x_{1}+P(x_{2},x_{3}),bx_{2}+cx_{3}+d,b^{\prime}x_{2}+c^{\prime}x_{3}+d^{\prime})$.
By Lemma 5.6(ii,iii), after possibly composing with an element of
$N_{\alpha}$, we can assume that $P$ is homogeneous of degree $m$, and
$d=d^{\prime}=0$. If $P$ is trivial, then by Corollary 3.7 we have (i) for
$b^{\prime}\neq 0$ or $\Gamma^{+}_{f}=\Gamma^{+}_{g}$ for $b^{\prime}=0$.
Analogously, if $P$ is nontrivial, then by Corollary 3.7 we have (ii) for
$b^{\prime}=0$ or $\Gamma^{+}_{f}\cap\Gamma^{+}_{g}=\\{q_{f}\\}$ for
$b^{\prime}\neq 0$. ∎

######  Lemma 5.8.

Let $0\leq\delta\leq 1$. Let $[\alpha],[\alpha^{\prime}]\in\nabla$ and let
$c,c^{\prime}$ be the projective line segments joining
$[\alpha],[\alpha^{\prime}]$ to
$[\delta\frac{\alpha_{1}}{\alpha_{2}},1,0],[\delta\frac{\alpha^{\prime}_{1}}{\alpha^{\prime}_{2}},1,0]$,
respectively. Then the translation of $(\nabla,|\cdot|)$ mapping $[\alpha]$ to
$[\alpha^{\prime}]$ maps $c$ to $c^{\prime}$.

###### Proof.

In the proof of [LP2]*lem 5.9 we showed that the reflections in the lines
parallel to the principal lines in $(\nabla,|\cdot|)$ extend to projective
maps of $P\mathbb{R}^{3}$ (preserving $\nabla$). Thus the translation $\tau$
mapping $[\alpha]$ to $[\alpha^{\prime}]$ maps the projective line through
$[\alpha]$ and $[0,0,1]$ to the projective line through $[\alpha^{\prime}]$
and $[0,0,1]$ and hence it maps $[\frac{\alpha_{1}}{\alpha_{2}},1,0]$ to
$[\frac{\alpha^{\prime}_{1}}{\alpha^{\prime}_{2}},1,0]$. Since $\tau$
preserves $[0,1,0]$ and $[1,0,0]$, as well as the cross-ratio on the
projective line through $[0,1,0]$ and $[1,0,0]$, we conclude that $\tau$ maps
$[\delta\frac{\alpha_{1}}{\alpha_{2}},1,0]$ to
$[\delta\frac{\alpha^{\prime}_{1}}{\alpha^{\prime}_{2}},1,0]$. ∎

###### Proof of Proposition 5.5.

Let $\gamma\subset\Gamma_{0}$ be a cycle of length $<2\pi+\frac{2\pi}{3}$. We
orient each edge of $\gamma$, so that its image in $I$ under $\rho_{+}$ is
oriented from $q$ to $s$. By Lemma 5.2, $\gamma$ decomposes into an even
number $j$ of oriented paths each of which lies in an $\Gamma^{+}_{f}$ and
ends with $s_{f}$. By Remark 5.4, each such path contributes at least
$\frac{\pi}{3}$ to the length of $\gamma$. Since $\gamma$ has length
$<2\pi+\frac{2\pi}{3}$, we have $j<8$ and so $j=2,4,$ or $6$. The cases
$j=2,4$ imply the first two bullet points of the proposition by Lemma 5.3.

Suppose now $j=6$, with
$\gamma=\gamma_{0}\cdot\gamma_{1}^{-1}\cdot\gamma_{2}\cdots\gamma_{5}^{-1}$,
where each $\gamma_{i}$ lies in $\Gamma^{+}_{f_{i}}$, for some
$f_{i}\in\mathrm{Stab}(\nu)$. Note that these paths contribute already $2\pi$
to the length of $\gamma$, and so for a contradiction we need to find
$\varepsilon_{0}$ bounding the excess contribution from below. By Remark 5.4,
points
$\rho_{+}(\gamma_{1}\cap\gamma_{2}),\rho_{+}(\gamma_{3}\cap\gamma_{4}),\rho_{+}(\gamma_{5}\cap\gamma_{0})$
correspond to directions towards
$[a,1,0],[a^{\prime},1,0],[a^{\prime\prime},1,0]$ for some $0\leq
a,a^{\prime},a^{\prime\prime}\leq m$. We need to find $\varepsilon_{0}$
independent of $m$ bounding below at least one of the angles at $[m,1,1]$
between the admissible lines in these directions and the direction towards
$[0,1,0]$. To do that, we will find $\delta>0$ independent of $m$ such that
$\max\\{a,a^{\prime},a^{\prime\prime}\\}\geq\delta m$. We can then assign
$\varepsilon_{0}$ to be the angle at $[m,1,1]$ between the projective lines in
the directions towards $[\delta m,1,0]$ and $[0,1,0]$, which is independent of
$m$ by Lemma 5.8.

By Lemma 5.6(i), we can suppose that all $f_{i}$ belong to
$\mathrm{Stab}^{\prime}(\nu)$. By Lemma 5.7, we have $f_{0}^{-1}f_{1}\in
h_{0}N_{\alpha},\ldots,f_{5}^{-1}f_{0}\in h_{5}N_{\alpha}$, where the second
and third components of $h_{i}=(x_{1}+P_{i},\cdot,\cdot)$ form a linear
automorphism in $x_{2},x_{3}$. Furthermore,

  * •

for $i=1,3,5,$ the polynomial $P_{i}$ is homogeneous in $x_{2},x_{3}$ of
degree $m$, and the third component of $h_{i}$ does not depend on $x_{2}$,
while

  * •

for $i=0,2,4,$ we have $P_{i}=0$ and the third component of $h_{i}$ depends on
$x_{2}$.

By Lemma 5.6(iii), the group $N_{\alpha}$ is normal in
$\mathrm{Stab}^{\prime}(\nu)$ and so by Lemma 5.6(ii) we have $h_{0}\cdots
h_{5}\sim_{\nu}\mathrm{id}$. Thus the first component of $h_{0}\cdots h_{5}$
equals $x_{1}+P$ for $P$ a polynomial in $x_{2},x_{3}$ of degree $<m$. Since
$P_{i}$ are homogeneous, we have $P=0$. By Lemma 5.7, we have that
$P_{1},P_{3},P_{5}$ are divisible by
$x_{3}^{m-a},x_{3}^{m-a^{\prime}},x_{3}^{m-a^{\prime\prime}}$, respectively.
Since the first component of $h_{0}\cdots h_{5}$ equals $x_{1}$, we obtain

$P_{1}(\cdot,\ell_{1})+P_{3}(\cdot,\ell_{2})+P_{5}(x_{2},x_{3})=0,$

where $\ell_{1},\ell_{2},x_{3}$ are pairwise independent linear forms in
$\mathbbm{k}[x_{2},x_{3}]$. Now we apply Lemma 4.2 with

$r=m-a,\,\ell_{1}^{r}R=P_{1},\,t=m-a^{\prime},\,\ell_{2}^{t}T=P_{3},\,u=m-a^{\prime\prime},\,x_{3}^{u}Q=P_{5}.$

Thus one of $a,a^{\prime},a^{\prime\prime}$ is $\geq
m-\big{\lfloor}\frac{2m+1}{3}\big{\rfloor}$, and so by ($\spadesuit$ ‣ 4) in
Section 4 it is $\geq\frac{m}{4}$. We can then set $\delta=\frac{1}{4}$. ∎

######  Example 5.9.

In the notation of the proof of Proposition 5.5, let $m=2$, let

$h_{4}=(x_{1},x_{3},x_{2}),\,h_{2}=(x_{1},x_{3},x_{2}+x_{3}),\,h_{0}=(x_{1},x_{2},x_{3}-x_{2}),$
$P_{5}=-x_{3}^{2},\,P_{3}=x_{3}^{2},\,P_{1}=x_{3}(x_{3}-2x_{2}).$

Then the identity $x_{2}^{2}-x_{3}^{2}+(x_{2}+x_{3})(x_{3}-x_{2})$ implies
$h_{0}\cdots h_{5}=\mathrm{id}$. The paths $\gamma_{i}$ for $i=1,2,3,4$ are of
length $\frac{\pi}{3}$, while the paths $\gamma_{0},\gamma_{5}$ are of length
$\frac{\pi}{2}$ [LP2]*lem 5.9. Thus $\gamma$ is a cycle in $\Gamma$ of length
$2\pi+\frac{\pi}{3}$.

######  Remark 5.10.

Let $L_{0},L_{1}$ be any non-principal admissible lines in $\nabla$. Then
there is an isometry $(\nabla,|\cdot|)$ mapping $L_{0}$ to $L_{1}$. Indeed, we
can assume without loss of generality that the projective intervals
$L_{0},L_{1}$ have endpoints $x_{0}^{+},x_{1}^{+}$ in the open projective
interval from $[1,0,0]$ to $[0,1,0]$ and $x_{0}^{-},x_{1}^{-}$ in the open
projective interval from $[1,0,0]$ to $[0,0,1]$. Let $x_{i}$ be the
intersection point of the projective line through $[0,0,1]$ and $x_{i}^{+}$
and the projective line through $[0,1,0]$ and $x_{i}^{-}$. Then as in the
proof of Lemma 5.8, the translation of $(\nabla,|\cdot|)$ mapping $x_{0}$ to
$x_{1}$ maps $x^{\pm}_{0}$ to $x^{\pm}_{1}$, and hence maps $L_{0}$ to
$L_{1}$.

###  5.3. Links of vertices with weight in $\mathrm{int}\,\nabla^{+}$

Figure 9.

Let $\nu=\nu_{[\alpha]}$ with
$[\alpha]=[\alpha_{1},\alpha_{2},\alpha_{3}]\in\mathrm{int}\,\nabla^{+}$. Let
$\Gamma_{0}$ be the link of $\nu$ in $\mathbf{X}$. We keep the notation
$\rho_{+}$ for the induced map from $\Gamma_{0}$ to the cycle $S$ that is the
link of $[\alpha]$ in $\nabla^{+}$. Let $s,q\in S$ be the directions towards
$[0,\alpha_{2},\alpha_{3}]$ or towards $[1,0,0]$, respectively. Let $I\subset
S$ be the path defined by
$\frac{\alpha^{\prime}_{2}}{\alpha^{\prime}_{3}}\geq\frac{\alpha_{2}}{\alpha_{3}}$.
If $s$ is a vertex of $S$, then let $e^{s}\subset S$ be the edge containing
$s$ lying inside $I$. See Figure 9. For $f\in\mathrm{Stab}(\nu)$, let
$\Gamma_{f}\subset\Gamma_{0}$ be the cycle that is the link of $\nu$ in
$\mathbf{E}_{f}$, and let $s_{f},q_{f},e^{s}_{f}\subset\Gamma_{f}$ be the
preimages under $\rho_{+}$ of $s,q,e^{s},$ respectively.

######  Example 5.11.

  1. (a)

Let

$h_{0}=(x_{1},x_{2}-cx_{3}^{p},x_{3}),\ h_{2}=(x_{1},x_{2}+cx_{3}^{p},x_{3}),$
$h_{1}=(x_{1}+x_{2}^{a^{\prime}}P^{\prime}(x_{2},x_{3}),x_{2},x_{3}),\
h_{3}=(x_{1}+x_{2}^{a}P(x_{2},x_{3}),x_{2},x_{3}),$

where $c,P,P^{\prime}\neq 0$. Suppose that we have $h_{0}\cdots
h_{3}=\mathrm{id}$, and for $i=0,\ldots,3$, let $f_{i}=h_{0}\cdots h_{i}$,
where $f_{3}=\mathrm{id}$. By Corollary 3.7, we obtain a cycle in the link of
$\nu$ decomposing into paths in $\Gamma_{f_{i}}$, examples of which are
illustrated in Figure 10(a), left and right. We proved in [LP2]*page 40 that
we have $p(a+a^{\prime})\leq m$ and consequently the cycle has length $\geq
2\pi$ [LP2]*lem 5.9. Equality cases include, for example for $m=10p$, the
configurations where $c=1$, and $a=10,P=1,a^{\prime}=0$ (illustrated on the
right), or $a=3,P=(x_{2}+x_{3}^{p})^{7},a^{\prime}=7,$ (illustrated on the
left).

  2. (b)

Let $m=p+\beta$ with $0<\beta<p$, and let $\alpha=(m,p,1)$. Let

$h_{0}=(x_{1},x_{2}-c^{\prime}x_{3}^{p},x_{3}),\
h_{2}=(x_{1},x_{2}+(c^{\prime}-c)x_{3}^{p},x_{3}),\
h_{4}=(x_{1},x_{2}+cx_{3}^{p},x_{3}),$
$h_{1}=(x_{1}-bcx_{2}x_{3}^{\beta},x_{2},x_{3}),\
h_{3}=(x_{1}+bc^{\prime}x_{2}x_{3}^{\beta},x_{2},x_{3}),\
h_{5}=(x_{1}+b(c-c^{\prime})x_{2}x_{3}^{\beta},x_{2},x_{3}),$

where $c,c^{\prime},c^{\prime}-c,b\neq 0$. Note that we have $h_{0}\cdots
h_{5}=\mathrm{id}$. For $i=0,\ldots,5$, let $f_{i}=h_{0}\cdots h_{i}$, where
$f_{5}=\mathrm{id}$. By Corollary 3.7, we obtain a cycle in the link of $\nu$
decomposing into paths in $\Gamma_{f_{i}}$ illustrated in Figure 10(b).

  3. (c)

As in (b), let $m=p+\beta$ with $0<\beta<p$, and let $\alpha=(m,p,1)$. Let

$h_{0}=(x_{1}-bcx_{3}^{m},x_{2}-cx_{3}^{p},x_{3}),\
h_{2}=(x_{1},x_{2}+cx_{3}^{p},x_{3}),$
$h_{1}=(x_{1}+bx_{2}x_{3}^{\beta},x_{2},x_{3}),\
h_{3}=(x_{1}-bx_{2}x_{3}^{\beta},x_{2},x_{3}),$

where $c,b\neq 0$. Note that we have $h_{0}\cdots h_{3}=\mathrm{id}$. For
$i=0,\ldots,3$, let $f_{i}=h_{0}\cdots h_{i}$, where $f_{3}=\mathrm{id}$. By
Corollary 3.7, we obtain a cycle in the link of $\nu$ decomposing into paths
in $\Gamma_{f_{i}}$ illustrated in Figure 10(c).

######  Proposition 5.12.

There is $\varepsilon_{0}>0$ independent of $\alpha$ such that each cycle in
$\Gamma_{0}$ of length $<2\pi+\varepsilon_{0}$

  * •

equals $\Gamma_{f}$, or

  * •

is the closure of
$(\Gamma_{f}\cup\Gamma_{g})\setminus(\Gamma_{f}\cap\Gamma_{g})$, or

  * •

is a cycle in Example 5.11(a), or

  * •

is the cycle in Example 5.11(b), or

  * •

is the cycle in Example 5.11(c),

up to a translation by an element in $\mathrm{Stab}(\nu)$ in the last three
bullet points.

Figure 10.

###### Proof.

Let $\gamma\subset\Gamma_{0}$ be a cycle of length $<2\pi+\frac{2\pi}{3}$. We
orient each edge of $\gamma$ with interior not containing an $s_{f}$ or a
$q_{f}$, so that its image in $S$ under $\rho_{+}$ is oriented from $q$ to
$s$. We divide each edge of $\gamma$ containing an $s_{f}$ (respectively,
$q_{f}$) into two edges oriented towards $s_{f}$ (respectively, away from
$q_{f}$). As in the previous subsections, $\gamma$ decomposes into an even
number $j$ of oriented paths each of which ends with an $s_{f}$. Each such
path is contained in an $\Gamma_{g}$ and contributes at least $\frac{\pi}{3}$
to the length of $\gamma$. Since $\gamma$ has length $<2\pi+\frac{2}{3}\pi$,
we have $j<8$ and so $j=2,4,$ or $6$. The case $j=2$ implies one of the first
two bullet points of the proposition. In the remaining cases we have that
$\rho_{+}(\gamma)$ is disjoint from $q$.

Consider now the case $j=4$, with the paths of $\gamma$ oriented towards
$s_{f}$ and $s_{g}$. In the second paragraph of the proof of [LP2]*prop 7.12
we observed that if $s$ is a vertex, then $e^{s}_{f}$ is the only edge of
$\Gamma_{0}$ containing $s_{f}$ whose image under $\rho_{+}$ lies in $I$. Thus
if $\gamma$ contains $e^{s}_{f}$, then the restriction of $\rho_{+}$ to
$\gamma$ is locally injective at $s_{f}$. Since $\rho_{+}(\gamma)$ is disjoint
from $q$, it follows that $\gamma$ contains also $e_{g}^{s}$. This implies the
second bullet point of the proposition. The same conclusion applies when $s$
is not a vertex. If $\gamma$ contains neither $e^{s}_{f}$ nor $e_{g}^{s}$,
then it is obtained by concatenating paths in four distinct $\Gamma_{f}$, each
of which contains $s_{f}$ as an endpoint. By [LP2]*page 39, this brings us to
Example 5.11(a).

Finally, we suppose $j=6$, with the paths of $\gamma$ oriented towards
$s_{f},s_{g},$ and $s_{h}$. Note that the six paths contribute already $2\pi$
to the length of $\gamma$, and so for a contradiction we need to find
$\varepsilon_{0}$ bounding the excess contribution from below. Suppose first
that $\gamma$ contains none of $e^{s}_{f},e^{s}_{g},e^{s}_{h}$, as in Figure
10(b). Since $s$ is a vertex, by [LP2]*rem 4.9 we have $[\alpha]=[m,p,1]$ for
some integers $m>p>1$. We have a decomposition
$\gamma=\gamma_{0}\cdot\gamma_{1}^{-1}\cdots\gamma_{5}^{-1}$, where each
$\gamma_{i}$ lies in $\Gamma_{f_{i}}$, for some $f_{i}\in\mathrm{Stab}(\nu)$.
By [LP2]*rem 4.10, points
$\rho_{+}(\gamma_{1}\cap\gamma_{2}),\rho_{+}(\gamma_{3}\cap\gamma_{4}),\rho_{+}(\gamma_{5}\cap\gamma_{0})$
correspond to directions towards
$[m-pa,0,1],[m-pa^{\prime},0,1],[m-pa^{\prime\prime},0,1]$ for some $0\leq
a,a^{\prime},a^{\prime\prime}\leq\big{\lfloor}\frac{m}{p}\big{\rfloor}.$

For $\frac{m}{p}\geq 2$, we will find $\varepsilon_{0}$ bounding below at
least one of the angles between the admissible lines in these directions and
the direction towards $[0,0,1]$. These angles are equal to the opposite ones:
the angles between the admissible lines in the directions towards
$[a,1,0],[a^{\prime},1,0],[a^{\prime\prime},1,0]$ and the direction towards
$[m,p,0]$. Thus by Lemma 5.8, we will find appropriate $\varepsilon_{0}$ as
long as there exists $\delta<1$ independent of $m$ such that
$\min\\{a,a^{\prime},a^{\prime\prime}\\}\leq\delta\frac{m}{p}$.

By [LP2]*lem 4.11(i), we can suppose that all $f_{i}$ lie in $M_{\alpha}$. By
[LP2]*lem 4.12, we have
$f_{0}^{-1}f_{1}\sim_{\nu}h_{0},\ldots,f_{5}^{-1}f_{0}\sim_{\nu}h_{5}$, where
$f\sim_{\nu}g$ means $\Gamma_{f}=\Gamma_{g}$, and $h_{i}$ have the following
form:

$h_{0}=(x_{1},x_{2}-c^{\prime}x_{3}^{p},x_{3}),\
h_{2}=(x_{1},x_{2}+c^{\prime\prime}x_{3}^{p},x_{3}),\
h_{4}=(x_{1},x_{2}+cx_{3}^{p},x_{3}),$ $h_{1}=(x_{1}+P_{1},x_{2},x_{3}),\
h_{3}=(x_{1}+P_{3},x_{2},x_{3}),\ h_{5}=(x_{1}+P_{5},x_{2},x_{3}).$

Here $c,c^{\prime},c^{\prime\prime}\neq 0$ and $P_{i}(x_{2},x_{3})$ are
homogeneous polynomials of weighted degree $m$ with $x_{2},x_{3}$ of weights
$p,1$. By [LP2]*lem 4.11(ii,iii), the elements $\sim_{\nu}\mathrm{id}$ of
$M_{\alpha}$ are exactly the elements of $M_{\alpha}$ of form
$(x_{1}+P(x_{2},x_{3}),x_{2}+Q(x_{3}),x_{3})$, where $-\nu(P)<m,\deg
Q=-\nu(Q)<p$, and they form a normal subgroup $N_{\alpha}<M_{\alpha}$.
Consequently, we have $h_{0}\cdots h_{5}\sim_{\nu}\mathrm{id}$ and so
$h_{0}\cdots h_{5}$ has the above form. Since $P_{i}$ are homogeneous of
degree $m$, $P$ is also homogeneous of degree $m$ and so we have
$c^{\prime\prime}=c^{\prime}-c,P=0$ and

$P_{1}(x_{2}+c^{\prime}x_{3}^{p},x_{3})+P_{3}(x_{2}+cx_{3}^{p},x_{3})+P_{5}(x_{2},x_{3})=0.$

By [LP2]*lem 4.12(i), we have that $P_{1},P_{3},P_{5}$ are divisible by
$x_{2}^{a},x_{2}^{a^{\prime}},x_{2}^{a^{\prime\prime}}$, respectively. Thus we
can apply Lemma 4.3 with $r=a^{\prime\prime},t=a^{\prime},u=a$, and set
$\delta=\frac{3}{4}$.

For $\frac{m}{p}<2,$ if one of $a,a^{\prime},a^{\prime\prime}$ equals $0$,
then $\delta=\frac{3}{4}$ works as well. Otherwise, we have
$a=a^{\prime}=a^{\prime\prime}=1$, which by Remark 4.4 brings us exactly to
Example 5.11(b).

Since $\rho_{+}(\gamma)$ is disjoint from $q$, it passes through $e^{s}$ an
even number of times. Thus it remains to consider the case where $\gamma$
contains $e^{s}_{f},e^{s}_{g}$, but not $e^{s}_{h}$, as in Figure 10(c). Then
we can assume
$\gamma_{0}\cdot\gamma_{1}^{-1}\subset\Gamma_{f_{0}},\gamma_{2}\cdot\gamma_{3}^{-1}\subset\Gamma_{f_{1}},\gamma_{4}\subset\Gamma_{f_{2}},\gamma_{5}\subset\Gamma_{f_{3}}$.
By [LP2]*rem 4.10, points
$\rho_{+}(\gamma_{1}\cap\gamma_{2}),\rho_{+}(\gamma_{3}\cap\gamma_{4}),\rho_{+}(\gamma_{5}\cap\gamma_{0})$
correspond to the directions towards
$[a,1,0],[m-pa^{\prime},0,1],[m-pa^{\prime\prime},0,1]$ for some $0\leq
a,a^{\prime},a^{\prime\prime}\leq\big{\lfloor}\frac{m}{p}\big{\rfloor}$. By
Lemma 5.8, for $\frac{m}{p}\geq 2$ we will find appropriate $\varepsilon_{0}$
as long as there exists $\delta<1$ independent of $m$ such that
$\min\\{\frac{m}{p}-a,a^{\prime},a^{\prime\prime}\\}\leq\delta\frac{m}{p}$.

By [LP2]*lem 4.12, we have
$f_{0}^{-1}f_{1}\sim_{\nu}h_{0},\ldots,f_{3}^{-1}f_{0}\sim_{\nu}h_{3}$, where

$h_{0}=(x_{1}+P_{0},x_{2}+c^{\prime}x_{3}^{p},x_{3}),\
h_{2}=(x_{1},x_{2}+cx_{3}^{p},x_{3}),$ $h_{1}=(x_{1}+P_{1},x_{2},x_{3}),\
h_{3}=(x_{1}+P_{3},x_{2},x_{3}),$

where $c,c^{\prime}\neq 0$ and $P_{i}(x_{2},x_{3})$ are homogeneous
polynomials of weighted degree $m$ with $x_{2},x_{3}$ of weights $p,1$. The
identity $h_{0}\cdots h_{3}\sim_{\nu}\mathrm{id}$ yields $c^{\prime}=-c$ and

$P_{0}(x_{2}+cx_{3}^{p},x_{3})+P_{1}(x_{2}+cx_{3}^{p},x_{3})+P_{3}(x_{2},x_{3})=0.$

By [LP2]*lem 4.12(i,iii) we have that $P_{0},P_{1},P_{3}$ are divisible by
$x_{3}^{m-pa},x_{2}^{a^{\prime}},x_{2}^{a^{\prime\prime}}$, respectively. Thus
we can apply Lemma 4.5 with $r=a^{\prime\prime},t=a^{\prime},u=m-pa$, and set
$\delta=\frac{4}{5}$.

For $\frac{m}{p}<2,$ if one of $a^{\prime},a^{\prime\prime}$ equals $0$ or
$a=1$, then $\delta=\frac{4}{5}$ works as well. Otherwise, we have
$a^{\prime}=a^{\prime\prime}=1$, and $a=0$ which by Remark 4.6 brings us
exactly to Example 5.11(c). ∎

##  6\. Diagrams

In the entire section, let $X$ be a simply connected combinatorial $2$-complex
(see Section 3.3 for the definition). A _disc_ (resp.  _half-plane_) _diagram_
$D$ is a combinatorial complex homeomorphic to a $2$-disc (resp. a half-
plane). A combinatorial map $\phi\colon D\to X$ from a disc (resp. half-plane)
diagram to $X$ is a _disc_ (resp.  _half-plane_) _diagram in $X$_. We say that
$\phi$ is _reduced_ if it is a local embedding at open edges. The restriction
of $\phi$ to $\partial D$ is the _boundary_ of $\phi$.

######  Remark 6.1.

Given an embedded combinatorial closed path $\gamma$ in $X$, there exists a
reduced disc diagram $\phi\colon D\to X$ with boundary $\gamma$. This is
essentially van Kampen’s lemma [vK], see [JS]*Lem 1.6 for a complete proof for
the case of $X$ a simplicial complex, which implies the case of arbitrary $X$.

###  6.1. Relative disc diagrams

Often, we will consider the following diagrams that are in general not
combinatorial.

######  Definition 6.2.

Let $D$ be a disc diagram with a subcomplex $I\subseteq\partial D$. A map
$\phi\colon D\to X$ is a _disc diagram in $X$ relative to $I$_ (or, simply, a
_relative disc diagram in $X$_), if:

  * •

the restriction of $\phi$ to any open cell $\sigma_{D}$ of $D$ is an embedding
into an open cell $\sigma_{X}$ of $X$,

  * •

this restriction is a homeomorphism onto $\sigma_{X}$ if the closure
$\overline{\sigma}_{D}$ is disjoint from $I$,

  * •

if $\sigma_{D}$ is an edge not contained in $I$, then $\sigma_{X}$ is an edge.

Analogously we define a _half-plane diagram in $X$ relative to $I=\partial
D$_, or, simply, a _relative half-plane diagram in $X$_. Again, a relative
disc or half-plane diagram $\phi$ is _reduced_ if it is is a local embedding
at open edges.

######  Lemma 6.3.

Let $\gamma$ be an embedded closed path in $X$, intersecting $X^{1}$, and that
can be subdivided into finitely many segments, the interior of each of which
lies in a single open cell of $X$. Then there exists a reduced relative (to
$\partial D$) disc diagram $\phi\colon D\to X$ with boundary $\gamma$.

In the proof we will need:

######  Remark 6.4.

Let $\sigma$ be an open $2$-disc, and let $g\colon\sigma\to\sigma$ be a local
embedding (hence a local homeomorphism) that is proper, i.e. that extends to a
continuous map on the one-point compactification ${S}^{2}$ of $\sigma$ mapping
the point ${S}^{2}\setminus\sigma$ to itself. Then $g$ is a covering map, and
hence a homeomorphism. This implies the following.

Let $\overline{\sigma}=\sigma\cup\partial\sigma$ be a closed $2$-disc, with
$\partial\sigma$ a concatenation of paths $\beta_{1}\cdots\beta_{k}$. Let
$g^{\prime}\colon\sigma\to\sigma$ be a local embedding that extends to a
continuous map $\overline{g}^{\prime}$ from $\overline{\sigma}$ to ${S}^{2}$
mapping only the endpoints of $\beta_{j}$ to ${S}^{2}\setminus\sigma$. Suppose
that $\overline{g}^{\prime}$ is a local embedding at each interior point of
each $\beta_{j}$ and that the restriction of $\overline{g}^{\prime}$ to the
interior of each $\beta_{j}$ is an embedding. Then the map $g^{\prime}$ is an
embedding. Indeed, we can extend $g^{\prime}$ to $g$ above by attaching to
$\sigma$ additional discs along $\beta_{j}$ and mapping them under $g$ to the
discs in $S^{2}$ bounded by $\overline{g}^{\prime}_{|\beta_{j}}$.

###### Proof of Lemma 6.3.

We subdivide $X$ along $\gamma$. The vertices of the new combinatorial
structure on $X$ are contained in the original $1$-skeleton. By Remark 6.1,
there is a reduced disc diagram $\phi\colon D\to X$ with boundary $\gamma$,
with respect to the new combinatorial structure. Assume that $D$ has the
minimal number of $2$-cells among all such disc diagrams.

We will now discuss the original combinatorial structure on $X$. First, we
remove from the combinatorial structure of $D$ all the edges $e$ of $D$
outside $\partial D$ with $\phi(e)\not\subset X^{1}$.

Second, let $v$ be an interior vertex of $D$ with $\phi(v)\notin X^{0}$, hence
$\phi(v)\in X^{1}\setminus X^{0}$. After possibly cutting and reglueing along
distinct edges starting at $v$ with the same image under $\phi$ (after which
$\phi$ is still reduced by the minimality assumption) we can assume that
$\phi$ a local embedding at $v$. Consequently, there are exactly two edges of
$D$ containing $v$, and they combine to an edge embedding under $\phi$ in an
edge of $X$. We then remove $v$ from the combinatorial structure of $D$. We
repeat this procedure for all such $v$.

The map $\phi$ is a local embedding on the resulting cells of $D$. Note that
each cell of $D$ is simply connected since otherwise it would have $\partial
D$ as a boundary component, which would contradict the assumption that
$\gamma$ intersected $X^{1}$. By Remark 6.4, we have that $\phi$ is a disc
diagram relative to $\partial D$. ∎

Note that in this article we will only be applying Lemma 6.3 to $\gamma$
satisfying Theorem 6.10, where the cutting and reglueing procedure above is
not necessary.

######  Definition 6.5.

Suppose that $X$ has a _piecewise smooth Euclidean metric_. This means that
each $2$-cell of $X$ is equipped with the Riemannian metric of a subset of the
Euclidean plane bounded by a piecewise smooth closed path, with nonzero
interior angles, and that these Riemannian metrics agree on commmon edges. We
also assume that for each vertex $v$ of $X$, all the cells of $X$ containing
$v$ have only finitely many possible isometry types, and so in particular $X$
is complete (see e.g. [NOP]*§1).

We define the links of the vertices in $X$ as in Section 5. Occasionally, we
will discuss the links of points that are not vertices. The link of a point
$x$ in the interior of an edge $e$ has two vertices corresponding to the two
components of $e\setminus x$, and edges of length $\pi$ corresponding to the
$2$-cells containing $e$. The link of a point in the interior of a $2$-cell is
a circle of length $2\pi$.

Let $\phi\colon D\to X$ be a reduced disc or half-plane diagram relative to
$I$. If the restriction of $\phi$ to each edge $\sigma_{D}\subset I$ with
$\sigma_{X}$ a $2$-cell is a geodesic in $\sigma_{X}$, then we can pull back
the piecewise smooth Euclidean metric from $X$ to a _degenerate_ piecewise
smooth Euclidean metric on $D$. This means that we allow the angles in $D$ to
be zero.

Furthermore, if $X$ is $\mathrm{CAT}(0)$, then by [NOP]*§2, we have that $D$
satisfies the local conditions of [BaBu]*Thm 7.1 implying $\mathrm{CAT}(0)$ in
the non-degenerate case. Since the only angles in $D$ that are possibly equal
zero are formed by pairs of edges one of which lies in $\partial D$ and has
zero geodesic curvature, it is easy to find an embedding of $D$ as a convex
subspace of $D$ with a (non-degenerate) piecewise smooth Euclidean metric also
satisfying the conditions of [BaBu]*Thm 7.1. Consequently, $D$ is
$\mathrm{CAT}(0)$ with both metrics.

We define the link of a vertex of such $D$ as in the non-degenerate case, but
we do not include the edges corresponding to the zero angles and we do not
include their vertices corresponding to the edges in $\partial D$.

We say that a disc or half-plane diagram $D$ is _locally Euclidean_ if each
interior point of $D$ has a neighbourhood isometric to a Euclidean disc.

######  Remark 6.6.

Suppose that $X$ as in Definition 6.5 is $\mathrm{CAT}(0)$.

  1. (i)

If $\phi\colon D\to X$ is a reduced relative disc diagram that is locally
Euclidean, then $\phi$ is a local isometric embedding at each interior point
of $D$ (see e.g. the second to last paragraph of the proof of [OP]*Lem 4.1).
In particular, if $D$ is isometric to a convex subset of $\mathbb{R}^{2}$,
then by [BH]*II.4.14 $\phi$ is an isometric embedding, and so $\phi(D)$ is
convex.

  2. (ii)

Let $D^{\pm}\subset X$ be embedded relative disc diagrams, and let $R$ be a
connected component of $D^{+}\cap D^{-}$. Suppose that $D^{\pm}$ are isometric
to convex subsets of $\mathbb{R}^{2}$. Then by (i) we have that $R$ is convex
in each $D^{\pm}$.

  3. (iii)

Generalising (i), let $R_{1},R_{2}\subset X$ be embedded relative disc
diagrams isometric to convex subsets of $\mathbb{R}^{2}$. Suppose that for
each point $x$ of $R_{1}\cup R_{2}$, the set $\mathcal{U}$ of the directions
at $x$ of the geodesics in $R_{1}\cup R_{2}$ has diameter $\leq\pi$ in the
path metric induced on $\mathcal{U}$ from the link of $x$. (We refrain from
calling $\mathcal{U}$ the link of $R_{1}\cup R_{2}$, since the piecewise
Euclidean metric on $R_{1}\cup R_{2}$ might be degenerate and $R_{1}\cup
R_{2}$ is not a disc diagram.) We claim that then $R_{1}\cup R_{2}$ is locally
convex (hence convex) in $X$.

Indeed, this follows from the fact that for $x_{i}\in R_{i}\setminus R_{3-i}$,
the geodesic between $x_{1}$ and $x_{2}$ in $R_{1}\cup R_{2}$, which passes
through $x\in R_{1}\cap R_{2}$, must be a geodesic in $X$, since otherwise the
geodesic triangle $x_{1}xx_{2}$ would have Alexandrov angle $<\pi$ at $x$. By
the diameter hypothesis and [NOP]*Lem 2.1, the shortest path between the
directions corresponding to $xx_{1}$ and $xx_{2}$ in the link of $x$ in $X$
would have to be contained in $\mathcal{U}$, contradicting the assumption that
$x_{1}xx_{2}$ is a geodesic in $R_{1}\cup R_{2}$.

###  6.2. Flat lemma

A _flat in $X$_ is an isometrically embedded Euclidean plane in $X$.

######  Lemma 6.7.

Suppose that $X$ as in Definition 6.5 is $\mathrm{CAT}(0)$. Then for each
circle $\omega$ of length $2\pi$ isometrically embedded in
$\partial_{\infty}X$ with the angle metric, there is a flat $F\subset X$ with
$\partial_{\infty}F=\omega$.

Note that for $X$ proper, Lemma 6.7 is [Leeb]*Prop 2.1.

###### Proof.

Choose a basepoint $x_{0}\in X$ and seven geodesic rays
$r_{0},r_{1},\ldots,r_{6}\colon[0,\infty)\to X$ starting at $x_{0}$
representing consecutive points on $\omega$ at distance $\frac{2\pi}{7}$ in
$\partial_{\infty}X$. Furthermore, for each integer $n\geq 0$, and
$i=0,\ldots,6$, let $\gamma^{n}_{i}$ be the geodesic in $X$ between $r_{i}(n)$
and $r_{i+1}(n)$, where $r_{7}=r_{0}$.

Consider a closed path $\gamma$ of the form
$r_{i+1}[n+1,n]\cdot(\gamma_{i}^{n})^{-1}\cdot
r_{i}[n,n+1]\cdot\gamma_{i}^{n+1}$, for some $i=0,\ldots,6$, and $n\geq 0$.
Let $B_{i}^{n}$ be the closed ball of radius $1+|\gamma^{i}_{n}|/2$ centred at
the midpoint of $\gamma^{i}_{n}$. Since $X$ is $\mathrm{CAT}(0)$, we have that
$B_{i}^{n}$ is convex. Since $B_{i}^{n}$ contains
$r_{i+1}[n+1,n]\cdot(\gamma_{i}^{n})^{-1}\cdot r_{i}[n,n+1]$, it contains the
entire $\gamma$. By [OP]*Rem 2.4, we have that $\gamma$ is contained in the
union of finitely many cells of $X$. Thus there is a union $K_{i}^{n}$ of
finitely many cells of $X$ intersecting $B_{i}^{n}$, such that $\gamma$ is
contained in $K_{i}^{n}$ and contractible in $K_{i}^{n}$.

By [BH]*II.9.8(4), for each $i$ we have $\frac{|\gamma^{n}_{i}|}{n}\to
2\sin\frac{\pi}{7}<1$. Consequently, the distance from $B_{i}^{n}$ to $x_{0}$
becomes arbitrary large as $n\to\infty$. In particular, each vertex of $X$
belongs to only finitely many $K^{n}_{i}$ and so the union $K$ of all
$K_{i}^{n}$ is proper. Since $X$ is $2$-dimensional, $K$ is locally
$\mathrm{CAT}(0)$ by [NOP]*§2 and [BaBu]*Thm 7.1.

Let $\widetilde{K}\to K$ be the universal cover of $K$, which is a complete
proper $\mathrm{CAT}(0)$ space. Let $\widetilde{x}_{0}\in\widetilde{K}$ be a
lift of $x_{0}$, and for $i=0,\ldots,6,$ let $\widetilde{r}_{i}$ be the lift
of $r_{i}$ to $\widetilde{K}$ starting at $\widetilde{x}_{0}$. Since we
included $K_{i}^{n}$ in $K$, the geodesics $\gamma_{i}^{n}$ lift as well to
geodesics $\widetilde{\gamma}_{i}^{n}$ connecting $\widetilde{r}_{i}(n)$ and
$\widetilde{r}_{i+1}(n)$. Thus by [BH]*II.9.8(4) we obtain that consecutive
$\widetilde{r}_{i}$ represent points
$[\widetilde{r}_{i}]\in\partial_{\infty}\widetilde{K}$ at distance
$\frac{2\pi}{7}$. Note that since $\widetilde{K}\to K\to X$ does not increase
distances, we have that the distance between each $[\widetilde{r}_{i}]$ and
$[\widetilde{r}_{i+2}]$ is $\frac{4\pi}{7}$. Consequently, the geodesics
joining consecutive $[\widetilde{r}_{i}]$ in $\partial_{\infty}\widetilde{K}$
form a locally (hence globally, since $\partial_{\infty}\widetilde{K}$ is
$\mathrm{CAT}(1)$) isometrically embedded circle
$\omega_{K}\subset\partial_{\infty}\widetilde{K}$ of length $2\pi$. By
[Leeb]*Prop 2.1, there is a flat $F\subset\widetilde{K}$ with boundary
$\omega_{K}$. By Remark 6.6(i), $F\to\widetilde{K}\to K\to X$ is an isometric
embedding. The boundary of the image of $F$ in $X$ contains $[r_{i}]$, and
thus coincides with $\omega$. ∎

###  6.3. Overlaps and unions

Let $\phi^{\pm}\colon D^{\pm}\to X$ be two embedded (and hence reduced)
relative disc diagrams. Suppose that we have a connected subset
$\beta\subset\phi^{+}(D^{+})\cap\phi^{-}(D^{-})$. The _overlap of $\phi^{\pm}$
at $\beta$_ is the connected component $R$ of
$\phi^{+}(D^{+})\cap\phi^{-}(D^{-})$ containing $\beta$. Denote
$R^{\pm}=(\phi^{\pm})^{-1}(R)\subseteq D^{\pm}$.

Let $\widehat{D}$ be the space obtained from the disjoint union of
$\overline{D^{+}\setminus R^{+}}$ and $\overline{D^{-}\setminus R^{-}}$ by
identifying the subsets $P^{+}\subseteq\mathrm{fr}\,R^{+}$ (the topological
boundary of $R^{+}$ in $D^{+}$) and $P^{-}\subseteq\mathrm{fr}\,R^{-}$
consisting of points with the same image under $\phi^{+}$ and $\phi^{-}$. In
other words, we have
$P^{\pm}=\mathrm{fr}\,R^{\pm}\cap(\phi^{\pm})^{-1}\phi^{\mp}(\mathrm{fr}\,R^{\mp})$.

The _union of $\phi^{\pm}$ at $\beta$_ is the map
$\widehat{\phi}\colon\widehat{D}\to X$ whose restriction to
$\overline{D^{\pm}\setminus R^{\pm}}$ equals $\phi^{\pm}$. If each pair
$(\overline{D^{\pm}\setminus R^{\pm}},P^{\pm})$ is _standard_ , i.e.
homeomorphic to the pair consisting of the closed unit disc and the closed
unit half-circle, then $\widehat{D}$ is homeomorphic to a closed disc and
$\widehat{\phi}$ is a local embedding but possibly not an embedding. See
Figures 11 and 12.

Figure 11. A first possibility for $\phi^{+}(D^{+}),\phi^{-}(D^{-})$, and
$\widehat{D}$, where $P^{\pm}=\mathrm{fr}\,R^{\pm}$. Examples of the sets
$I^{\pm}$ discussed in Lemma 6.8 are shown in bold.

Note that the following lemma is trivial for $\phi^{\pm}$ combinatorial.

######  Lemma 6.8.

Let $\phi^{\pm}\colon D^{\pm}\to X$ be embedded disc diagrams relative to
$I^{\pm}\subset\partial D^{\pm}$. Let $\beta$ be a connected component of
$\phi^{+}(\partial D^{+})\cap\phi^{-}(\partial D^{-})$ with
$J^{\pm}=(\phi^{\pm})^{-1}(\beta)\subseteq I^{\pm}$. Let
$\widehat{\phi}\colon\widehat{D}\to X$ be the union of $\phi^{\pm}$ at
$\beta$. Suppose that each pair $(\overline{D^{\pm}\setminus
R^{\pm}},P^{\pm})$ is standard, that $\widehat{\phi}(\partial\widehat{D})$
intersects $X^{1}$, and that the following condition $(*)$ holds for $w^{\pm}$
the first or the last points of $P^{\pm}$:

We have $\phi^{+}(w^{+})=\phi^{-}(w^{-})\in X^{1}$, or, for the maximal
subpaths $f^{\pm}$ in $\partial(D^{\pm}\setminus R^{\pm})\setminus P^{\pm}$
incident to $w^{\pm}$ mapped under $\phi^{\pm}$ to the same open $2$-cell of
$X$ as $w^{\pm}$, the path $\phi^{+}(f^{+})\cdot\phi^{-}(f^{-})$ is embedded
in $X$.

Then $\widehat{D}$ can be given a combinatorial structure such that
$\widehat{\phi}$ is a disc diagram relative to the closure $\widehat{I}$ of
the union of $I^{\pm}\setminus J^{\pm}$.

Figure 12. A second possibility for $\phi^{+}(D^{+}),\phi^{-}(D^{-})$, and
$\hat{D}$, where $P^{+}\subsetneq\mathrm{fr}\,R^{+}$. Here a subset of $I^{-}$
becomes contained in $\overline{D^{+}\setminus R^{+}}\subset\widehat{D}$.

###### Proof.

We start with the combinatorial structure on each $D^{\pm}$ subdivided along
$\mathrm{fr}\,R^{\pm}\subset D^{\pm}$, and at the endpoints of $P^{\pm}$. Let
$\widehat{D}_{0}$ be the disc diagram obtained from glueing $D^{\pm}$ along
$J^{\pm}$, and let $\widehat{\phi}_{0}\colon\widehat{D}_{0}\to X$ be the map
with restrictions $\phi^{\pm}$ to $D^{\pm}$. Note that
$\partial\widehat{D}_{0}$ is the closure of the union of $\partial
D^{\pm}\setminus J^{\pm}$ and contains $\widehat{I}$. Furthermore,
$\widehat{D}$ and $\widehat{\phi}$ are obtained from $\widehat{D}_{0}$ and
$\widehat{\phi}_{0}$ by cancelling the $2$-cells of $R^{\pm}$, which gives a
natural embedding $\widehat{I}\subseteq\partial\widehat{D}$. Let $\widehat{J}$
be the subset of $J^{\pm}$ remaining in $\widehat{D}$.

Note that the restrictions of $\phi^{\pm}$ to $\overline{D^{\pm}\setminus
R^{\pm}}$, which are also restrictions of $\widehat{\phi}$, are disc diagrams
in $X$ relative to the intersection of their boundary with
$\widehat{J}\cup\widehat{I}$. This follows from the fact that the remaining
part of $\partial\overline{D^{\pm}\setminus R^{\pm}}$ is mapped under
$\phi^{\pm}$ into $X^{1}$ by the definition of $R^{\pm}$. However,
$\widehat{\phi}$ might no longer be a relative disc diagram since the cells of
$\widehat{J}$ become interior in $\widehat{D}$. To correct this, we remove
from the combinatorial structure of $\widehat{D}$ the edges and the interior
vertices contained in $\widehat{J}$ that are not mapped under $\widehat{\phi}$
into $X^{1}$.

After this correction, the restriction $g^{\prime}$ of the map
$\widehat{\phi}$ to each open $2$-cell $\sigma_{\widehat{D}}$ of $\widehat{D}$
has still image in a single open cell of $X$. Note that $\sigma_{\widehat{D}}$
is simply connected since otherwise it would have $\partial\widehat{D}$ as a
boundary component, which would contradict the assumption that
$\widehat{\phi}(\partial\widehat{D})$ intersected $X^{1}$. Since $\phi^{\pm}$
were relative disc diagrams, by the definition of $R^{\pm}$ we have that
$g^{\prime}$ is a local embedding. Since $\phi^{\pm}$ were embeddings, we can
apply Remark 6.4 to $g^{\prime}$, possibly using condition $(*)$ if one of
$\beta_{j}=f^{+}\cdot f^{-}$. This implies that $g^{\prime}$ is an embedding,
and even a homeomorphism for $\overline{\sigma}_{\widehat{D}}$ disjoint from
$\widehat{I}$.

Finally, we remove from the combinatorial structure of $\widehat{D}$ the
interior vertices that are not mapped under $\widehat{\phi}$ into $X^{0}$, and
possibly the endpoints of $P^{\pm}$ if they do not belong to $\widehat{I}$ and
are not mapped under $\widehat{\phi}$ into $X^{0}$. Then $\widehat{\phi}$
becomes a disc diagram relative to $\widehat{I}$. ∎

###  6.4. Curvature

Let $D$ be a disc diagram with a degenerate piecewise smooth Euclidean metric.
For a vertex $v\in D$, the _angle_ at $v$ is the sum of the Riemannian angles
at $v$ of all $2$-cells of $D$ containing $v$. We analogously define _angles_
at $v$ between a pair of edges containing $v$. If $v\in\mathrm{int}\,D$, then
its _curvature_ $\kappa(v)$ is $2\pi$ minus its angle. If $v\in\partial D$,
then its _boundary curvature_ $\kappa_{\partial}(v)$ is $\pi$ minus its angle.
If $e$ is an edge of $D$, then its _curvature_ $\kappa(e)$ is the integral of
its geodesic curvature in the (one or two) $2$-cells of $D$ containing it. In
particular, if $e$ is a Riemannian geodesic in the $2$-cells containing it,
then $\kappa(e)=0$. From adding up the classical Gauss–Bonnet formula for the
$2$-cells of $D$, we obtain the following (see e.g. [BaBu]*§2.3).

######  Theorem 6.9.

Let $D$ be a disc diagram. Then

$\sum_{v\in\mathrm{int}D}\kappa(v)+\sum_{v\in\partial
D}\kappa_{\partial}(v)+\sum_{e\subset D}\kappa(e)=2\pi.$

We have the following consequence of [Stad]*Thm 2. For the reader’s
convenience, we include the proof in our $2$-dimensional setup. A similar
proof was earlier provided in [Bader]*Thm 12.1 in the case where each edge $e$
of $X$ is a Riemannian geodesic in the $2$-cells containing it.

######  Theorem 6.10.

Suppose that $X$ is $\mathrm{CAT}(0)$. Let $\gamma\colon S^{1}\to X$ be an
embedded closed path consisting of $3$ or $4$ geodesic segments. Then each
reduced relative disc diagram $\phi\colon D\to X$ with boundary $\gamma$ is
embedded.

###### Proof.

Step $1$. $\phi$ is a local embedding.

Let $V\subset D$ denote the set of the (at most $4$) endpoints of the geodesic
segments in $\partial D$. Since $X$ is $\mathrm{CAT}(0)$, each vertex
$v\in\mathrm{int}\,D$ and each edge $e\not\subset\partial D$ has nonpositive
curvature. Since $\gamma$ is locally geodesic outside $V$, each
$e\subset\partial D$ or $v\in\partial D\setminus V$ has also nonpositive
(boundary) curvature. Finally, each vertex of $V$ has curvature $\leq\pi$. (In
fact, it is $<\pi$ or one of the edges containing this vertex has negative
curvature.) Let $\mathcal{K}$ denote the sum of all these curvatures.

Let $e\neq f$ be oriented edges starting at a vertex $v$ of $D$, and suppose
$\phi(e)=\phi(f)$. If $v\in\mathrm{int}\,D$, then, since $X$ is
$\mathrm{CAT}(0)$, both angles at $v$ between $e$ and $f$ are $\geq 2\pi$, so
the angle at $v$ is $\geq 4\pi$. Thus $\kappa(v)\leq-2\pi$, and so
$\mathcal{K}<4\pi-2\pi$, which contradicts Theorem 6.9. If $v\in\partial
D\setminus V$, then let $\theta_{1}+\theta_{2}+\theta_{3}$ be the angle at
$v$, decomposed to indicate the angles at $v$ between $\partial D,e,f,$ and
$\partial D$ again. Since $X$ is $\mathrm{CAT}(0)$, we have $\theta_{2}\geq
2\pi$. Since $v\in\partial D\setminus V$, we have
$\theta_{1}+\theta_{3}\geq\pi$. Consequently, $\kappa_{\partial}(v)\leq-2\pi$,
which is again a contradiction. Finally, if $v\in V$, then
$\kappa_{\partial}(v)\leq-\pi$, and so $\mathcal{K}<3\pi-\pi$, contradiction.

Step $2$. $\phi$ is an embedding.

Suppose that $\beta\subset D$ is an embedded combinatorial path from $w_{1}$
to $w_{2}$ such that the restriction of $\phi$ to $\beta$ is an embedding
except that $\phi(w_{1})=\phi(w_{2})$. Let $\psi\colon D^{\prime}\to X$ be a
reduced relative disc diagram with boundary $\phi_{|\beta}$, guaranteed by
Lemma 6.3. Suppose that among such $\beta$ and $\psi$, the disc diagram
$D^{\prime}$ has the minimal possible number of $2$-cells.

Let $v^{\prime}\in\partial D^{\prime}\setminus\psi^{-1}\phi(w_{1})$ be a
vertex with $\kappa_{\partial}(v^{\prime})>0$ and let $e^{\prime},f^{\prime}$
be the edges of $\partial D^{\prime}$ containing $v^{\prime}$. Let
$e,v,f\subset\beta$ be such that
$\phi(e)=\psi(e^{\prime}),\phi(v)=\psi(v^{\prime}),\phi(f)=\psi(f^{\prime})$.
By the minimality assumption, the images of the restrictions of $\phi$ and
$\psi$ to the open neighbourhoods of $e,e^{\prime}$ in any $2$-cells
containing $e,e^{\prime}$, are distinct, and the same property holds for
$f,f^{\prime}$. Thus, if $v\in\mathrm{int}\,D$, then since $X$ is
$\mathrm{CAT}(0)$, both angles in $D$ between $e$ and $f$ are
$\geq\pi+\kappa_{\partial}(v^{\prime})$, and so
$\kappa(v)\leq-2\kappa_{\partial}(v^{\prime})$. If $v\in\partial D\setminus
V$, then $\theta_{2}\geq\pi+\kappa_{\partial}(v^{\prime})$ and
$\theta_{1}+\theta_{3}\geq\kappa_{\partial}(v^{\prime})$. Consequently,
$\kappa_{\partial}(v)\leq-2\kappa_{\partial}(v^{\prime})$. Finally, if $v\in
V$, then
$\kappa_{\partial}(v)\leq-\kappa_{\partial}(v^{\prime})\leq\pi-2\kappa_{\partial}(v^{\prime})$.

Let now $e^{\prime}\subset\partial D^{\prime}$ be an edge with
$\kappa(e^{\prime})>0$, and let $e\subset\beta$ be such that
$\phi(e)=\psi(e^{\prime})$. Then $\phi(e)$ is not a geodesic and so
$e\not\subset\partial D$. Moreover, since $X$ is $\mathrm{CAT}(0)$, we have
$\kappa(e)\leq-2\kappa(e^{\prime})$.

By Theorem 6.9 applied to $D^{\prime},$ the sum of all the curvatures of edges
$e^{\prime}\subset\partial D^{\prime}$ and vertices $v^{\prime}\in\partial
D^{\prime}\setminus\psi^{-1}\phi(w_{1})$ is at least $\pi$. By the previous
paragraph, each positive contribution $\kappa_{\partial}(v^{\prime})$ or
$\kappa(e^{\prime})$ to that sum decreases the maximal possible curvature of
$v,e$ in $D$ by at least $2\kappa_{\partial}(v^{\prime})$ or
$2\kappa(e^{\prime})$. Thus $\mathcal{K}<4\pi-2\pi$, which is a contradiction.
∎

###  6.5. Frillings

The following operation will be needed to discuss the limits of relative disc
diagrams.

######  Definition 6.11.

Let $D$ be a half-plane diagram. A _frilling_ of $D$ is a combinatorial
complex $D^{*}$ obtained from $D$ via the following subdivision and quotient
map $D\to D^{*}$. First, we subdivide some $2$-cells of $D$ along new edges
(and their vertices) each of which has exactly one endpoint in $\partial D$.
Then, we quotient some of the edges intersecting $\partial D$ to vertices,
including all the new edges. Finally, we collapse some of the $2$-cells.

Here a _collapse_ of an open $2$-cell $\sigma\subset D$, such that
$\overline{\sigma}\cap\partial D$ is a nontrivial path $\tau$, is a
deformation retraction of $D$ onto $D\setminus(\sigma\cup\mathrm{int}\ \tau)$.

Figure 13. An example of a frilling.

See Figure 13(a), where we subdivide $D$ along the dashed edge, we quotient
this edge along with the other red edges, and then we collapse the two
indicated $2$-cells $\sigma$, where the paths $\tau$ are indicated in green.
The resulting frilling $D^{*}$ is illustrated in Figure 13(b).

######  Remark 6.12.

We denote by $\partial D^{*}$ the image of $\partial D$ in $D^{*}$. Note that
if the preimage in $\partial D$ of each point of $\partial D^{*}$ is compact,
then $D^{*}$ is a half-plane diagram $D^{\prime}$ wedged at points of
$\partial D^{\prime}$ with combinatorial complexes that are contractible
(which may exist only if $\partial D^{*}$ is not an embedded line) or
homeomorphic to $2$-spheres. In our applications in further sections we will
always have $D^{\prime}=D^{*}$.

######  Construction 6.13.

Let $D$ be a half-plane diagram. Suppose that $D$ is an increasing union of
disc diagrams $D_{n}$ with $I_{n}=D_{n}\cap\partial D$. We then say that
$(D_{n},I_{n})$ _converge_ to $(D,\partial D)$.

Suppose that additionally we have relative disc diagrams $\phi_{n}\colon
D_{n}\to X$ satisfying the assumptions (i–iii) below. In the construction
below we define their _limit_ $\phi^{*}\colon D^{*}\to X$, where $D^{*}$ is a
particular frilling of $D$. The reason for passing to the frilling is that it
will turn out that if the hypothesis of Remark 6.12 holds, then the
restriction $\phi^{\prime}$ of $\phi^{*}\colon D^{*}\to X$ to $D^{\prime}$ is
a relative half-plane diagram.

For future reference, whenever we say that $\phi^{*}\colon D^{*}\to X$ (or
$\phi^{\prime}\colon D^{\prime}\to X$) is the _limit_ of $\phi_{n}$, then in
particular we mean that $(D_{n},I_{n})$ can be identified (by combinatorial
isomorphisms) with subcomplexes of a half-plane diagram $D$ converging to
$(D,\partial D)$, that $\phi_{n}$ satisfy the assumptions (i–iii), and that
$D^{*}$ (or $D^{\prime}$) is the particular frilling described below.

The assumptions that we make on $\phi_{n}$ are the following.

  1. (i)

The attaching map of each cell in $X$ is injective, and the boundary of each
$\phi_{n}$ intersects $X^{1}$ at least twice (which implies that the attaching
map of each cell in $D_{n}$ is injective, since otherwise the boundary of a
$2$-cell would contain $\partial D$ and would map into $X\setminus X^{1}$
except for one point).

  2. (ii)

For each cell $\sigma_{D}$ in $D$, the cell $\sigma_{X}$ in Definition 6.2 is
the same for all $\phi_{n}$ such that $\sigma_{D}$ lies in $D_{n}$.

  3. (iii)

For each $2$-cell $\sigma_{D}$ whose closure intersects $\partial D$, the
closed paths $\phi_{n|\partial\sigma_{D}}$ converge to a closed path
$\phi_{\sigma_{D}}\colon\partial\sigma_{D}\to\overline{\sigma}_{X}$.
Moreorover, for
$\phi^{*}_{\sigma_{D}}\colon\partial\sigma_{D}^{*}\to\overline{\sigma}_{X}$,
obtained from $\phi_{\sigma_{D}}$ by quotienting to points the edges of
$\partial\sigma_{D}$ on which $\phi_{\sigma_{D}}$ is constant, we have that

     * •

the restriction of $\phi^{*}_{\sigma_{D}}$ to the subset of
$\partial\sigma_{D}^{*}$ that is the image of $\partial\sigma_{D}\cap\partial
D$ is an embedding, and

     * •

$\phi^{*}_{\sigma_{D}}(\partial\sigma_{D}^{*})$ bounds a finite (possibly
empty) union of open discs in $\sigma_{X}$.

Then the _limit_ $\phi^{*}\colon D^{*}\to X$ is the following map whose domain
$D^{*}$ is the following frilling $D^{*}$ of $D$. To start, on each cell
$\sigma_{D}$ of $D$ disjoint from $\partial D$ (which is thus a cell of
$D^{*}$) we define $\phi^{*}$ to be a homeomorphism onto $\sigma_{X}$. On each
remaining edge $\sigma_{D}$ of $D$ not contained in $\partial D$, we define
$\phi^{*}$ to be a homeomorphism onto the subset of the edge $\sigma_{X}$
bounded by the limits of $\phi_{n}(\partial\sigma_{D})$. In the case where
these two limits coincide, we quotient $\sigma_{D}$ to a vertex in $D^{*}$.

Consider now a $2$-cell $\sigma_{D}$ of $D$ whose closure intersects $\partial
D$. If $\phi^{*}_{\sigma_{D}}$ is an embedding on the entire
$\partial\sigma_{D}^{*}$, then we define $\phi^{*}$ on $\sigma_{D}$ to be a
homeomorphism onto the subset of the $2$-cell $\sigma_{X}$ bounded by
$\phi^{*}_{\sigma_{D}}(\partial\sigma_{D}^{*})$. Then the $2$-cell
$\sigma_{D}$ remains a $2$-cell of $D^{*}$, and we identify
$\partial\sigma_{D}^{*}$ with its boundary, and $\phi^{*}_{\sigma_{D}}$ with
the restriction of $\phi^{*}$ to its boundary. Otherwise, we denote by
$\tau_{i},\tau_{i}^{\prime}$ the maximal subpaths of
$\partial\sigma_{D}\cap\partial
D,\overline{\partial\sigma_{D}\setminus\partial D}$ with the same image under
$\phi_{\sigma_{D}}$, for $\tau_{i}$ not an endpoint of a connected component
of $\partial\sigma_{D}\cap\partial D$. We

  * •

subdivide the $2$-cell $\sigma_{D}$ along edges starting (resp. ending) at an
endpoint of $\tau_{i}$ (resp. the corresponding endpoint of
$\tau_{i}^{\prime}$),

  * •

quotient these edges to vertices, and

  * •

collapse the resulting $2$-cells bounded by nontrivial
$\tau_{i},\tau^{\prime}_{i}$.

This allows us to extend $\phi^{*}$ to an embedding on each of the $2$-cells
of $D^{*}$ resulting from $\sigma_{D}$ and on the image in
$\partial\sigma_{D}^{*}$ of each edge of $\partial\sigma_{D}\cap\partial D$
that was not removed during a collapse. Consequently, if the hypothesis of
Remark 6.12 holds, then the restriction $\phi^{\prime}$ of $\phi^{*}\colon
D^{*}\to X$ to $D^{\prime}$ is a relative half-plane diagram.

See Figure 14, where on the top we illustrate a part of the complex $X$, and
where $\phi_{1}(D_{1})$ is located above the straight line $\phi_{1}(I_{1})$
in (a). Choosing $\phi_{n}$ so that $\phi_{n}(I_{n})$ converges to the
straight line in (b), we have that $(D_{n},I_{n})$ converge to $(D,\partial
D)$ from Figure 13(a), with limit $\phi^{*}=\phi^{\prime}$ whose domain is the
frilling $D^{*}$ from Figure 13(b).

Figure 14.

##  7\. Diagrams in $\mathbf{X}$

In this section we describe the additional structure of the relative disc or
half-plane diagrams $D\to X$ for $X=\mathbf{X}$ or $X=\nabla^{+}$.

Note that by Lemma 3.3(iii), each geodesic segment $\gamma$ in $\mathbf{X}$
passes through only finitely many chambers. Since admissible lines form a
locally finite set in $(\nabla^{+},|\cdot,\cdot|)$ [LP2]*lem 4.1, and bound
convex half-spaces [LP2]*lem 5.1, each geodesic segment in $\nabla^{+}$
intersects the union of admissible lines in a union of a finite set of points
and a finite set of subsegments. Thus there are finitely many connected
components of intersections of $\gamma$ with the open cells of $\mathbf{X}$,
and Lemma 6.3 can be applied to piecewise geodesic closed paths in
$\mathbf{X}$. Furthermore, the attaching maps of the cells in $\mathbf{X}$ and
$\nabla^{+}$ are injective as required in Construction 6.13(i).

###  7.1. Limits in $\nabla^{+}$

######  Remark 7.1.

Suppose in Construction 6.13 that $X=\nabla^{+}$ and all $\phi_{n}$ are the
compositions of reduced relative disc diagrams $\widetilde{\phi}_{n}$ in
$\mathbf{X}$ with $\rho_{+}$. Furthermore, suppose that $\partial D$ is a
concatenation $I_{a}\cdot I_{b}\cdot I_{c}$ and we have $\epsilon>0$ and a
positive function $\delta\colon I_{a}\cup I_{c}\to\mathbb{R}$ such that

  * •

the restrictions of $\widetilde{\phi}_{n}$ to the intersections of
$I_{a},I_{b},I_{c}$ with $I_{n}$ are geodesics concatenated at an Alexandrov
angle $\geq\epsilon$, and

  * •

for each $n$, and each $t\in I_{a}\cap I_{n}$, we have
$d_{\mathbf{X}}\big{(}\widetilde{\phi}_{n}(t),\widetilde{\phi}_{n}(I_{c}\cap
I_{n})\big{)}\geq\delta(t)$, and the same condition holds if we interchange
$I_{a}$ with $I_{c}$.

Then:

  1. –

For each $2$-cell $\sigma_{D}$, the restriction of $\phi_{n}$ to
$\partial\sigma_{D}\cap\partial D$ converges to a disjoint union of finitely
many (possibly constant) piecewise geodesics. Thus the maps
$\phi^{*}_{\sigma_{D}}$ satisfy the assumption of Construction 6.13(iii).
Furthermore, the preimage of each point under $\partial D\to\partial D^{*}$ is
connected.

  2. –

All $D_{n}$ are locally $\mathrm{CAT}(0)$, and so $D^{*}$ is locally
$\mathrm{CAT}(0)$. Thus, by Theorem 6.9, we have that $D^{*}$ contains no
subcomplexes homeomorphic to $2$-spheres.

  3. –

Consequently $D^{\prime}=D^{*},\phi^{\prime}=\phi^{*}$, and $\partial
D^{\prime}$ is a path. Note that $\phi^{\prime}_{|\partial D^{\prime}}$ is
obtained from $\phi_{|\partial D}$ by quotienting to points the segments on
which $\phi_{|\partial D}$ is constant.

  4. –

Since there is a uniform bound on the number of $2$-cells traversed by a
geodesic in $\mathbf{X}$ contained in the star of a vertex $v$ of $\mathbf{X}$
with given $\rho_{+}(v)\in\nabla^{+}$ [OP]*Claim in §2, the preimage of each
point under $\partial D\to\partial D^{\prime}$ is compact, and so Remark 6.12
applies. We conclude that $\partial D^{\prime}$ is a line and $D^{\prime}$ is
a half-plane diagram.

######  Definition 7.2.

Let $\phi^{\prime}\colon D^{\prime}\to\nabla^{+}$ be a relative half-plane
diagram that is the limit of $\phi_{n}$ that are the compositions with
$\rho_{+}$ of reduced relative disc diagrams $\widetilde{\phi}_{n}\colon
D_{n}\to\mathbf{X}$. Let $J\subset\partial D$ be homeomorphic with
$[0,\infty)$.

Let $r\colon J\to\mathbf{X}$ be a geodesic ray and suppose that the
restrictions of $\widetilde{\phi}_{n}$ to $\partial D_{n}\cap J$ coincide with
the restrictions of $r$. We then say that $\phi^{\prime}$ is _bordered_ by
$r$. This terminology is justified since then the sequence $\phi_{n}$ is
constant on $J$, and so the map $D\to D^{\prime}$ is the identity on $J$. We
may thus identify $J$ with a subset of $\partial D^{\prime}$, and the
restriction of $\phi^{\prime}$ to $J$ equals $\rho_{+}\circ r$.

Let $r\colon J\to\mathbf{X}$ be a geodesic ray disjoint from the vertex set
$\mathbf{X}^{0}$ and transverse to the $1$-skeleton $\mathbf{X}^{1}$. Suppose
that the restrictions of $\widetilde{\phi}_{n}$ to $J\cap\partial D_{n}$ are
geodesics that converge to $r$. We then say that $\phi^{\prime}$ is _weakly
bordered_ by $r$. In that case, we can arrange $\phi^{\prime}$ to be bordered
by $r$ after modifying $\widetilde{\phi}_{n}$ on the cells intersecting $J$.

Below we describe how does the operation of passing to the limit of relative
disc diagrams behave with respect to the operation of the union.

######  Remark 7.3.

Suppose that $(D^{\pm}_{n},I^{\pm}_{n})$ converge to half-plane diagrams
$(D^{\pm},\partial D^{\pm})$, with frillings $D^{*\pm}=D^{{}^{\prime}\pm}$ and
$\phi^{{}^{\prime}\pm}\colon D^{{}^{\prime}\pm}\to\nabla^{+}$ the limits of
the compositions $\phi^{\pm}_{n}$ of embedded relative disc diagrams
$\widetilde{\phi}^{\pm}_{n}\colon D_{n}^{\pm}\to\mathbf{X}$ with $\rho_{+}$.
Let $J^{\pm}\subset\partial D^{\pm}$ be rays with
$\beta_{n}=\widetilde{\phi}^{+}_{n}(J^{+}\cap\partial
D_{n}^{+})=\widetilde{\phi}^{-}_{n}(J^{-}\cap\partial D^{-}_{n})$ a connected
component of the intersection of $\widetilde{\phi}^{\pm}_{n}(\partial
D^{\pm}_{n})$. Let $\widehat{\phi}_{n}\colon\widehat{D}_{n}\to\mathbf{X}$ be
the unions of $\widetilde{\phi}_{n}^{+}$ and $\widetilde{\phi}_{n}^{-}$ at
$\beta_{n}$ with overlaps $R_{n}$.

After passing to subdiagrams and a subsequence, for all $n$ we have
$(\widetilde{\phi}^{\pm}_{n})^{-1}(R_{n})=R^{\pm}\cap D^{\pm}_{n}$ for some
fixed $R^{\pm}\subseteq D^{\pm}$. Let $\widehat{D}$ be the space obtained from
the disjoint union of $\overline{D^{\pm}\setminus R^{\pm}}$ by identifying the
subsets $P^{\pm}\subseteq\mathrm{fr}\,R^{\pm}$ consisting of points with the
same image under $\widetilde{\phi}^{\pm}_{n}$ (this does not depend on $n$).

Suppose that each pair $(\overline{D^{\pm}\setminus R^{\pm}},P^{\pm})$ is
homeomorphic to the pair consisting of the closed upper half-plane in
$\mathbb{R}^{2}$ and the negative real axis, so that we can assume that each
$(\overline{D_{n}^{\pm}\setminus R_{n}^{\pm}},P_{n}^{\pm})$ is standard.
Suppose that the condition $(*)$ of Lemma 6.8 holds for $P_{n}^{\pm}$, and so
$\widehat{\phi}_{n}$ are disc diagrams relative to appropriate
$\widehat{I}_{n}$.

Then $\widehat{D}$ has a combinatorial structure such that
$(\widehat{D}_{n},\widehat{I}_{n})$ converge to
$(\widehat{D},\partial\widehat{D})$. Moreover, we have a frilling
$\widehat{D}\to\widehat{D}^{\prime}$ and a relative half-plane diagram
$\widehat{\phi}^{\prime}\colon\widehat{D}^{\prime}\to\nabla^{+}$ that is the
limit of $\rho_{+}\circ\widehat{\phi}_{n}$. Note that
$\widehat{D}\to\widehat{D}^{\prime}$ coincides with $D^{\pm}\to
D^{{}^{\prime}\pm}$ on the cells contained entirely in $D^{+}$ or $D^{-}$, and
$\widehat{\phi}^{\prime}$ coincides with $\phi^{{}^{\prime}\pm}$ on their
quotients.

Furthermore, if $\phi^{\prime\pm}$ are both bordered by a geodesic ray with
domains $J^{\pm}$, and $R^{\pm}=J^{\pm}$, then each cell $\sigma$ of
$\widehat{D}^{\prime}$ with $\overline{\sigma}$ disjoint from
$\partial\widehat{D}^{\prime}$, considered as a cell of $\widehat{D}$, is a
union of cells $\sigma_{\sharp}$ of $D^{\pm}$, and
$\widehat{\phi}^{\prime}_{|\sigma}=\widehat{\phi}_{|\sigma}$ is the union of
$\phi^{{}^{\prime}\pm}_{|\sigma_{\sharp}}=\phi^{\pm}_{|\sigma_{\sharp}}$.

######  Remark 7.4.

More generally, in Remark 7.3 we can suppose that (see Figure 15):

  * •

$P^{\pm}$ is a (possibly empty or infinite) union of bounded paths $P^{k\pm}$
and an unbounded path $P^{\infty\pm}$, pairwise disjoint except possibly at
the endpoints, and

  * •

$\overline{D^{-}\setminus R^{-}}$ is a union of disc diagrams $D^{k}$, and a
half-plane diagram $D^{\infty}$, pairwise disjoint except possibly at the
endpoints of $P^{k-}$.

We require that

  * •

each $(\overline{D^{+}\setminus R^{+}},P^{\infty+}),(D^{\infty},P^{\infty-})$
is homeomorphic to the pair consisting of the closed upper half-plane in
$\mathbb{R}^{2}$ and the negative real axis, and

  * •

each $(\overline{D^{+}\setminus R^{+}},P^{k+})$ is homeomorphic to the pair
consisting of the closed upper half-plane in $\mathbb{R}^{2}$ and a bounded
interval in its boundary, and

  * •

each $(D^{k},P^{k-})$ is standard.

Figure 15.

We can then assume that $(\overline{D_{n}^{\pm}\setminus
R_{n}^{\pm}},P_{n}^{\pm})$ have analogous structure, where
$\overline{D_{n}^{+}\setminus R_{n}^{+}}$ and $D_{n}^{\infty}$ are disc
diagrams. Suppose that the endpoints of $P^{\pm}_{n}$ are mapped under
$\phi^{\prime\pm}_{n}$ into the $1$-skeleton of $\nabla^{+}$. Then
$\widehat{\phi}_{n}$ are disc diagrams relative to appropriate
$\widehat{I}_{n}$ and the last two paragraphs of Remark 7.3 hold as well.

###  7.2. Folding locus

######  Definition 7.5.

Let $\phi\colon D\to\nabla^{+}$ be a relative disc or half-plane diagram. We
say that $\phi$ is _$\mathbf{X}$ -reduced_ if at a neighbourhood of each
vertex of $D$ it is the composition of a reduced relative disc diagram in
$\mathbf{X}$ and the map $\rho_{+}$.

The _folding locus_ $L\subset D$ of $\phi$ is the closure of the union of open
edges $e$ in $D\setminus\partial D$ that

  * •

do not map to $\partial\nabla^{+}$ under $\phi$, but

  * •

at which $\phi$ is not a local embedding.

If such an edge $e$ is mapped under $\phi$ into a principal line, then we
_orient_ $e$ in the direction opposite to a vertex of $\nabla$ (see Figure
16).

For example, since in Remark 7.3 the relative disc diagrams
$\widehat{\phi}_{n}$ are local embeddings, we have that each
$\rho^{+}\circ\widehat{\phi}_{n}$, and consequently $\widehat{\phi}^{\prime}$,
is $\mathbf{X}$-reduced.

Figure 16.

From Propositions 5.1, 5.5, and 5.12 we deduce the following, where
$\varepsilon_{0}$ is the minimum of the constants in Propositions 5.5 and
5.12.

######  Corollary 7.6.

Let $\phi\colon D\to\nabla^{+}$ be an $\mathbf{X}$-reduced relative disc or
half-plane diagram and let $v\in D\setminus\partial D$ be a vertex of the
folding locus $L\subset D$. Suppose that the angle at $v$ in $D$ is
$<2\pi+\varepsilon_{0}$. Then the intersection of $L$ with the star of $v$ in
$D$ has one of the forms in Figure 17.

The angles at $v$ in (a),(b),(e) and the two top angles in (f) are the obvious
multiples of $\frac{\pi}{3}$. Both angles in (c) are $>\pi$ and in (d) they
are $\geq\pi$. The two bottom angles in (f), (g), the two top angles in (h),
as well as all the angles in (i) are $>\frac{\pi}{3}$. The two bottom angles
in (h) are $>\frac{2\pi}{3}$. The sum of the top and bottom angle at each side
of (g) is $\geq\pi$.

Figure 17.

###### Proof.

The third and second bullet point in Proposition 5.1 correspond to (a) and
(b), respectively. The second bullet point in Propositions 5.5 and 5.12
corresponds to (b), (c), and (d). The third bullet point in Proposition 5.12
corresponds to (e), (f), and (g). The fifth and fourth bullet points in
Proposition 5.12 correspond to (h), (i), respectively. ∎

######  Remark 7.7.

  1. (i)

If in Corollary 7.6 we have an edge $e$ of $L$ containing $v$ that is oriented
towards $v$ (resp. non-oriented but not as in Figure 17(h)), then there is an
edge $e^{\prime}$ of $L$ containing $v$ at angle $\geq\pi$ from $e$ that is
oriented from $v$ (resp. or non-oriented).

  2. (ii)

Thus if at each interior vertex of $D$ the angle is $<2\pi+\varepsilon_{0}$,
then each oriented edge of $L$ is contained in a sequence of oriented edges of
$L$ forming an oriented geodesic ray or segment ending at $\partial D$.
Furthermore, each non-oriented edge $e$ of $L$ is contained in a geodesic of
$L$ whose each end is unbounded or lies at $\partial D$ or at $v$ in Figure
17(h).

  3. (iii)

Let $c\subset\nabla^{+}$ be the (typically non-geodesic) ray that is a
parametrisation of an admissible line
$\alpha_{1}=m_{2}\alpha_{2}+m_{3}\alpha_{3}$. Then there is a subray
$c^{\prime}$ of $c$ such that any other admissible line in $\nabla^{+}$ with
unbounded intersection with $\alpha_{1}\geq m_{2}\alpha_{2}+m_{3}\alpha_{3}$
is disjoint from $c^{\prime}$ or has form $\alpha_{2}=m\alpha_{3}$.

  4. (iv)

Consequently, if in part (i) in Figure 17(b,c,d) we have $\phi(e)\subset
c^{\prime}$, with $\phi(v)$ separating $\phi(e)$ from the unbounded end of
$c^{\prime}$, then $\phi(e^{\prime})$ is the consecutive edge on $c^{\prime}$,
and we have Figure 17(b) or (d).

######  Definition 7.8.

For any point $[\alpha]\in\nabla^{+}$, let $S_{[\alpha]}$ denote its unit
tangent space in the Euclidean plane $(\nabla,|\cdot,\cdot|)$, which can be
identified with the link of $[\alpha]$ in $\nabla$. Furthermore, let
$S^{\angle}_{[\alpha]}$ denote the quotient of $S_{[\alpha]}$ by the action of
the order $6$ dihedral group generated by the reflections in the lines through
$[\alpha]$ parallel to the principal lines. Parallel transporting in
$(\nabla,|\cdot,\cdot|)$, we can identify all $S_{[\alpha]}$ and all
$S^{\angle}_{[\alpha]}$, which we denote then shortly $S$ and $S^{\angle}$.
The direction towards (resp. away from) the vertices of $\nabla$ determines
the _principal point_ (resp.  _antiprincipal point_) of $S^{\angle}$. The
union of these two points is denoted $\partial S^{\angle}\subset S^{\angle}$.

Consider now an $\mathbf{X}$-reduced relative disc or half-plane diagram
$\phi\colon D\to\nabla^{+}\subset\nabla$ with folding locus $L\subset D$. For
$x\in D$, let $S_{x}$ be its link in $D$, let $\phi_{x}\colon S_{x}\to
S_{\phi(x)}$ be the map induced by $\phi$ between the links, and let
$\phi^{\angle}_{x}$ be the composition of $\phi_{x}$ with $S_{\phi(x)}\to
S^{\angle}_{\phi(x)}$.

Suppose now that $D$, with the degenerate piecewise Euclidean metric pulled
back from $\nabla^{+}$, is a convex subset of the Euclidean plane
$\mathbb{R}^{2}$. For $x\in D$, let $\mathbb{S}_{x}$ be its unit tangent space
in $\mathbb{R}^{2}$. Parallel transporting, we can identify all
$\mathbb{S}_{x}$, which we denote then as $\mathbb{S}$. For $x\notin
L\cup\phi^{-1}(\partial\nabla^{+})$, the map $\phi_{x}$ extends to an isometry
$\phi_{x}\colon\mathbb{S}=\mathbb{S}_{x}\to S_{\phi(x)}=S$. Since
$D\subset\mathbb{R}^{2}$, all edges of $L$ are mapped under $\phi$ to
principal lines, and hence their directions are mapped to $\partial
S^{\angle}$ in $S^{\angle}$. Thus it can be easily seen that
$\phi^{\angle}_{x}$ does not depend on $x$, and we can denote it
$\phi^{\angle}$.

Below, a _positively_ (resp.  _negatively_) _oriented_ geodesic ray in the
folding locus consists of edges that are all oriented towards (resp. away
from) the unbounded end of the ray.

######  Lemma 7.9.

Let $\phi\colon D\to\nabla^{+}$ be an $\mathbf{X}$-reduced relative half-plane
diagram with interior vertex angles $<2\pi+\varepsilon_{0}$. Let $r$ be a
positively oriented geodesic ray in the folding locus $L\subset D$.

  1. (i)

Then $\phi(r)$ is unbounded.

  2. (ii)

Assume additionally that $r$ is a connected component of $L$. Then there is
$M$ depending only on the first edge of $\phi(r)$, such that for the subray
$r^{\prime}$ obtained from $r$ by removing its initial length $M$ segment, we
have that $\phi(r^{\prime})$ is contained in a principal line.

The same statements hold for $r$ negatively oriented.

Finally, if $r$ is a geodesic ray in $D$ disjoint from $L$, with the tangent
direction of $r$ at some (hence each) point $x$ of $r$ sent to $\partial
S^{\angle}$ under the map $\phi^{\angle}_{x}$ from Definition 7.8, then
$\phi(r)$ is unbounded as well.

###### Proof.

If $r$ is positively oriented, then each edge of $\phi(r)$ is directed away
from $[1,0,0],[0,1,0],$ or $[0,0,1]$. The change of the direction might occur
only at $\partial\nabla^{+}$ or at $\phi(v)$ for a vertex $v$ described in
Figure 17(e,h). We have the following possibilities, as illustrated in Figure
18:

Figure 18. Image of a positively oriented ray in $\nabla^{+}$. Numbers (1),
(2), (3) correspond to the cases in the proof of Lemma 7.9.

  1. (1)

At $\partial\nabla^{+}$, the direction away from $[1,0,0]$ changes to the
direction away from $[0,1,0]$ or the direction away from $[0,1,0]$ changes to
the direction away from $[0,0,1]$.

  2. (2)

At $\phi(v)$ as in Figure 17(e) (which cannot occur under the additional
assumption in (ii)), the direction away from $[0,0,1]$ changes to the
direction away from $[0,1,0]$. In that case $\phi(v)$ has
$\frac{\alpha_{1}}{\alpha_{2}}$ an integer $\geq 2$.

  3. (3)

At $\phi(v)$ as in Figure 17(h) (which also cannot occur in (ii)), the
direction away from $[0,1,0]$ changes to the direction away from $[1,0,0]$. In
that case $\phi(v)$ has $\frac{\alpha_{1}}{\alpha_{2}}<2$.

Moving away from $[0,1,0]$ increases $\frac{\alpha_{1}}{\alpha_{2}}$.
Consequently, for a subray $r^{\prime}\subset r$ each edge of
$\phi(r^{\prime})$ is directed away from $[0,1,0]$ (which eventually does not
happen in (ii)) or $[0,0,1]$. Thus the orthogonal projection w.r.t.
$|\cdot,\cdot|$ of $\phi(r)$ onto the principal line $\alpha_{2}=\alpha_{3}$
diverges to $\infty$.

In $r$ is negatively oriented, for a subray $r^{\prime}\subset r$ each edge of
$\phi(r^{\prime})$ is directed towards $[0,1,0]$ (which eventually does not
happen in (ii)) or $[1,0,0]$. Thus the orthogonal projection w.r.t.
$|\cdot,\cdot|$ of $\phi(r)$ onto the principal line $\alpha_{1}=\alpha_{2}$
diverges to $\infty$.

The last assertion is proved similarly. ∎

Below, a _non-oriented component_ $\mathcal{L}$ of the folding locus $L$ is a
connected component of the union of the open non-oriented edges of $L$ and the
vertices in Figure 17(d,g).

######  Lemma 7.10.

There is a continuous function
$M\colon\big{(}0,\frac{\pi}{3}\big{)}\to\mathbb{R}$ such that the following
holds. Let $\phi\colon D\to\nabla^{+}$ be an $\mathbf{X}$-reduced relative
half-plane diagram with interior vertex angles $<2\pi+\varepsilon_{0}$. Let
$\mathcal{L}\subset L$ be a non-oriented component, let $x\in\mathcal{L}$, and
let ${u}\in S_{x}$ be the direction of an edge of $\mathcal{L}$. If
$\phi^{\angle}_{x}({u})\in S^{\angle}$ is at angle $\varepsilon$ from the
principal point, then the connected component of $\mathcal{L}\setminus x$, to
which $u$ is tangent

  * •

has length $<M(\varepsilon)$ and ends at $\partial D$, or

  * •

contributes $\leq-\varepsilon$ to the sum of the curvatures of the interior
edges and the interior vertices of $D$.

###### Proof.

Consider the arc-length parametrisation $\mathcal{L}\colon[0,T)\to D$
(possibly $T=\infty$) of the union of $x$ and the connected component of
$\mathcal{L}\setminus x$ tangent to $u$, with $\mathcal{L}(0)=x$. Let
${u}(t)\in S_{\mathcal{L}(t)}$ be the direction of $\mathcal{L}[t,T)$. In the
case where $T<\infty$ and $\lim_{t\to T}\mathcal{L}(t)$ is a vertex in the
interior of $D$, which we call $\mathcal{L}(T)$, we have the configuration of
Figure 17(c,f,h,i). For (c,f,i), we define additionally ${u}(T)\in
S_{\mathcal{L}(T)}$ to be the direction of the edge $e$ of $L$ opposite at
$\mathcal{L}(T)$ to the non-oriented edge of $\mathcal{L}$ (Remark 7.7(i)).
For (h) we denote by $e$ the edge outside $L$ indicated in Figure 19 whose
direction ${u}(T)\in S_{\mathcal{L}(T)}$ has
$\phi^{\angle}_{\mathcal{L}(T)}\big{(}{u}(T)\big{)}$ principal.

Figure 19.

We claim that $t\to\phi^{\angle}_{\mathcal{L}(t)}\big{(}{u}(t)\big{)}\in
S^{\angle}$ is a decreasing function, where the interval $S^{\angle}$ of
length $\frac{\pi}{3}$ is parametrised by the angle from the principal point.
Furthermore, the sum of the integral of the geodesic curvature and the
exterior angles at each side of an interval $\mathcal{L}(t_{1},t_{2})$ (resp.
$\mathcal{L}(t_{1},t_{2}]$, including possibly the exterior angle at
$\mathcal{L}(T)$ of $\mathcal{L}\cdot e$ for $t_{2}=T$), is at least as large
as the angle between
$\phi^{\angle}_{\mathcal{L}(t_{1})}\big{(}u(t_{1})\big{)}$ and
$\lim_{t\nearrow t_{2}}\phi^{\angle}_{\mathcal{L}(t)}\big{(}{u}(t)\big{)}$
(resp. $\phi^{\angle}_{\mathcal{L}(t_{2})}\big{(}{u}(t_{2})\big{)}$).

Indeed, for $\mathcal{L}(t_{1},t_{2})$ a single edge this follows from the
definition of geodesic curvature and the position of non-principal admissible
lines in $\nabla$. Furthermore, for $t\in(t_{1},t_{2}]$ with $\mathcal{L}(t)$
a vertex, we have the configuration of Figure 17(d,g) for $t<T$ or Figure
17(c,f,h,i) for $t=T$. Except for (h,i), in all these cases the angle between
$\lim_{t^{\prime}\nearrow
t}\phi^{\angle}_{\mathcal{L}(t^{\prime})}\big{(}{u}(t^{\prime})\big{)}$ and
$\phi^{\angle}_{\mathcal{L}(t)}\big{(}{u}(t)\big{)}$ is the exterior angle at
$\mathcal{L}(t)$ of $\mathcal{L}\cdot e$. In case (i), and on one side in case
(h) the exterior angle is even larger. This justifies the claim.

Let now $\mathcal{L}^{\prime}\colon(-\infty,+\infty)\to\nabla$ be any non-
principal admissible line, all of which are isometric by Remark 5.10. Let $K$
be the path in $\mathcal{L}^{\prime}$ of points where the direction of
$\mathcal{L}^{\prime}$ maps under $S\to S^{\angle}$ to the interval
$\big{[}\frac{\varepsilon}{2},\varepsilon\big{]}$. Let $M=M(\varepsilon)$ be
the length of $K$.

Without loss of generality, we can assume that $\phi(x)$ is the starting point
of $K$. Furthermore, by the monotonicity in the claim and by Remark 5.10,
consecutive edges of $\phi\big{(}\mathcal{L}[0,T)\big{)}$ can be mapped by the
translations of $\nabla$ to disjoint (except at the endpoints) edges of
$\mathcal{L}^{\prime}$, ordered consistently. Consequently, if $\mathcal{L}$
has length $\geq M$, then the minimal subpath $K^{\prime}$ of
$\mathcal{L}^{\prime}$ containing all of these edges has also length $\geq M$.
Then $K^{\prime}$ contains a point where the direction of
$\mathcal{L}^{\prime}$ maps under $S\to S^{\angle}$ into
$[0,\frac{\varepsilon}{2}]$. By the claim, the sum of the curvatures at each
side of $\mathcal{L}$ is $\leq-\frac{\varepsilon}{2}$, as desired. If
$\mathcal{L}$ has length $<M$ but ends in the interior of $D$, then the image
under $S\to S^{\angle}$ of the direction of $\phi(e)$ above is the principal
point and so by the claim the sum of the curvatures at each side of
$\mathcal{L}$ is $\leq-\varepsilon$. ∎

##  8\. Rays

In this section, we discuss particular geodesic rays in $\mathbf{X}$ that
represent some points in $\partial_{\infty}\mathbf{X}$ fixed by elements
conjugate into $C$ or $B^{\prime}$.

###  8.1. Principal rays

Let $a\geq 1$. An _$a$ -principal ray_ is a geodesic ray in a chamber
$\mathbf{E}^{+}_{f}$ that is the reparametrisation of the curve
$\nu_{f,[ta,t,1]}$, where $t\geq t_{0}$ for some $t_{0}\geq 1$. An _$a$
-antiprincipal ray_ is a geodesic ray in a chamber $\mathbf{E}^{+}_{f}$ that
is the reparametrisation of the curve $\nu_{f,[t,a,1]}$, where $t\geq t_{0}$
for some $t_{0}\geq a$. A geodesic ray is _principal_ (resp.  _antiprincipal_)
if it is an $a$-principal (resp. $a$-antiprincipal) ray for some $a\geq 1$.
See Figure 20.

Figure 20. Principal and antipricipal rays in a chamber $\mathbf{E}^{+}_{f}$.