# Genericity of sublinearly Morse directions in general metric spaces

Yulan Qing  Department of Mathematics
University of Tennessee at Knoxville
1403 Circle Drive, Knoxville, TN 37916, USA<EMAIL_ADDRESS>and  Wenyuan Yang
Beijing International Center for Mathematical Research
Peking University
Beijing 100871, China P.R<EMAIL_ADDRESS>
###### Abstract.

In this paper, we show that for any proper statistically convex-cocompact
actions on proper metric spaces, the sublinearly Morse boundary has full
Patterson-Sullivan measure in the horofundction boundary.

###### 2000 Mathematics Subject Classification:

Primary 20F65, 20F67

(W.Y.) Partially supported by National Key R & D Program of China
(SQ2020YFA070059) and National Natural Science Foundation of China (No.
12131009 and No.12326601)

###### Contents

  1. 1 Introduction
  2. 2 Preliminary
    1. 2.1 Notations
    2. 2.2 Contracting subsets
    3. 2.3 Horofunction boundary
    4. 2.4 Sublinearly Morse boundaries
    5. 2.5 Convergence boundary
    6. 2.6 Regularly contracting geodesic rays
    7. 2.7 Statistically convex-cocompact actions
  3. 3 Conformal density on convergence boundary
    1. 3.1 Patterson-Sullivan measures on convergence boundary
    2. 3.2 Shadow Lemma
    3. 3.3 Conical points
  4. 4 Genericity of sublinearly Morse directions
    1. 4.1 Growth tightness of elements without frequent barriers
    2. 4.2 Non-frequently contracting segments have non-frequent barriers
    3. 4.3 Completion of the proof of Theorem A

##  1\. Introduction

Suppose that a group $G$ admits a proper and isometric action on a proper
geodesic metric space $(\mathrm{X},d)$. The group $G$ is assumed to be non-
elementary: there is no finite index subgroup isomorphic to the integer group
$\mathbb{Z}$ or the trivial group.

The contracting property captures a key feature of quasi-geodesics in Gromov
hyperbolic spaces, rank-1 geodesics in CAT(0) spaces, thick geodesics in
Teichmüller spaces, and many others. In recent years, this notion and its
variants have been proven fruitful in the study of general metric spaces.

Let $A$ be a closed subset of $\mathrm{X}$, and $\pi_{A}:\mathrm{X}\to A$ be
the nearest-point projection map. We say that $A$ is $C$–contracting for
$C\geq 0$ if

$\mathrm{diam}(\pi_{A}(B))\leq C$

for any metric ball $B$ disjoint from $A$. An element of infinite order is
called contracting, if it acts by translation on a contracting quasi-geodesic.

Contracting boundary for CAT(0) spaces was introduced by Charney-Sultan [2] as
quasi-isometric invariant, and has attracted active research interests in
recent years ([3, 14]). It is observed that the contracting boundary is
measurably negligible in harmonic measures and conformal measures. The
underlying set of the contracting boundary consists of the endpoints of
contracting geodesic rays in the visual boundary.

On the other hand, Qing and Rafi [14] showed that a much larger class of
sublinearly Morse geodesics equipped with an appropriate topology (called the
sublinearly Morse boundary $\partial_{\kappa}\mathrm{X}$) is still quasi-
isometrically invariant. See Definition 2.13. It is of interest to demonstrate
that the sublinearly Morse boundary is indeed larger than its QI-invariant
predecessors. In [8], the authors showed that a subset of the sublinearly
Morse directions that is _regularly contracting_ (see Definition 2.16) is
generic in the Patterson-Sullivan measure on visual boundary of CAT(0) spaces
under some conditions, and on the Thurston boundary of Teichmüller spaces.

The motivation of this paper is to extend the result of [8] to the conformal
measure on horofunction boundary of any proper metric space. Recently, a
general theory of a conformal density has been developed by Coulon and the
second-named author independently in [4, 23] on horofunction boundary in
presence of contracting elements. A key tool in this generalization is
assuming that the given $G\curvearrowright X$ is a proper, _statistically
convex-cocompact (SCC)_ action, introduced by the second-named author in [20],
with a contracting element. Except the negatively curved examples which lies
out of the interests of this paper, we mention:

  * •

CAT(0)-groups, including right-angled Coxeter/Artin groups, act geometrically
on CAT(0) spaces with rank-1 elements.

  * •

Mapping class groups act on Teichmüller space endowed with Teichmüller metric.

Using different and more geometrically flavored tool, we show that sublinearly
Morse directions form a full measure set with respect to conformal measures on
horofunction boundary.

Let $\partial_{h}{\mathrm{X}}$ be the horofunction boundary. Every sublinearly
Morse equivalent class $[\gamma]$ in $\partial_{\kappa}\mathrm{X}$ contains a
representative of geodesic ray $\gamma$, defining a horofunction $\gamma^{+}$
in $\partial{\mathrm{X}}$. By abuse of language, we also call the set of
images
$\\{\gamma^{+}\in\partial_{h}{\mathrm{X}}:[\gamma]\in\partial_{\kappa}\mathrm{X}\\}$
_sublinearly Morse directions_ in $\partial_{h}{\mathrm{X}}$ 111We do not
claim here the assignment $[\gamma]\to\gamma^{+}$ is well-defined, up to
finite difference or sublinear difference.. Our main result says that such
directions form a generic set with respect to conformal measures.

######  Theorem ATheorem 4.1.

Let $G\curvearrowright\mathrm{X}$ be a proper, non-elementary SCC action on a
proper geodesic metric space $\mathrm{X}$ with a contracting element. Let
$\mu_{o}$ be the $\omega_{G}$–dimensional conformal density on the
horofunction boundary $\partial_{h}{\mathrm{X}}$. Then the set of sublinearly
Morse directions is a $\mu_{o}$–full measure subset.

If $\mathrm{X}$ is a proper CAT(0) space, then there is a unique geodesic ray
representative, so gives an injective map from sublinearly Morse boundary
$\partial_{\kappa}\mathrm{X}$ into the visual boundary (homeomorphic to
$\partial_{h}{\mathrm{X}}$). We thus obtain the following result. This removes
some technical assumption (e.g. geodesically complete) on $\mathrm{X}$ in the
recent result of [8, Theorem 1.1], which is required to apply the results of
Ricks [16].

######  Theorem B.

Let $G\curvearrowright\mathrm{X}$ be a proper, non-elementary SCC action on a
proper CAT(0) space $\mathrm{X}$ with a rank-1 element. Let $\mu_{o}$ be the
$\omega_{G}$–dimensional conformal density on the the visual boundary
$\partial_{h}{\mathrm{X}}$. Then the set of sublinearly Morse directions is
$\mu_{o}$–full measure.

At last, let us emphasize that Theorem A holds for any convergence boundary
including Thurston boundary of Teichmüller spaces. In particular, this gives a
different proof of this result in [8]. We refer the reader to Theorem 4.1 for
the precise statement.

### Historic background and related works

In his celebrated proof of now called Mostow rigidity theorem, G. Mostow
carried out a two-step strategy as follows:

  1. (1)

Quasi-isometry induces an homeomorphisms called boundary map between visual
boundaries,

  2. (2)

Proving the boundary map that is quasi-conformal is actually conformal by
using ergodicity of geodesic flows. Absence in dimension 2, the non-
singularity of this map sending a positive Lebesgue measure to the target
turns out to be a remarkable feature in this approach.

Sought for a quasi-isometric invariant boundary has been an active research
topic, from a perspective of quasi-isometric classification in geometric group
theory. The famous counter-example of Croke-Kleiner [5] on the visual boundary
of CAT(0) spaces generates the interests to the subclass of hyperbolic-like
directions as the quasi-isometric invariant boundary, as shown in the fore-
mentioned works of Charney-Sultan [2] and Qing-Rafi-Tiozzo [15].

Stimulated by Step 2, it is natural to ask whether the Q.I. invariant boundary
under construction is generic in a measure of interest. Lebesgue measure on
the visual boundary of rank-1 symmetric spaces is an important instance of
conformal measures considered in this paper. First construction of such
measures appeared in the seminal work of Patterson [12] and has been further
developed by Sullivan, setting the track for fruitful research in last several
decades. Recently, the theory of conformal density on the horofunction
boundary is generalized to groups with contracting element independently by
Coulon [4] and by the second-named author in [23]. This is the main setup of
the paper. We refer the reader to these papers and references therein for a
detailed discussion.

Statistically convex-cocompact actions of groups has received wide attention
in the last few years. The concept was introduced in [20] to generalize the
idea of convex-cocompact subgroups in Kleinian groups to non-hyperbolic
settings. Independently, SCC actions were also introduced by Schapira-Tapie
[17] under the terminology of strongly positively recurrent actions (or
manifolds) in dynamical system. The class of SCC actions encompasses many
examples, but forms a strict subclass of actions with purely exponential
growth. In many circumstances, the latter is equivalent to the so-called
positively recurrent actions in [13].

Another measure of interest is harmonic measure arising from random walk on
Poisson boundary. Recently Qing-Rafi-Tiozzo [15] showed that, when
$\kappa=\log t$, the $\kappa$-boundary of the Cayley graph of the mapping
class group can be identified with the Poisson boundary of the associated
random walks. More general results concerning stationary measures were
recently announced by Inhyeok Choi, who in place of our ergodic theoretic and
boundary techniques uses a pivoting technique developed by Gouëzel. It will be
interesting to understand that under suitable conditions and restricted to the
correcting compactification, whether the sublinearly Morse directions that
supports the hitting measure and the ones that supports the Patterson-Sullivan
measures are singular or equivalent to each other.

Let us close the introduction by mentioning the following conjecture of
Sullivan [18]:

######  Conjecture 1.1.

Suppose that a discrete action $G\curvearrowright\mathbb{H}^{n}$ is of
divergence type. Let $\mu_{o}$ be the corresponding Patterson-Sullivan measure
supported on $\partial\mathbb{H}^{n}$. Then the following set of boundary
points $\xi\in\partial\mathbb{H}^{n}$ for which the geodesic ray
$\gamma=[o,\xi]$ satisfies

$\lim_{t\to\infty}\frac{d(\gamma(t),Go)}{t}=0$

is generic.

This was confirmed by Sullivan [18, Corollary 19] if the Bowen-Margulis
measure on geodesic flow is finite. It follows by definition that this limit
tends to $0$ along regularly contracting rays, so the genericty of regularly
contracting rays would imply the conjecture. In [8, Thm 5.1], it is proved
that the genericty of regularly contracting rays holds for any negatively
curved Riemannian manifold with finite BM measure on geodesic flow.

Note that if the action $G\curvearrowright\mathrm{X}$ is of divergence type,
then $\mu_{o}$ is supported on conical limit points of $G$ by Hopf-Tsuji-
Sullivan dichotomy. The conjecture then predicts it is further supported on
the “sublinear” limit points (without assuming SCC action or finiteness of BMS
measures etc). Therefore, if a variant of Sullivan’s conjecture holds for the
set of regularly contracting rays, then Theorem A would be valid under a
proper action of divergence type. See [7, Sec. 6] for other implication of
Sullivan’s conjecture.

#### Organization of the paper

We present relevant background in Section 2. Section 3 collects all tools from
the recently developed conformal density theory on convergence boundary.
Section 4 is devoted to the proof of Theorem A.

##  2\. Preliminary

###  2.1. Notations

Let $(\mathrm{X},d)$ be a proper geodesic metric space. Let $A$ be a closed
subset of $\mathrm{X}$ and $x$ be a point in $\mathrm{X}$. By $d(x,A)$ we mean
the set-distance between $x$ and $A$, i.e.

$d(x,A):=\inf\big{\\{}d(x,y):y\in A\big{\\}}.$

Let

$\pi_{A}(x):=\big{\\{}y\in A:d(x,y)=d(x,A)\big{\\}}$

be the set of nearest-point projections from $x$ to $A$. Since $\mathrm{X}$ is
a proper metric space, $\pi_{A}(x)$ is non empty. We refer to $\pi_{A}(x)$ as
the _projection set_ of $x$ to $A$. Define

$\textbf{d}_{A}(x,y)=\mathrm{diam}(\pi_{A}(x)\cup\pi_{A}(y)).$

If a set $A$ is countable then we use $\sharp A$ to denote the number of
elements in $A$.

Let $\alpha:[a,b]\to\mathrm{X}$ be a path with arc-length parametrization from
the initial point $\alpha^{-}=\alpha(a)$ to the terminal point
$\alpha^{+}=\alpha(b)$. If $x=\alpha(s)$ and $y=\alpha(t)$ are two points on
$\alpha$, $[x,y]_{\alpha}$ denotes the parametrized subpath of $\alpha$ going
from $x$ to $y$, that is, the restriction of $\alpha$: $[s,t]\to\mathrm{X}$.
We also write $\alpha[s,t]$ simply to denote the segment restricting on the
interval $[s,t]$. Let $[x,y]$ denote a geodesic segment (not necessarily
unique) between $x,y\in\mathrm{X}$.

If $\beta$ is a geodesic ray emanating from the base-point, then $\beta|_{{\sf
r}}$ denote the point on $\beta$ that is distance ${\sf r}$ from ${o}$.

A continuous path $\alpha$ is called a $(q,Q)$–quasi-geodesic for constants
$q\geq 1$, $Q>0$ if for any rectifiable subpath $\beta$,

$\ell(\beta)\leq q\cdot d(\beta^{-},\beta^{+})+Q$

where $\ell(\beta)$ denotes the length of $\beta$. Equivalently, if
$\alpha:[a,b]\subseteq\mathbb{R}\to\mathrm{X}$ is parametrized with arclength
parametrization, for any $s,t\in[a,b]$, we have

$\frac{|s-t|}{{\sf q}}-Q\leq d\big{(}\alpha(s),\alpha(t)\big{)}\leq
q\,|s-t|+Q.$

The additional assumption that quasi-geodesics are continuous is not
necessary, but it is added for convenience and to make the exposition simpler.
When $X$ is a geodesic metric space, one can always adjust a quasi-geodesic
ray slightly to make it continuous (see [1, Lemma III.1.11]).

Denote by $\alpha\cdot\beta$ (or simply $\alpha\beta$) the concatenation of
two paths $\alpha,\beta$ provided that $\alpha^{+}=\beta^{-}$.

We say that a subset $A$ is $r$–close to $B$ if $A$ is contained in the
$r$–neighborhood of $B$.

Let $f,g$ be real-valued functions. Then $f\prec_{c_{i}}g$ means that there is
a constant $C>0$ depending on parameters $c_{i}$ such that $f<Cg$. The symbol
$\succ_{c_{i}}$ is defined similarly, and $\asymp_{c_{i}}$ means both
$\prec_{c_{i}}$ and $\succ_{c_{i}}$ are true. The constant $c_{i}$ will be
omitted if it is a universal constant.

###  2.2. Contracting subsets

######  Definition 2.1 (Contracting subset).

For given $C\geq 1$, a subset $U$ in $\mathrm{X}$ is called $C$–contracting if
for any geodesic $\gamma$ with $d(\gamma,U)\geq C$, we have

$\mathrm{diam}(\pi_{U}(\gamma))\leq C.$

A collection of $C$–contracting subsets is referred to as a $C$–contracting
system.

A subset $U\subseteq\mathrm{X}$ is called quasi-convex if for a function
$\sigma:\mathbb{R}^{+}\to\mathbb{R}^{+}$: given $(q,Q)\geq(1,0)$, any
$(q,Q)$–quasi-geodesic with endpoints in $U$ lies in $N_{\sigma((q,Q))}(U)$.

######  Lemma 2.2.

Let $U$ be a $C$-contracting subset. Then

  1. (1)

There exists $\sigma=\sigma(C)$ such that $U$ is $\sigma$–quasi-convex.

  2. (2)

For any $r>0$, there exists $\hat{C}=\hat{C}(C,r)$ such that a subset
$V\subseteq\mathrm{X}$ having Hausdorff distance at most $r$ to $U$ is
$\hat{C}$–contracting.

  3. (3)

For any $\lambda\geq 1,Q\geq 0$, there exists $\hat{C}=\hat{C}(\lambda,Q,C)$
so that any subpath of a $(\lambda,Q)$–quasi-geodesic $U$ is
$\hat{C}$–contracting.

  4. (4)

There exists $\hat{C}=\hat{C}(C)$ such that $d_{U}(y,z)\leq d(y,z)+\hat{C}$
for any $y,z\in\mathrm{X}$.

###### Proof.

The assertion (3) is proved in [21, Prop 2.2.3]. The others are
straightforward applications of contracting property. ∎

In this paper, we are interested in a contracting system $\mathscr{F}$ with
$\tau$–bounded intersection property for a function $\tau:\mathbb{R}_{\geq
0}\to\mathbb{R}_{\geq 0}$ if the following holds

$\forall U,V\in\mathscr{F}:\quad\mathrm{diam}({N_{r}(U)\cap
N_{r}(V)})\leq\tau(r)$

for any $r\geq 0$. This is, in fact, equivalent to a bounded projection
property of $\mathscr{F}$: there exists a constant $B>0$ such that the
following holds

$\mathrm{diam}(\pi_{U}(V))\leq B$

for any $U\neq V\in\mathscr{F}$. See [22, Lemma 2.3].

An infinite order element $h\in G$ is called contracting if the subgroup
$\langle h\rangle$ acts by translation on a contracting quasi-geodesic. The
set of contracting elements is preserved under conjugacy.

######  Lemma 2.3.

[20, Lemma 2.11] Let $h\in G$ be a contracting element. Then it is contained
in the following unique maximal elementary subgroup

$E(h)=\\{g\in G:\exists
n\in\mathbb{N}_{>0},(\;gh^{n}g^{-1}=h^{n})\;\lor\;(gh^{n}g^{-1}=h^{-n})\\}.$

Keeping in mind the basepoint $o\in\mathrm{X}$, the axis of $h$ is defined as
the following quasi-geodesic

(1) $\mathrm{Ax}(h)=\\{fo:f\in E(h)\\}.$

Notice that $\mathrm{Ax}(h)=\mathrm{Ax}(k)$ and $E(h)=E(k)$ for any
contracting element $k\in E(h)$.

###  2.3. Horofunction boundary

We recall the construction of horofunction boundary, which are endowed with
the so-called finite and sublinear difference partitions.

Fix a basepoint $o\in\mathrm{X}$. For each $y\in\mathrm{X}$, we define a
Lipschitz map $b_{y}:\mathrm{X}\to\mathrm{X}$ by

$\forall x\in\mathrm{X}:\quad b_{y}(x)=d(x,y)-d(o,y).$

This family of $1$–Lipschitz functions sits in the set of continuous functions
on $\mathrm{X}$ vanishing at $o$. Endowed with the compact-open topology, the
Arzela-Ascoli Lemma implies that the closure of $\\{b_{y}:y\in\mathrm{X}\\}$
gives a compactification of $\mathrm{X}$. The complement, denote by
$\partial_{h}{\mathrm{X}}$, of $\mathrm{X}$ is called the horofunction
boundary.

A Buseman cocycle $B_{\xi}:\mathrm{X}\times\mathrm{X}\to\mathbb{R}$
(independent of $o$) is given by

$\forall x_{1},x_{2}\in\mathrm{X}:\quad
B_{\xi}(x_{1},x_{2})=b_{\xi}(x_{1})-b_{\xi}(x_{2}).$

The topological type of horofunction boundary is independent of the choice of
basepoints. Every isometry $\phi$ of $\mathrm{X}$ induces a homeomorphism on
$\overline{\mathrm{X}}$:

$\forall
y\in\mathrm{X}:\quad\phi(\xi)(y):=b_{\xi}(\phi^{-1}(y))-b_{\xi}(\phi^{-1}(o)).$

According to the context, both $\xi$ and $b_{\xi}$ are used to denote the
boundary points.

####  Finite difference relation.

Two horofunctions $b_{\xi},b_{\eta}$ have $K$–finite difference for $K\geq 0$
if the $L^{\infty}$–norm of their difference is $K$–bounded:

$\|b_{\xi}-b_{\eta}\|_{\infty}\leq K.$

The locus of $b_{\xi}$ consists of horofunctions $b_{\eta}$ so that
$b_{\xi},b_{\eta}$ have $K$–finite difference for some $K$ depending on
$\eta$. The loci $[b_{\xi}]$ of horofunctions $b_{\xi}$ form a finite
difference equivalence relation $[\cdot]$ on $\partial_{h}{\mathrm{X}}$. The
locus $[\Lambda]$ of a subset $\Lambda\subseteq\partial_{h}{\mathrm{X}}$ is
the union of loci of all points in $\Lambda$. We say that $\Lambda$ is
saturated if $[\Lambda]=\Lambda$.

Say that a sequence of subsets $X_{n}$ is escaping if $d(o,X_{n})\to\infty$.
We summarize the following properties of horofunction boundary in [23, Sec.
5], which are called the convergence property of the horofunction boundary.

####  Sublinear difference relation.

It will be also useful to consider another bigger relation than the finite
difference relation. We say two horofunctions
$b_{\xi},b_{\eta}:[0,\infty)\to\mathrm{X}$ have sublinear difference if

(2) $\lim_{n\to\infty}\sup_{d(o,x)\geq
n}\frac{|b_{\xi}(x)-b_{\eta}(x)|}{d(o,x)}=0.$

Since two horofunction representatives of a given point
$\xi\in\partial{\mathrm{X}}$ differ by a constant for different basepoints,
this relation is independent on the choice of basepoint. The sublinear
difference relation is an equivalence relation. We denote by $[\xi]_{s}$ the
equivalent class of $\xi\in\partial{\mathrm{X}}$, and
$[\partial_{h}{\mathrm{X}}]_{s}$ the resulting quotient space of
$\partial_{h}{\mathrm{X}}$. It is clear that $[\xi]\subseteq[\xi]_{s}$, so
$[\partial_{h}{\mathrm{X}}]$ is also quotient of
$[\partial_{h}{\mathrm{X}}]_{s}$. In particular, Theorem 2.15 holds for
sublinear equivalence relation as well, where $[\cdot]$ could be replaced with
$[\cdot]_{s}$.

Any geodesic ray $\alpha$ tends to a unique point, denoted by $\alpha^{+}$, at
the horofunction boundary $\partial_{h}{\mathrm{X}}$. Namely, the associated
horofunction is as follows

$\alpha^{+}(x):=\lim_{t\to\infty}[d(\alpha(t),x)-t]$

######  Lemma 2.4.

Let $\alpha,\beta:[0,\infty)\to\mathrm{X}$ be two geodesic ways issuing from
the same base-point $o\in\mathrm{X}$, ending at
$\alpha^{+},\beta^{+}\in\partial_{h}{\mathrm{X}}$ respectively. If there are
two sequences of real numbers $s_{n},r_{n}\to\infty$ satisfying

(3) $\displaystyle\limsup_{n\to\infty}r_{n}/s_{n}<1$ (4) $\displaystyle\forall
n\gg 0,\;\limsup_{t\to\infty}d(o,[\alpha(s_{n}),\beta(t)])\leq r_{n}$

then $[\alpha^{+}]_{s}\neq[\beta^{+}]_{s}$.

###### Proof.

Let $x_{n}=\alpha(s_{n})\in\alpha$ so we have $\alpha^{+}(x_{n})=-s_{n}$. We
are going to prove that $x_{n}$ violates (2). For small enough $\epsilon>0$,
according to (4), there are $t_{m}\to\infty$ so that

$d(o,[\alpha(s_{n}),\beta(t_{m})])\leq r_{n}+\epsilon$

so by triangle inequality,

$|d(\alpha(s_{n}),\beta(t_{m}))-s_{n}-t_{m}|\leq 2(r_{n}+\epsilon)$

Thus, letting $s_{n}\to\infty$ and $\epsilon\to 0$,

$\displaystyle\limsup_{s_{n}\to\infty}\frac{|\beta^{+}(x_{n})-\alpha^{+}(x_{n})|}{s_{n}}$
$\displaystyle=\limsup_{s_{n}\to\infty}\frac{|\beta^{+}(x_{n})+s_{n}|}{s_{n}}$
$\displaystyle=\frac{|\lim_{t_{m}\to\infty}[d(\beta(t_{m}),x_{n})-t_{m}-s_{n}]+2s_{n}|}{s_{n}}>0$

where the last inequality uses (3). Hence,
$[\alpha^{+}]_{s}\neq[\beta^{+}]_{s}$ follows by definition. ∎

######  Lemma 2.5.

Let $\alpha,\beta$ be two geodesic rays ending at
$[\alpha^{+}]\neq[\beta^{+}]$ respectively. Assume that $\alpha$ contains
infinitely many disjoint $C$–contracting segments of length $L$ in its
$r$–neighborhood. If $L\gg 0$, then $[\alpha^{+}]_{s}\neq[\beta^{+}]_{s}$.

###### Proof.

Pick up two unbounded sequences of $x_{n}\in\alpha$ and $y_{n}\in\beta$. By
assumption, $\alpha$ contains infinitely many $C$–contracting segments $p_{m}$
of length $L$. By the $C$–contracting property of $p_{m}$, if
$[x_{n},y_{n}]\cap N_{C}(p_{m})=\emptyset$ for infinitely many $p_{m}$, then
$\beta\cap N_{C}(p_{m})\neq\emptyset$. As $p_{m}$ is escaping, this implies
the intersection of $\alpha$ and $N_{r+C}(\beta)$ is unbounded. By definition
of horofunctions, $[\alpha^{+}]=[\beta^{+}]$ are in the same finite difference
class, contradicting the assumption. Consequently, there must exist some
$p_{m}$ so that $[x_{n},y_{n}]\cap N_{C}(p_{m})\neq\emptyset$. Again the
$C$–contracting property of $p_{m}$ shows $[x_{n},y_{n}]$ intersects
$N_{C}(p_{m})$ as $n\to\infty$. Setting $D=d(o,p_{m})+r+C$, we obtain
$d(o,[x_{n},y_{n}])\leq D$, where $D$ is independent of $n$. It thus follows
from Lemma 2.4 that $[\xi]_{s}\neq[\eta]_{s}$. The proof is complete. ∎

For hyperbolic spaces, the sublinear and finite difference relation turns out
to be the same on $\partial_{h}{\mathrm{X}}$.

######  Corollary 2.6.

If $\mathrm{X}$ is a Gromov hyperbolic geodesic space, then
$[\partial_{h}{\mathrm{X}}]_{s}$ is homeomorphic to the Gromov boundary
$\partial{\mathrm{X}}$.

###### Proof.

Any geodesic ray in a hyperbolic space is uniformly contracting. The
conclusions follows immediately from Lemma 2.5. ∎

###  2.4. Sublinearly Morse boundaries

Now we introduce a large class of quasi-geodesic rays that are quasi-isometry
invariant. Intuitively, these quasi-geodesics have a weak Morse-like property.
To begin with, we fix a function that is sublinear in the following sense:

#### Sublinear functions

We fix a function

$\kappa\colon\thinspace[0,\infty)\to[1,\infty)$

that is monotone increasing, concave and sublinear, that is

$\lim_{t\to\infty}\frac{\kappa(t)}{t}=0.$

Note that using concavity, for any $a>1$, we have

(5) $\kappa(at)\leq
a\left(\frac{1}{a}\,\kappa(at)+\left(1-\frac{1}{a}\right)\kappa(0)\right)\leq
a\,\kappa(t).$

We say a quantity ${\sf D}$ _is small compared to a radius ${\sf r}>0$_ if

(6) ${\sf D}\leq\frac{{\sf r}}{2\kappa({\sf r})}.$

######  Remark.

The assumption that $\kappa$ is increasing and concave makes certain arguments
cleaner, otherwise they are not really needed. One can always replace any
sublinear function $\kappa$, with another sublinear function
$\overline{\kappa}$ so that $\kappa(t)\leq\overline{\kappa}(t)\leq{\sf
C}\,\kappa(t)$ for some constant ${\sf C}$ and $\overline{\kappa}$ is monotone
increasing and concave. For example, define

$\overline{\kappa}(t)=\sup\Big{\\{}\lambda\kappa(u)+(1-\lambda)\kappa(v)\,\mathbin{\Big{|}}\,\
0\leq\lambda\leq 1,\ u,v>0,\ \text{and}\ \lambda u+(1-\lambda)v=t\Big{\\}}.$

The requirement $\kappa(t)\geq 1$ is there to remove additive errors in the
definition of $\kappa$–contracting geodesics.

######  Definition 2.7 ($\kappa$–neighborhood).

Recall that, for $x\in X$, we have $\lVert x\rVert=d(o,x)$. To simplify
notation, we often drop $\lVert{\mathchoice{\mkern 1.0mu\mbox{\raise
2.2pt\hbox{$\centerdot$}}\mkern 1.0mu}{\mkern 1.0mu\mbox{\raise
2.2pt\hbox{$\centerdot$}}\mkern 1.0mu}{\mkern 1.5mu\centerdot\mkern
1.5mu}{\mkern 1.5mu\centerdot\mkern 1.5mu}}\rVert$. That is, for $x\in X$, we
define

$\kappa(x):=\kappa(\lVert x\rVert).$

For a closed set $Z$ and a constant ${\sf n}$, define the $(\kappa,{\sf
n})$–neighbourhood of $Z$ to be

$\mathcal{N}_{\kappa}(Z,{\sf
n})=\Big{\\{}x\in\mathrm{X}\,\mathbin{\Big{|}}\,d(x,Z)\leq{\sf
n}\cdot\kappa(x)\Big{\\}}.$

$o$$\tau$$x$$x_{b}$${\sf n}\cdot\kappa(x)$$\lVert x\rVert$$(\kappa,{\sf
n})$–neighbourhood of $\tau$ Figure 1. A $\kappa$–neighbourhood of a geodesic
ray $\tau$ with multiplicative constant ${\sf n}$.

In this paper, $Z$ is either a geodesic or a quasi-geodesic. That is, we can
write $\mathcal{N}_{\kappa}(\tau,{\sf n})$ to mean the $(\kappa,{\sf
n})$–neighborhood of the image of the geodesic ray $\tau$. Or, we can use
phrases like “the quasi-geodesic $\beta$ is $\kappa$–contracting” or “the
geodesic $\tau$ is in a $(\kappa,{\sf n})$–neighbourhood of the geodesic $c$”.

We recall the definition of $\kappa$–contracting and $\kappa$–Morse sets from
[15].

######  Definition 2.8 ($\kappa$–Morse I).

We say a closed subset $Z$ of $\mathrm{X}$ is _$\kappa$ –Morse_ if there is a
function

${\sf m}_{Z}\colon\thinspace\mathbb{R}_{+}^{2}\to\mathbb{R}_{+}$

so that if $\beta\colon\thinspace[s,t]\to\mathrm{X}$ is a $(q,Q)$–quasi-
geodesic with end points on $Z$ then

$\beta[s,t]\subseteq\mathcal{N}_{\kappa}\big{(}Z,{\sf m}_{Z}(q,Q)\big{)}.$

We refer to ${\sf m}_{Z}$ as a _Morse gauge_ for $Z$. Without loss of
generality one can assume

(7) ${\sf m}_{Z}(q,Q)\geq\max(q,Q).$

######  Definition 2.9 ($\kappa$–Morse II).

We say a closed subset $Z$ of $\mathrm{X}$ is _$\kappa$ –Morse II_ if there is
a function ${\sf m}_{Z}\colon\thinspace\mathbb{R}^{2}\to\mathbb{R}$ such that,
for every constants ${\sf r}>0$, ${\sf n}>0$ and every sublinear function
$\kappa^{\prime}$, there is an ${\sf R}={\sf R}(Z,{\sf r},{\sf
n},\kappa^{\prime})>0$ where the following holds: Let
$\eta\colon\thinspace[0,\infty)\to\mathrm{X}$ be a $(q,Q)$–quasi-geodesic ray
so that ${\sf m}_{Z}(q,Q)$ is small compared to ${\sf r}$, let $t_{\sf r}$ be
the first time $\lVert\eta(t_{\sf r})\rVert={\sf r}$ and let $t_{\sf R}$ be
the first time $\lVert\eta(t_{\sf R})\rVert={\sf R}$. Then

$d\big{(}\eta(t_{\sf R}),Z\big{)}\leq{\sf n}\cdot\kappa^{\prime}({\sf
R})\quad\Longrightarrow\quad\eta[0,t_{\sf
r}]\subseteq\mathcal{N}_{\kappa}\big{(}Z,{\sf m}_{Z}(q,Q)\big{)}.$

It is also natural to generalize the notion of contracting to the sublinear
setting:

######  Definition 2.10 ($\kappa$–contracting).

For a closed subspace $Z$ of $\mathrm{X}$, we say $Z$ is _$\kappa$
–contracting_ if there is a constant ${\sf c}_{Z}$ so that, for every
$x,y\in\mathrm{X}$

$d(x,y)\leq
d(x,Z)\quad\Longrightarrow\quad\textbf{d}_{Z}\big{(}x,y\big{)}\leq{\sf
c}_{Z}\cdot\kappa(x).$

${o}$ Figure 2. A sublinearly contracting geodesic ray

The following theorem summarizes the relation of the above three notions.

######  Theorem 2.11 ([15]).

Let $\mathrm{X}$ be a proper geodesic metric space and let $\tau$ be a quasi-
geodesic ray in $\mathrm{X}$. Then

  1. (1)

$\tau$ is $\kappa$–Morse I if and only if $\tau$ is $\kappa$–Morse II. Thus we
say a set is kappa-Morse if it is either $\kappa$–Morse I or $\kappa$–Morse
II.

  2. (2)

If $\tau$ is $\kappa$–contracting then $\tau$ is $\kappa$–Morse.

  3. (3)

If $\tau$ is $\kappa$-Morse then it is $\kappa^{\prime}$-contracting.

Analogous to Lemma 2.3 (2), $\kappa$–Morse is a property than can also be
established by approximity. That is to say, if a quasi-geodesic ray $\alpha$
sublinearly fellow travel a $\kappa$–Morse ray $\beta$, then $\alpha$ is also
$\kappa$–Morse. We now make this precise. First assume without loss of
generality that a quasi-geodesic ray is a continuous path. Define

$\displaystyle\alpha_{r}:=\\{\alpha(t_{0})\,|\,$
$\displaystyle\alpha(t_{0})\in(\alpha\cap X\backslash B(o,r)),\text{ and for
any other }$ $\displaystyle\text{}\alpha(t)\in\alpha\cap X\backslash
B(o,r),\text{ we have }t_{0}\leq t\\}$

We say two quasi-geodesic rays _sublinearly fellow travel_ if

$\lim_{r\to\infty}\frac{d(\alpha_{r},\beta_{r})}{r}=0.$

######  Theorem 2.12.

[15] Let $\alpha$ be a $(q_{1},Q_{2})$–quasi-geodesic ray and let $\beta$ be a
$(q_{2},Q_{2})$–quasi-geodesic ray that is $\kappa$–Morse. If $\alpha$ and
$\beta$ sublinearly fellow travel, then there exists a $\kappa$–neighborhood
depending only on $(q_{1},Q_{1})$ and $(q_{2},Q_{2})$ such that

$\alpha\in N_{\kappa}\big{(}\beta,m((q_{1},Q_{1}),(q_{2},Q_{2}))\big{)}.$

Furthermore, $\alpha$ is a $\kappa$–Morse ray.

Lastly, a quasi-geodesic is called _sublinearly Morse_ if it is $\kappa$–Morse
for some sublinearly growing function $\kappa$.

######  Definition 2.13 (Sublinearly Morse boundary).

Given a sublinear function $\kappa$, let $\partial_{\kappa}X$ denote the set
of equivalence classes of $\kappa$–Morse quasi-geodesics. Equipped with a
coarse version of cone topology, we call this set the _$\kappa$ –Morse
boundary_ of $\mathrm{X}$ and denote it $\partial_{\kappa}\mathrm{X}$ (for
more details, see [15]).

It is shown in [15] that $\mathrm{X}\cup\partial_{\kappa}\mathrm{X}$ with the
coarse cone topology is a QI-invariant space and a metrizable topological
space.

###  2.5. Convergence boundary

In this subsection, we discuss an axiomatic approach introduced in [23] to the
boundary of proper geodesic metric spaces in presence of contracting subsets.
The motivating examples are Gromov boundary of hyperbolic spaces and visual
boundary of CAT(0) spaces. Before describing more examples in Examples, we
need to introduce a few terminologies.

Let $(\mathrm{X},d)$ be a proper metric space admitting an isometric action of
a non-elementary countable group $\Gamma$ with a contracting element. Consider
a metrizable compactification
$\overline{\mathrm{X}}:=\partial{\mathrm{X}}\cup\mathrm{X}$, so that
$\mathrm{X}$ is open and dense in $\overline{\mathrm{X}}$. We also assume that
the action of $\textrm{Isom}(\mathrm{X})$ extends by homeomorphism to
$\partial{\mathrm{X}}$.

We equip $\partial{\mathrm{X}}$ with a $\textrm{Isom}(\mathrm{X})$–invariant
partition $[\cdot]$: $[\xi]=[\eta]$ implies $[g\xi]=[g\eta]$ for any
$g\in\textrm{Isom}(\mathrm{X})$. The locus $[Z]$ of a subset
$Z\subseteq\partial{\mathrm{X}}$ is the union of all $[\cdot]$–classes of
$\xi\in Z$. We say that $\xi$ is minimal if $[\xi]=\\{\xi\\}$, and a subset
$U$ is $[\cdot]$–saturated if $U=[U]$.

We say that $[\cdot]$ restricts to be a closed partition on a
$[\cdot]$–saturated subset $U\subseteq\partial{\mathrm{X}}$ if $x_{n}\in
U\to\xi\in\partial{\mathrm{X}}$ and $y_{n}\in U\to\eta\in\partial{\mathrm{X}}$
are two sequences with $[x_{n}]=[y_{n}]$, then $[\xi]=[\eta]$. (Possibly
$\xi,\eta$ are not in $U$ anymore.) If $U=\partial{\mathrm{X}}$, this is
equivalent to say that the relation $\\{(\xi,\eta):[\xi]=[\eta]\\}$ is a
closed subset in $\partial{\mathrm{X}}\times\partial{\mathrm{X}}$, so the
quotient space $[\partial{\mathrm{X}}]$ is Hausdorff. In general, $[\cdot]$
may be not closed over the whole $\partial{\mathrm{X}}$ (say the horofunction
boundary with finite difference relation), but is closed on certain
interesting subsets as requested in Assumption (C) below.

With respect to the given partition, we say that a sequence of points $x_{n}$
tend (resp. accumulates) to $[\xi]$ if the limit point (resp. any accumulate
point) is contained in $[\xi]$. This implies that $[x_{n}]$ tends or
accumulates to $[\xi]$ in the quotient space $[\partial{\mathrm{X}}]$. So, an
infinite ray $\gamma$ terminates at a point in $[\xi]\in\partial{\mathrm{X}}$
if any sequence of points on $\gamma$ accumulates in $[\xi]$.

Recall that $\Omega_{o}(A)=\\{x\in\mathrm{X}:[o,x]\cap A\neq\emptyset\\}$ is
the cone of a subset $A\subseteq\mathrm{X}$ with light source at $o$. A
sequence of subsets $A_{n}$ is called escaping if $d(o,A_{n})\to\infty$ for
some (or any) $o\in\mathrm{X}$.

######  Definition 2.14.

We say that $(\overline{\mathrm{X}},[\cdot])$ is a convergence
compactification if the following assumptions hold.

  1. (A)

Any contracting geodesic ray $\gamma$ accumulates into a closed subset $[\xi]$
for some $\xi\in\partial{\mathrm{X}}$ such that any sequence of
$y_{n}\in\mathrm{X}$ with escaping projections $\pi_{\gamma}(y_{n})$ tends to
$[\xi]$.

  2. (B)

Let $\\{\gamma_{n}\subseteq\mathrm{X}:n\geq 1\\}$ be an escaping sequence of
$C$–contracting quasi-geodesics for some $C>0$. Then for any given
$o\in\mathrm{X}$, there exist a subsequence of
$\\{A_{n}:=\Omega_{o}(\gamma_{n}):n\geq 1\\}$ (still denoted by $A_{n}$) and
$\xi\in\partial{\mathrm{X}}$ such that $A_{n}$ accumulates into $[\xi]$:

Any convergent sequence of points $x_{n}\in A_{n}$ tends to a point in
$[\xi]$.

  3. (C)

The set $\mathcal{C}$ of non-pinched points $\xi\in\partial{\mathrm{X}}$ is
non-empty. If $x_{n},y_{n}\in\mathrm{X}$ are two sequence of points converging
to $[\xi]$, then $[x_{n},y_{n}]$ is an escaping sequence of geodesic segments.

$\pi_{\gamma}(y_{n})$$y_{n}$$[\xi]$$o$$\gamma_{1}$$[\xi]$$\gamma_{n}$$y_{1}\in\Omega_{o}(\gamma_{1})$$y_{n}\in\Omega_{o}(\gamma_{n})$$o$$o$$x_{n}$$y_{n}$$[\xi]\subseteq\mathcal{C}$$\Longrightarrow$$d(o,[x_{n},y_{n}])\rightarrow\infty$(A)(B)(C)
Figure 3. Illustrate Assumptions (A)(B)(C) in Definition 2.14

######  Examples.

The first three convergence boundaries are equipped with a maximal partition
$[\cdot]$ (i.e. each $[\cdot]$-class is singleton). See [23] for more details.

  1. (1)

Hyperbolic space $\mathrm{X}$ with Gromov boundary $\partial{\mathrm{X}}$,
where all boundary points are non-pinched.

  2. (2)

CAT(0) space $\mathrm{X}$ with visual boundary $\partial{\mathrm{X}}$
(homeomorphic to horofunction boundary), where all boundary points are non-
pinched.

  3. (3)

The Cayley graph $\mathrm{X}$ of a relatively hyperbolic group with Bowditch
or Floyd boundary $\partial{\mathrm{X}}$, where conical limit points are non-
pinched.

If $\mathrm{X}$ is infinitely ended, we could also take $\partial{\mathrm{X}}$
as the end boundary with the same statement.

  4. (4)

Teichmüller space $\mathrm{X}$ with Thurston boundary $\partial{\mathrm{X}}$,
where $[\cdot]$ is given by Kaimanovich-Masur partition [10] and uniquely
ergodic points are non-pinched.

  5. (5)

Any proper metric space $\mathrm{X}$ with horofunction boundary
$\partial{\mathrm{X}}$, where $[\cdot]$ is given by finite difference
partition and all boundary points are non-pinched. If $\mathrm{X}$ is the
cubical CAT(0) space, the horofunction boundary is exactly the Roller
boundary. If $\mathrm{X}$ is the Teichmüller space with Teichmüller metric,
the horofunction boundary is the Gardiner-Masur boundary ([11, 19]).

From these examples, we see that the convergence boundary is not unique for a
fixed proper metric space. In applications, the horofunction boundary provides
a non-trivial convergence boundary for any proper action.

######  Theorem 2.15.

[23, Theorem 1.1] The horofunction boundary is a convergence boundary with
finite difference relation $[\cdot]$, where all boundary points are non-
pinched.

As finite difference partition is finer than the sublinear difference
partition, we see by definition that the above conclusion holds for sublinear
difference partition as well.

###  2.6. Regularly contracting geodesic rays

The notion of the regularly contracting segment introduced in [8] shall be
recalled in the subsection. This is a metric notion without involving any
group action.

For any $\theta\in[0,1]$, if $\gamma$ is a geodesic, a $\theta$–segment means
a connected and open subsegment of $\gamma$ with length $\theta\ell(\gamma)$.

######  Definition 2.16.

Fix $r,C>0$.

  1. (1)

Given $L>0,\theta\in(0,1]$, we say that a geodesic $\gamma$ is
$(r,C,L)$–contracting at $\theta$–frequency if every $\theta$–segment of
$\gamma$ contains a subsegment of length $L$ that is $r$–close to a
$C$–contracting geodesic. That is to say, for any $0<t<(1-\theta)\ell(\gamma)$
there is an interval of time $[s-L,s+L]\subset[t,t+\theta\ell(\gamma)]$ and a
$C$–contracting geodesic $p$ such that,

(8) $u\in[s-L,s+L]\qquad\Longrightarrow\qquad d(\gamma(u),p)\leq r.$

  2. (2)

A geodesic ray $\gamma\colon\thinspace[0,\infty)\to\mathrm{X}$ is
$(r,C,L)$–contracting at $\theta$–frequency if there is an $R_{0}>0$ such that
any initial segment of $\gamma$ of length at least $R_{0}$ (i.e. the segment
$\gamma[0,t]$ for any $t\geq R_{0}$) is $(r,C,L)$–contracting at
$\theta$–frequency.

  3. (3)

A geodesic ray $\gamma\colon\thinspace[0,\infty)\to\mathrm{X}$ is _$(r,C)$
–regularly contracting_ if $\tau$ is $(r,C,L)$–contracting at
$\theta$–frequency for each $L>0$ and $\theta\in(0,1]$.

A bi-infinite geodesic $\gamma$ is regularly contracting if the rays
$t\to\gamma(t)$ and $t\to\gamma(-t)$ are both regularly contracting.

Let us note the following connection with convergence boundary.

######  Lemma 2.17.

Let $(\mathrm{X},\partial{\mathrm{X}})$ be a convergence compactification.
Then a $(r,C,L)$–contracting geodesic ray $\gamma$ at $\theta$–frequency
accumulates into a unique $[\cdot]$–class denoted by $[\gamma^{+}]$ in
$[\partial{\mathrm{X}}]$.

###### Proof.

Let $\gamma$ be an $(r,C,L)$–contracting geodesic ray at $\theta$–frequency,
where $p_{n}$ is a sequence of the $C$–contracting geodesic segments
satisfying (8). Up to taking subsegment of length $L$, we may assume that
$p_{n}$ is escaping.

Apply Definition 2.14(B) to the sequence of $(C+1)$-contracting quasi-
geodesics $N_{r}(p_{n})$. We thus obtain a unique $[\cdot]$–class in
$\partial{\mathrm{X}}$, denoted by $[\gamma^{+}]$, and that
$\Omega_{o}(N_{r}(p_{n}))$ accumulates to $[\gamma^{+}]$. In particular, any
sequence of points on $\gamma$ are eventually contained in
$\Omega_{o}(N_{r}(p_{n}))$, so tend to $[\gamma^{+}]$. The proof is complete.
∎

As fore-mentioned, regularly contracting rays are the set of
$\kappa$–contracting rays.

######  Theorem 2.18.

[8] If $\tau$ is regularly contracting, then it is $\kappa$–contracting for
some sublinear function $\kappa$. In particular, it is also $\kappa$–Morse.

Let us equip the space $\mathrm{X}$ of interest with a convergence boundary
$\partial{\mathrm{X}}$. A universal choice is to consider the horofunction
boundary endowed with finite or sublinear difference partition. However, the
discussion in what follows equally works for any convergence boundary, e.g. in
Examples Examples.

By Lemma 2.17, let $\mathcal{FC}(\theta,r,C,L)$ be the set of all
$[\cdot]_{s}$–classes $[\xi]\in[\partial{\mathrm{X}}]$ so that some geodesic
ray $\gamma$ in $\mathrm{X}$ starting at $o$ and ending at $[\xi]$ is
$(r,C,L)$–contracting at $\theta$–frequency.

Accordingly, let $\mathcal{RC}(r,C)$ be the set of all $[\cdot]$–classes
$[\xi]\in[\partial{\mathrm{X}}]$ so that some geodesic ray $\gamma$ in
$\mathrm{X}$ starting at $o$ and ending at $[\xi]_{s}$ is $(r,C)$–regular
contracting.

######  Corollary 2.19.

For any $r,C>0$, the following holds

$\displaystyle\mathcal{RC}(r,C)=\bigcap_{L\in\mathbb{N}}\left[\bigcap_{\theta\in(0,1]\cap\mathbb{Q}}\mathcal{FC}(\theta,r,C,L)\right].$

where $\mathbb{Q}$ denotes the set of rational numbers.

###  2.7. Statistically convex-cocompact actions

In this subsection, we first recall a class of statistically convex-cocompact
actions introduced in [20].

Consider the ball of radius $n$ centered at $o\in\mathrm{X}$:

(9) $N(o,n)=\\{v\in Go:d(o,v)\leq n\\}$

Fix $\Delta\geq 1$. Consider the annulus of radius $n$ centered at
$o\in\mathrm{X}$:

(10) $A(o,n,\Delta)=\\{v\in Go:|d(o,v)-n|\leq\Delta\\}$

Let $\Gamma$ be a subset of $G$. The _critical exponent_ of $\Gamma$ is
defined as

$\omega_{\Gamma}=\limsup_{n\to\infty}\frac{\log{\sharp(N(o,n)\cap\Gamma
o)}}{n}.$

Given constants $0\leq M_{1}\leq M_{2}$, let $\mathcal{O}_{M_{1},M_{2}}$ be
the set of element $g\in G$ such that there exists some geodesic $\gamma$
between $N_{M_{2}}(o)$ and $N_{M_{2}}(go)$ with the property that the interior
of $\gamma$ lies outside $N_{M_{1}}(\Gamma o)$.

$N_{M_{1}}(Go)$$B(o,M_{2})$$o$$B(go,M_{2})$$go$$\gamma\cap
N_{M_{1}}(Go)=\emptyset$ Figure 4. Illustration of
$\mathcal{O}_{M_{1},M_{2}}$.

######  Definition 2.20 (SCC action).

If there exist positive constants $M_{1},M_{2}>0$ such that
$\omega_{\mathcal{O}_{M_{1},M_{2}}}<\omega_{G}<\infty$, then the proper action
of $G$ on $\mathrm{X}$ is called statistically convex-cocompact (SCC).

######  Remark.

The idea to define the set $\mathcal{O}_{M_{1},M_{2}}$ is to look at the
action of the fundamental group of a finite volume negatively curved Hadamard
manifold on its universal cover. It is then easy to see that for appropriate
constants $M_{1},M_{2}>0$, the set $\mathcal{O}_{M_{1},M_{2}}$ coincides with
the union of the orbits of cusp subgroups up to a finite Hausdorff distance.
The assumption in SCC actions was called the parabolic gap condition by
Dal’bo, Otal and Peigné in [6].

When the SCC action contains a contracting element, the definition is
independent of the basepoint (see [20, Lemma 6.2]).

A key notion in studying growth gaps is that of a barrier-free element
introduced in [20].

######  Definition 2.21.

Fix constants $r,M>0$.

  1. (1)

Given $r>0$ and $f\in G$, we say that a geodesic $\gamma$ contains an
$(r,f)$–barrier if there exists an element $t\in G$ so that

(11) $\max\\{d(t\cdot o,\gamma),d(t\cdot fo,\gamma)\\}\leq r.$

If no such $t\in G$ exists so that (11) holds, then $\gamma$ is called
$(r,f)$–barrier-free.

  2. (2)

An element $g\in G$ is $(r,M,f)$–barrier-free if there exists an
$(r,f)$–barrier-free geodesic between $N_{M}(o)$ and $N_{M}(go)$.

We have chosen two parameters $M_{1},M_{2}$ so that the definition of a
statistically convex-cocompact action is flexible and easy to verify. It is
enough to take $M_{1}=M_{2}=M$ in our use. Henceforth, we set
$\mathcal{O}_{M}:=\mathcal{O}_{M,M}$ for ease of notation.

Given $r,M>0$ and any $f\in G$, let $\mathcal{V}_{r,M,f}$ be the collection of
all $(r,M,f)$–barrier-free elements in $G$. The following results will be
crucially used in next sections.

######  Proposition 2.22.

[20, Theorems B & C] If $G$ admits a SCC action on a proper geodesic space
$(\mathrm{X},d)$ with a contracting element, then

  1. (1)

$G$ has purely exponential growth.

  2. (2)

Let $M_{0}$ be the constant in the definition of SCC action, then for any
$M>M_{0}$, there exists $r=r(M)>0$ such that $\mathcal{V}_{r,M,f}$ is
exponentially negligible for any $f\in G$.

##  3\. Conformal density on convergence boundary

Consider a proper action $G\curvearrowright\mathrm{X}$ with a contracting
element, endowed with a convergence compactification
$\overline{\mathrm{X}}=\mathrm{X}\cup\partial{\mathrm{X}}$. This section
recalls the general setup of quasi-conformal density supported on
$\partial{\mathrm{X}}$, and explains Patterson’s famous construction of such
density from the proper action $G\curvearrowright\mathrm{X}$.

###  3.1. Patterson-Sullivan measures on convergence boundary

Let $\mathcal{M}^{+}(\overline{\mathrm{X}})$ be the set of finite positive
Borel measures on $\overline{\mathrm{X}}:=\partial{\mathrm{X}}\cup\mathrm{X}$,
on which $G$ acts by push-forward:

$g_{\star}\mu(A)=\mu(g^{-1}A)$

for any Borel set $A$. Let $\mathcal{C}$ be the (non-empty) set of non-pinched
points in $\partial{\mathrm{X}}$ in Definition 2.14(C).

######  Definition 3.1.

Let $\omega\in[0,\infty[$. The following map

$\displaystyle\mu:\mathrm{X}\longrightarrow$
$\displaystyle\mathcal{M}^{+}(\overline{\mathrm{X}})$ $\displaystyle
x\longmapsto$ $\displaystyle\mu_{x}$

is called a $\omega$–dimensional $G$–quasi-equivariant quasi-conformal density
if for any $g,h\in G$ and any $x\in\mathrm{X}$, we have

(12) $\displaystyle\mu_{y}-\textrm{ a.e. }\xi\in\partial{\mathrm{X}}:$
$\displaystyle\quad\frac{dg_{\star}\mu_{x}}{d\mu_{x}}(\xi)\in[\frac{1}{\lambda},\lambda],$
(13) $\displaystyle\mu_{y}-\textrm{ a.e. }\xi\in\mathcal{C}:$
$\displaystyle\quad\frac{1}{\lambda}e^{-\omega
B_{\xi}(x,y)}\leq\frac{d\mu_{x}}{d\mu_{y}}(\xi)\leq\lambda e^{-\omega
B_{\xi}(x,y)}$

for a universal constant $\lambda\geq 1$. We normalize $\mu_{o}$ to be a
probability measure: its mass $\|\mu_{o}\|=\mu_{o}(\overline{\mathrm{X}})=1$.

We say that $\\{\mu_{x}:x\in\mathrm{X}\\}$ is non-trivial if $\mu_{x}$ is
supported on $\mathcal{C}$.

If $\lambda=1$ for (12), the map
$\mu:\mathrm{X}\to\mathcal{M}^{+}(\overline{\mathrm{X}})$ is $G$–equivariant,
that is, $\mu_{gx}=g_{\star}\mu_{x}$; equivalently, $\mu_{gx}(gA)=\mu_{x}(A)$.
If both $\lambda=1$, we call $\mu$ a conformal density.

#### Patterson-Sullivan measures

Choose a basepoint $o\in\mathrm{X}$. The Poincaré series for the action of
$G\curvearrowright\mathrm{X}$

$\mathcal{P}_{G}(s,x,y)=\sum\limits_{g\in G}e^{-sd(x,gy)},\;s\geq 0$

diverges at $0\leq s<\omega_{G}$ and converges at $s>\omega_{G}$. The action
of $G$ on $\mathrm{X}$ is called divergent if $\mathcal{P}_{G}(s,x,y)$
diverges at the critical exponent $\omega_{G}$. Otherwise, $G$ is called
convergent. Recall that $[\Lambda Go]$ is the limit set for the action
$G\curvearrowright\mathrm{X}$, i.e. the set of accumulation points of a
$G$-orbit, up to taking $[\cdot]$-closure.

We start to construct a family of measures
$\\{\mu_{x}^{s,y}\\}_{x\in\mathrm{X}}$ supported on $Gy$ for any given
$s>\omega_{G}$ and $y\in\mathrm{X}$. Assume that $\mathcal{P}_{G}(s,x,y)$ is
divergent at $s=\omega_{G}$. Set

(14) $\mu_{x}^{s,y}=\frac{1}{\mathcal{P}_{G}(s,o,y)}\sum\limits_{g\in
G}e^{-sd(x,gy)}\cdot{\mbox{Dirac}}{(gy)},$

where $s>\omega_{G}$ and $x\in\mathrm{X}$. Note that $\mu^{s,y}_{o}$ is a
probability measure supported on $Gy$. If $\mathcal{P}_{G}(s,x,y)$ is
convergent at $s=\omega_{G}$, the Poincaré series in (14) needs to be replaced
by a modified series as in [12].

Fix $y\in\mathrm{X}$. Choose $s_{i}\to\omega_{G}$ such that
$\mu_{x}^{s_{i},y}$ are convergent in $\mathcal{M}^{+}(\Lambda{G}o)$. The
Patterson-Sullivan measures $\mu_{x}^{y}=\lim\mu_{v}^{s_{i},y}$ are the limit
measures. Note that $\mu_{o}(\Lambda{G}o)=1$. In what follows, we usually
write $\mu_{x}=\mu_{x}^{o}$ for $x\in\mathrm{X}$.

######  Theorem 3.2.

Suppose that $G$ acts properly on a proper geodesic space $\mathrm{X}$
compactified with horofunction boundary $\partial_{h}{\mathrm{X}}$. Then the
family $\\{\mu_{x}:x\in\mathrm{X}\\}$ of Patterson-Sullivan measures is a
$\omega_{G}$-dimensional $G$-equivariant conformal density supported on
$[\Lambda Go]$.

In the sequel, we write PS-measures as shorthand for Patterson-Sullivan
measures.

###  3.2. Shadow Lemma

In what follows, we make the standing assumption:

######  Convention 3.3.

Let $F$ be a set of three (mutually) independent contracting elements
$f_{i},i=1,2,3$, that form a contracting system

(15) $\mathscr{F}=\\{g\cdot\mathrm{Ax}(f_{i}):g\in G\\}$

where the axis $\mathrm{Ax}(f_{i})$ defined in (1) depending on the choice of
a basepoint $o\in\mathrm{X}$ is $C$–contracting for some $C>0$.

We may often assume $d(o,fo)$ is large as possible, by taking sufficiently
high power of $f\in F$. The contracting constant $C$ is not effected.

Let $r>0$ and $x,y\in\mathrm{X}$. First of all, define the usual cone and
shadow:

$\Omega_{x}(y,r)\quad:=\quad\\{z\in\mathrm{X}:\exists[x,z]\cap
B(y,r)\neq\emptyset\\}$

and $\Pi_{x}(y,r)\subseteq\partial{\mathrm{X}}$ be the topological closure in
$\partial{\mathrm{X}}$ of $\Omega_{x}(y,r)$.

The partial shadows $\Pi_{o}^{F}(go,r)$ and cones $\Omega_{o}^{F}(go,r)$ given
in Definition 3.4 depends on the choice of a contracting system $\mathscr{F}$
as in (15). Without index $F$, $\Pi_{o}(go,r)$ denotes the usual shadow.

######  Definition 3.4 (Partial cone and shadow).

For $x\in\mathrm{X},y\in Go$, the $(r,F)$–cone $\Omega_{x}^{F}(y,r)$ is the
set of elements $z\in\mathrm{X}$ such that $y$ is an $(r,F)$–barrier for some
geodesic $[x,z]$.

The $(r,F)$–shadow $\Pi_{x}^{F}(y,r)\subseteq\partial{\mathrm{X}}$ is the
topological closure in $\partial{\mathrm{X}}$ of the cone
$\Omega_{x}^{F}(y,r)$.

The key fact in the theory of conformal density is the Sullivan’s shadow
lemma.

######  Lemma 3.5.

[23, Lemma 6.3] Let $\\{\mu_{x}\\}_{x\in\mathrm{X}}$ be a _nontrivial_
$\omega$–dimensional $G$–equivariant conformal density for some $\omega>0$
(i.e. supported on the set $\mathcal{C}\subseteq\partial{\mathrm{X}}$ of non-
pinched points). Then there exist $r_{0},L_{0}>0$ with the following property.

Assume that $d(o,fo)>L_{0}$ for each $f\in F$. For given $r\geq r_{0}$, there
exist $C_{0}=C_{0}(F),C_{1}=C_{1}(F,r)$ such that

$\begin{array}[]{rl}C_{0}\mathrm{e}^{-\omega\cdot
d(o,go)}\leq\mu_{o}(\Pi_{o}^{F}(go,r))\leq\mu_{o}(\Pi_{o}(go,r))\leq
C_{1}\mathrm{e}^{-\omega\cdot d(o,go)}\end{array}$

for any $go\in Go$.

As a corollary, we obtain a lower bound on the dimension of a conformal
density.

######  Lemma 3.6.

[23, Prop. 6.6] Under the assumption of Lemma 3.5, we have
$\omega\geq\omega_{G}$. Moreover, $\omega_{G}$ can be realized as stated in
Theorem 3.2.

###  3.3. Conical points

We give the definition of a conical point relative to the above
$C$–contracting system $\mathscr{F}$ in (3.3).

######  Definition 3.7.

A point $\xi\in\partial{\mathrm{X}}$ is called $(r,F)$–conical if for some
$x\in Go$, the point $\xi$ lies in infinitely many $(r,F)$–shadows
$\Pi_{x}^{F}(y_{n},r)$ for $y_{n}\in Go$. We denote by $\Lambda_{r}^{F}(Go)$
the set of $(r,F)$–conical points.

$g_{n}o$$g_{n}f_{n}o$$g_{n}\mathrm{Ax}(f_{n})$$o$$\Pi_{o}^{F}(g_{n}o,r)\ni\
\xi$$\leq r$$\leq r$ Figure 5. Conical points

The following useful property [23, Lemma 4.4] resembles the usual definition
of conical points in Kleinian groups.

######  Lemma 3.8.

Let $\xi\in\Lambda_{r}^{F}(Go)$. Then there exists $\hat{r}>0$ with the
following property. For any basepoint $o\in\mathrm{X}$ there exists a geodesic
ray $\gamma$ starting at $o$ ending at $[\xi]$ with infinitely many
$(\hat{r},F)$-barriers. That is,

  * •

there exist $g_{n}o\in G$ and $f_{n}\in F$ so that
$d(g_{n}o,\gamma),d(g_{n}f_{n}o,\gamma)\leq\hat{r}$.

Combined with Lemma 2.4, we obtain the following corollary.

######  Lemma 3.9.

The sublinear difference relation restricted on the set of conical points
coincides with finite difference relation.

At last, we recall the following fact saying that conical points are generic
for divergence type action. It is a part of the Hopf-Tsuji-Sullivan dichotomy
[23, Theorem 1.10], where the converse (easier direction) is also true.

######  Lemma 3.10.

If $G\curvearrowright\mathrm{X}$ is of divergence type, then $\mu_{o}$ charges
full measure on $[\Lambda_{r}^{F}(Go)]$ and every $[\xi]$–class is
$\mu_{o}$–null.

##  4\. Genericity of sublinearly Morse directions

This section is devoted to the proof of Theorem 4.1, recalled as follows.

######  Theorem 4.1.

Let $G\curvearrowright\mathrm{X}$ be a non-elementary SCC action with a
contracting element, equipped with a convergence boundary
$\partial{\mathrm{X}}$. Let $\\{\mu_{x}\\}_{x\in\mathrm{X}}$ be a
$\omega$–dimensional $G$–quasi-equivariant, quasi-conformal density for some
$\omega>0$, supported on the subset of non-pinched points
$\mathcal{C}\subseteq\partial{\mathrm{X}}$. Then the set of regularly
contracting rays has $\mu_{o}$–full measure.

###  4.1. Growth tightness of elements without frequent barriers

Let $\gamma$ be a geodesic segment. For any $\theta\in[0,1]$, a
$\theta$–segment of $\gamma$ means a connected and open subsegment of $\gamma$
with length $\theta\ell(\gamma)$. Recall that if $\gamma$ has
$(r,f)$–barriers, by the definition 2.21, there exists $t\in G$ so that

$to,tfo\in N_{r}(\gamma).$

Recall from Convention 3.3 that $F\subseteq G$ is a set of three independent
contracting elements, so that $C>0$ is the common contracting constant for
each $f\in F$. That is, the axes $\mathrm{Ax}(f)=E(f)\cdot o$ is a
$C$–contracting quasi-geodesic. For simplicity, we also assume that any
geodesic segment with two endpoints in $\mathrm{Ax}(f)$ is $C$–contracting (by
Lemma 2.2(2) and (3)).

######  Definition 4.2.

Fix $\theta\in(0,1]$, $r>0$ and $f\in G$. We say that a geodesic $\gamma$
contains $(r,f)$–barriers at $\theta$–frequency if for every $\theta$–segment
of $\gamma$ has $(r,f)$–barriers.

An element $g\in G$ has $(r,f)$–barriers at $\theta$–frequency if there exists
a geodesic $\gamma$ between $B(o,M)$ and $B(go,M)$ such that $\gamma$ has
$(r,f)$–barriers at $\theta$–frequency.

Fix any $\theta\in(0,1]$. Form now on,

######  Convention 4.3.

we fix a constant $r\geq C$ satisfy Proposition 2.22 and Lemma 3.5.

We need the following two elementary lemmas.

######  Lemma 4.4.

Fix $M>0$. There exist $\hat{r}=\hat{r}(C,r)$ and $L_{1}=L_{1}(C,M)>0$ with
the following property. Choose $h\in E(f)$ so that $d(o,ho)>L_{1}$. Let
$\alpha,\beta$ be two geodesics from $B(o,M)$ to $B(go,M)$. If $\alpha$
contains a $\theta$–segment with $(r,h)$–barriers, then $\beta$ contains some
$\theta$–segment with $(\hat{r},h)$–barriers.

###### Proof.

Let $p$ be a $\theta$–segment of $\alpha$ with an $(r,h)$–barrier. That is,
$d(to,p),d(tho,p)\leq r$ for some $t\in G$. Let $u,v$ be the entry and exit
points of $\alpha\cap N_{r}(t\mathrm{Ax}(f))$, so $[u,v]$ contains this
$(r,f)$–barrier $to$ and

$d(u,v)\geq d(o,fo)-2r\geq L_{1}-2r.$

As the shortest projection map to $t\mathrm{Ax}(f)$ is coarsely $1$–Lipschitz
by Lemma 2.2(4), $B(o,M)$ and $B(go,M)$ projects to a bounded set in
$t\mathrm{Ax}(f)$ of diameter $B$ depending on $M$ and $C$. If $\beta$ is
disjoint with $N_{r}(t\mathrm{Ax}(f))$ where $r\geq C$, we project $\beta$ to
$t\mathrm{Ax}(f)$ with diameter at most $C$, yielding that $d(u,v)\leq 2B+C$.
If we choose

$L_{1}\gg 2B+C+2r$

we would obtain a contradiction, whence $\beta$ has to intersect
$N_{r}(t\mathrm{Ax}(f))$.

Let $x$ and $y$ be the entry and exit points of $\beta$ in
$N_{r}(t\mathrm{Ax}(f))$. Arguing with a similar projection argument, we see
that $x,y$ are within $D$ distance to $u$ and $v$ respectively, where
$D=2B+2C$. By Lemma 2.2, $[x,y]$ is $\hat{C}$–contracting and the quasi-
convexity implies $[u,v]$ is $r^{\prime}$–close to $[x,y]$ for some
$r^{\prime}=r^{\prime}(\hat{C})$. Thus,

$d(to,[u,v]),d(tfo,[u,v])\leq\hat{r}:=r+r^{\prime}.$

Extending $[u,v]$ to be a $\theta$–segment of $\beta$, we proved that $\beta$
contains $(\hat{r},h)$–barriers in a $\theta$–segment. ∎

The following lemma will be used in the proof of Lemma 4.12. Roughly speaking,
it says that elements in $\mathcal{O}_{M_{1},M_{2}}$ have no $(r,f)$–barrier.

######  Lemma 4.5.

Fix any $M\geq C$. Let $\gamma$ be a geodesic with two endpoints in
$N_{M}(Go)$ and $\alpha$ be a component in the complement $\gamma\setminus
N_{M}(Go)$. Then there exists a constant $L_{2}>0$ depending only on $r,C$ so
that $\alpha\cap N_{r}(h\mathrm{Ax}(f))$ has diameter at most $L_{2}$ for any
$h\in G$.

###### Proof.

Let $x$ and $y$ be the entry and exit points of $\alpha$ in
$N_{r}(h\mathrm{Ax}(f))$ respectively. If $x^{\prime}$ and $y^{\prime}$ are
the corresponding projection points to $h\mathrm{Ax}(f)$, we have
$d(x,x^{\prime}),d(y,y^{\prime})\leq r$. By assumption, $\alpha\cap
N_{M}(Go)=\emptyset$. Noting $C\leq M$ and $\mathrm{Ax}(f)\subseteq Go$, we
have $\alpha\cap N_{C}(h\mathrm{Ax}(f))=\emptyset$. The $C$–contracting
property implies the projection of $\alpha$ to $h\mathrm{Ax}(f)$ has diameter
at most $C$. Consequently, we obtain $d(x^{\prime},y^{\prime})\leq C$ and thus
$d(x,y)\leq d(x^{\prime},y^{\prime})+2r\leq C+2r$. Setting $L_{0}=C+2r_{0}$
completes the proof. ∎

######  Convention 4.6.

We fix the constants $\hat{r}$ satisfying Lemma 4.4 and Lemma 4.5. We choose
$M\geq\max\\{\hat{r},C\\}$ as in the definition of SCC actions (2.20).

Fix an element $h\in E(f)$ where $f\in F$ is given as in Convention (3.3).

Let $\mathcal{W}(\theta,r,h)$ denote the set of elements $g$ in $G$ having no
$(r,h)$–barriers at $\theta$–frequency: for any geodesic $\gamma$ from
$B(o,M)$ to $B(go,M)$, $\gamma$ contains some $\theta$–segment without
$(r,h)$–barriers.

######  Lemma 4.7.

Let $G\curvearrowright\mathrm{X}$ be a non-elementary action with a
contracting element. Assume that $G\curvearrowright\mathrm{X}$ has purely
exponential growth. Then the set $\mathcal{W}(\theta,r,h)$ is growth
negligible.

In addition, if $G\curvearrowright\mathrm{X}$ is a SCC action, then the set
$\mathcal{W}(\theta,r,h)$ is growth tight.

###### Proof.

For simplicity, write $W:=\mathcal{W}(\theta,r,h)$ in this proof. Given $g\in
W$, let $\gamma$ be a fixed geodesic from $B(o,M)$ to $B(go,M)$ so that some
$\theta$–segment of $\gamma$, denoted by $\alpha$, is $(r,h)$–barrier-free.
That is to say, $\ell(\alpha)\geq\theta\ell(\gamma)$ and $\alpha$ contains no
$(r,h)$–barrier.

We may assume $\alpha$ is a maximal open segment with this property. We must
have the two endpoints of $\alpha$ lies in $N_{r}(Go)$, otherwise we can
extend $\alpha$ until reaching $N_{r}(Go)$. Thus, there exist $g_{1},g_{2}\in
G$ such that

$d(g_{1}o,\alpha_{-}),d(g_{2}o,\alpha_{+})\leq r,$

in particular, $\hat{g}=g_{1}^{-1}g_{2}$ is $(r,h)$–barrier-free by
definition. As a consequence, this implies that any $g\in W$ can be written as
a product of three elements $g=g_{1}\cdot\hat{g}\cdot(g_{2}^{-1}g)$ so that

$|d(o,go)-d(o,g_{1}o)-d(o,g_{2}o)-d(o,\hat{g}o)|\leq 4M.$

By Proposition 2.22, the set of elements $\hat{g}\in\mathcal{V}_{r,M,h}$ is
growth negligible: as $n\to\infty$,

$\frac{\sharp N(o,n)\cap\mathcal{V}_{r,M,h}}{\sharp N(o,n)}\to 0.$

If the action is SCC, then $\mathcal{V}_{r,M,h}$ is growth tight: for some
$\epsilon>0$,

$\frac{\sharp N(o,n)\cap\mathcal{V}_{r,M,h}}{\sharp
N(o,n)}\leq\mathrm{e}^{-\epsilon n}.$

Moreover, the element $\hat{g}$ in the above product takes a definite
proportion:

$d(o,\hat{g}o)\geq\ell(\alpha)-2M\geq\theta d(o,go)-2M.$

Since $G\curvearrowright\mathrm{X}$ has purely exponential growth, we have
$\sharp N(o,n)\asymp\mathrm{e}^{n\omega_{G}}$. By a similar argument as in [9,
Lemma 3.9], a straightforward computation shows that $W$ is growth tight if
the action is SCC, and is growth negligible in general. The lemma is then
proved. ∎

######  Corollary 4.8.

If the action is SCC, then the Poincaré series associated to
$W:=\mathcal{W}(\theta,r,h)$ is convergent at $s=\omega_{G}$. Namely,

$\sum_{g\in W}\mathrm{e}^{-\omega_{G}d(o,go)}<\infty.$

######  Remark.

We expect this same conclusion holds for proper action with purely exponential
growth.

Let $\mathcal{NFB}(\theta,r,h)$ denote the set of limit points
$\xi\in\partial{\mathrm{X}}$ that is contained in infinitely many shadows at
elements in $\mathcal{W}(\theta,r,h)$. Precisely, denoting
$W:=\mathcal{W}(\theta,r,h)$, $\mathcal{NFB}({\theta,r,h})$ is the limit
supper of the following sequence of sets

$\bigcup\limits_{v\in A(o,n,\Delta)\cap{W}}\Pi_{o}^{F}(v,r)$

where $A(o,n,\Delta)$ is defined in (10), as $n\to\infty$. Hence, the sequence

$\Lambda_{n}:=\bigcup\limits_{k\geq n}\left[\bigcup\limits_{v\in
A(o,k,\Delta)\cap{W}}\Pi_{o}^{F}(v,r)\right]$

tends to $\mathcal{NFB}(\theta,r,h)$.

We clarify the set $\mathcal{NFB}(\theta,r,h)$ with the following result,
which shall not be used in other places.

######  Lemma 4.9.

Let $g_{n}\in\mathcal{W}(\theta,r,h)$ be a sequence of elements without
$(r,h)$–barriers at $\theta$–frequency. Assume that $\xi$ is contained in a
sequence of shadows $\Pi_{o}^{F}(y_{n},r)$ where $y_{n}:=g_{n}o$. Then there
exists a geodesic ray $\gamma$ starting at $o$ and ending at $[\xi]$ such that
$\gamma$ contains no $(r,h)$–barriers at $\theta$–frequency.

###### Proof.

For $\xi\in\Pi_{o}^{F}(y_{n},r)$, there exists
$z_{m}\in\Omega_{o}^{F}(y_{n},r)\to\xi$ such that $[o,z_{m}]$ contains a
$(r,F)$–barrier at $y_{n}=g_{n}o$. That is to say, for some $f^{\prime}\in F$,

$d(g_{n}o,[o,z_{m}]),d(g_{n}f^{\prime}o,[o,z_{m}])\leq r.$

This inequality holds for any limit of $[o,z_{m}]$ by Ascoli-Arzela Lemma with
respect to the local uniform convergence topology. A Cantor diagonal argument
then provides a limiting geodesic ray $\gamma$, which has the above
$(r,F)$–barrier at every $y_{n}$. As $[o,y_{n}]$ has no $(r,f)$–barriers at
$\theta$–frequency, any limit $\gamma$ does so. It remains to note that
$\gamma$ ends at a point in $[\xi]$. To see this, first note that any limit
point $\eta$ of $y_{n}$ lies in $[\xi]$, as the direct computation shows the
difference $b_{\xi}$ and $b_{\eta}$ is uniformly bounded using the above
inequality. As $d(y_{n},\gamma)\leq r$, the same reasoning shows that the
Buseman function to $\gamma$ has finite difference to $b_{\eta}$. This then
concludes the proof. ∎

###  4.2. Non-frequently contracting segments have non-frequent barriers

We now relate the metric notion of frequently contracting segments (Def. 2.16)
to the analogous one, but involving a group action, with frequent barriers
(Def. 4.2). The title says the main result, Lemma 4.12, of this subsection.
First of all, we state a preparatory lemma.

######  Lemma 4.10.

Let $\gamma$ be a geodesic segment with $(r,h)$–barriers at $\theta$–frequency
for some $h\in E(f)$. Then $\gamma$ is $(r,C,L)$–contracting at
$\theta$–frequency, where $L=d(o,ho)$.

###### Proof.

The proof follows by unravelling the definitions. By definition, if $\gamma$
contains an $(r,h)$–barrier, then $\gamma$ contains a segment $[o,ho]$ in the
$r$–neighborhood of $\mathrm{Ax}(f)$, which is $C$–contracting. ∎

######  Lemma 4.11.

Let $h_{n}\in E(f)$ be a sequence of elements with $d(o,h_{n}o)\to\infty$. If
$\gamma$ is a geodesic ray with $(r,h_{n})$–barriers at $\theta$–frequency for
any $\theta\in(0,1]$, then $\gamma$ is $(r,C)$–regular contracting.

###### Proof.

As $\theta\in(0,1]$ is arbitrary and $d(o,ho)\to\infty$, it follows from Lemma
4.10 that $\gamma$ is $(r,C)$–frequently contracting. ∎

We now look at the set of $(r,C,L)$–contracting rays at $\theta$–frequency.
Precisely, let $\mathcal{FC}(\theta,r,C,L)$ consist of all
$[\cdot]_{s}$–classes $[\xi]_{s}\in[\partial{\mathrm{X}}]_{s}$ so that some
geodesic ray $\gamma$ in $\mathrm{X}$ starting at $o$ and ending at
$[\xi]_{s}$ is $(r,C,L)$–contracting at $\theta$–frequency.

The relation with the previously defined set $\mathcal{NFB}(\theta,r,f)$ with
no frequent $(r,f)$–barriers is explained in the following lemma. This is
crucial in the proof of Theorem Theorem A later on.

######  Lemma 4.12.

Assume that

  * •

$L$ is a constant bigger than $\max\\{L_{0},L_{1}\\}$, where $L_{0}$ are given
by Lemma 4.5 and $L_{1}$ are given by Lemma 4.4.

  * •

$h\in E(f)$ is an element with $d(o,ho)>2L$,

Then the complementary set
$\Lambda_{r}^{F}(Go)\setminus\mathcal{FC}(\theta,\hat{r},C,L)$ is contained in
the set $\mathcal{NFB}(\theta,r,h)$.

The relevance of $f\in F$ to the constant $C$ is that any segment in
$N_{r}(\mathrm{Ax}(f))$ is assumed to be $C$–contracting. See Convention 3.3.

###### Proof.

Let $\xi\in[\Lambda_{r}^{F}(Go)]$ be not contained in
$[\mathcal{FC}(\theta,\hat{r},C,L)]$. By definition, there exists a geodesic
ray $\gamma=[o,\xi]$ that is not $(r,C,L)$–contracting at $\theta$–frequency.
By definition, there exist a sequence of positive numbers $R_{n}\to\infty$
with the following property :

($\circledast$) $\displaystyle\gamma[0,R_{n}]\text{ contains a
}\theta\text{-interval }\alpha_{n}\text{ so that }$ $\displaystyle\text{no
subsegment of }\alpha_{n}\text{ with length }L\text{ is }\hat{r}\text{–close
to a }C\text{-contracting segment.}$

By Lemma 4.10, $\gamma[0,R_{n}]$ has no $(\hat{r},h)$–barriers at
$\theta$–frequency.

If the point $\gamma[R_{n}]$ lies in the closed neighborhood $N_{M}(Go)$,
there exists $g_{n}\in G$ such that $d(g_{n}o,\gamma[R_{n}])\leq M$. By Lemma
4.4, any geodesic from $B(o,M)$ to $B(g_{n}o,M)$ has no $(r,h)$–barriers at
$\theta$–frequency. By definition, $g_{n}$ has no $(r,h)$–barriers at
$\theta$–frequency, so $g_{n}\in\mathcal{W}(\theta,r,h)$ by definition.

Otherwise, we have $\gamma[R_{n}]$ lies outside $N_{M}(Go)$. The remainder of
the proof is then to find another time $R_{n}^{\prime}$ so that
$\gamma[0,R_{n}^{\prime}]$ has the above property (LABEL:NoBarrierPrpty)(i.e.
not $(r,C,L)$–contracting at $\theta$–frequency), and the new point
$\gamma[R_{n}^{\prime}]$ lies in $N_{M}(Go)$. To that end, our discussion is
divided into the following two cases. Write explicitly
$\alpha_{n}:=\gamma[s_{n},t_{n}]$ for some $0\leq s_{n}<t_{n}\leq R_{n}$.

Case 1. Assume that $\gamma[0,R_{n}]$ contains a point
$\gamma[R_{n}^{\prime}]$ in $N_{M}(Go)$ for some time $t_{n}\leq
R_{n}^{\prime}<R_{n}$. As $\alpha_{n}$ is contained in
$\gamma[0,R_{n}^{\prime}]$ and $R_{n}^{\prime}\leq R_{n}$ is an earlier time,
the segment $\gamma[0,R_{n}^{\prime}]$ has the above property
(LABEL:NoBarrierPrpty). Also, $d(g_{n}o,\gamma[R_{n}^{\prime}])\leq M$ for
some $g_{n}\in G$.

Case 2. After $\alpha_{n}=\gamma[s_{n},t_{n}]$, $\gamma[0,R_{n}]$ contains no
point in $N_{M}(Go)$. That is to say, for any $t\in[t_{n},R_{n}]$, we have
$d(\gamma[t],Go)>M$. In this case, we shall extend the segment
$\gamma[0,R_{n}]$ to a future time $R_{n}^{\prime}\geq R_{n}$, which still has
the above property (LABEL:NoBarrierPrpty), so that
$d(g_{n}o,\gamma[R_{n}^{\prime}])\leq M$ for some $g_{n}\in G$. See Figure 6.

Indeed, let $R_{n}^{\prime}>R_{n}$ be the least number so that
$\gamma[R_{n}^{\prime}]$ is contained in $N_{M}(Go)$. The number
$R_{n}^{\prime}<\infty$ exists, since $\gamma$ ends at $[\cdot]_{s}$–class of
a conical point $\xi\in[\Lambda_{r}^{F}(Go)]$, which according to Lemma 3.8
intersects $N_{\hat{r}}(Go)$ in a unbounded diameter.

Let $g_{n}\in G$ satisfy $d(g_{n}o,\gamma[R_{n}^{\prime}])\leq M$. We need to
verify

######  Claim.

$g_{n}$ has no $(r,h)$–barriers at $\theta$–frequency.

###### Proof of the claim.

The segment $\alpha_{n}^{\prime}:=\gamma[s_{n},R_{n}^{\prime}]$ is the union
of $\alpha_{n}$ and $\gamma[t_{n},R_{n}^{\prime}]$. As $\alpha_{n}$ is a
$\theta$–interval of $\gamma[0,R_{n}]$, it follows that $\alpha_{n}^{\prime}$
is a $\theta$–interval of $\gamma[0,R_{n}^{\prime}]$.

Now, assume by contradiction that $g_{n}$ has $(r,f)$–barriers at
$\theta$–frequency. By definition, any geodesic (say
$\gamma[0,R_{n}^{\prime}]$) between $B(o,M)$ and $B(g_{n}o,M)$ has
$(r,h)$–barriers at $\theta$–frequency. In particular, the $\theta$–interval
$\alpha_{n}^{\prime}$ has $(r,h)$–barriers. Hence, we have some $g\in G$ so
that $go,gho$ lie in the $r$–neighborhood of $\alpha_{n}^{\prime}$. That is,
$\alpha_{n}^{\prime}\cap N_{r}(g\mathrm{Ax}(f))$ has diameter at least
$d(o,ho)>2L$.

$s_{n}$$t_{n}$$R_{n}$$R^{\prime}_{n}$$\ell(\alpha_{n})=\theta\cdot
R_{n}$$\xi$$N_{M}(Go)$$h_{n}o$$o$$\alpha^{\prime}_{n}$ Figure 6. Case (2) in
the proof of Lemma 4.12

As $\gamma[t_{n},R_{n}^{\prime}]$ lies outside $N_{M}(Go)$, we obtain from
Lemma 4.5 that $\gamma[t_{n},R_{n}^{\prime}]\cap N_{r}(g\mathrm{Ax}(f))$ has
diameter at most $L_{0}$. Thus, $\alpha_{n}\cap N_{r}(g\mathrm{Ax}(f))$ has
diameter at least $2L-L_{0}>L$. By assumption, any segment with two endpoints
in $g\mathrm{Ax}(h)$ is $C$–contracting, we see that $\alpha_{n}$ contains a
segment of length $L$, that is $r$–close to a $C$–contracting segment. This is
a contradiction with (LABEL:NoBarrierPrpty), so we proved that
$g_{n}\in\mathcal{W}(\theta,r,h)$. ∎

In summary, we produced a sequence of elements
$g_{n}\in\mathcal{W}(\theta,r,h)$ so that $d(g_{n}o,\gamma)\leq C$. Hence,
$\xi$ lies in $\mathcal{NFB}(\theta,r,h)$. The lemma is proved. ∎

In order to prove that $\mathcal{FC}(\theta,r,C,L)$ is $\mu_{o}$–full, we only
need to prove that $\mathcal{NFB}(\theta,r,h)$ is $\mu_{o}$–null.

######  Lemma 4.13.

Assume that the action $G\curvearrowright\mathrm{X}$ is SCC. Fix
$\theta\in(0,1)$ and a contracting element $h\in E(f)$. Then the set
$\mathcal{NFB}(\theta,r,h)$ is $\mu_{o}$–null.

###### Proof.

In this proof, we write $W:=\mathcal{W}({\theta,r,h})$ for simplicity. The
following argument is standard, but we write it for the completeness.

By way of contradiction, assume that $\mu_{o}(\mathcal{NFB}({\theta,r,h}))>0$.
By assumption, $\mu_{o}(\Lambda_{n})\geq\frac{1}{2}\mu_{o}(\mathcal{NFB})>0$
for all $n\gg 0$. As $W\subseteq G$ is growth tight and $\omega$ is the
critical exponent of $G$, the following Poincaré series associated with $W$

$\mathcal{P}_{W}(\omega,o,o)=\sum_{g\in W}e^{-\omega d(o,go)}$

is convergent. By Lemma 3.5, we obtain a uniform lower bound on the partial
sum of $\mathcal{P}_{{W}}(\omega,o,o)$ for any $n\gg 0$:

$\bigcup\limits_{k\geq n}\left[\bigcup\limits_{v\in
A(o,k,\Delta)\cap{W}}e^{-\omega n}\right]\succ\mu_{o}(\Lambda_{n})$

thus contradicting to the convergence of $\mathcal{P}_{W}(\omega,o,o)$. This
shows that $\mathcal{NFB}$ is $\mu_{o}$–null. ∎

###  4.3. Completion of the proof of Theorem A

As shown in (2.19), the set of regularly contracting rays $\mathcal{RC}(r,C)$
for given $r,C>0$ is the countable intersection of
$\mathcal{FC}(\theta,r,C,L)$ over rationals $\theta\in\mathbb{Q}$ and integers
$L\in\mathbb{N}$.

According to Lemma 4.12, $\mathcal{FC}(\theta,r,C,L)$ is contained in
$\mathcal{NFB}(\theta,r,h)$ for some $h\in E(f)$, so is $\mu_{o}$–full for
every $L,\theta$ by Lemma 4.13. Hence, the countable intersection
$\mathcal{RC}(r,C)$ is $\mu_{o}$–full.

By Theorem 2.18, $\mathcal{RC}(r,C)$ is a subset of the sublinearly Morse
directions, thus the set of sublinearly Morse directions is $\mu_{o}$–full.

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