Using Patterson-Sullivan measures we investigate growth problems for groups acting on a metric space with a strongly contracting element.

Patterson-Sullivan measures, critical exponent, growth rate, contracting element, amenability.

Primary MSC Classification.

§ INTRODUCTION

Let $G$ be a group acting properly by isometries on a proper geodesic space $(X, \distV)$.
In particular $G$ is countable.
It will always be endowed with the discrete topology.
Its exponential growth rate measures the size of its orbits and is defined as

\begin{equation*}
\omega(G,X) = \limsup_{\ell \to \infty} \frac 1\ell \ln  \card{\set{g \in G}{\dist o{go} \leq \ell}}.
\end{equation*}

This number does depend on the space $X$.
Nevertheless, if the context is clear, we simply write $\omega_G$ instead of $\omega(G,X)$.
It is also the critical exponent of the Poincaré series of $G$ defined by

\begin{equation*}
\mathcal P_G(s) = \sum_{g \in G} e^{-s\dist o{go}},
\end{equation*}

that is $\mathcal P_G(s)$ diverges (converges) whenever $s < \omega_G$ ($s > \omega_G$).
If $G$ is the fundamental group of a hyperbolic manifold $M$ acting on the universal cover $X = \tilde M$, then $\omega_G$ has numerous interpretations: it is the entropy of the geodesic flow, the Hausdorff dimension of the radial limit set of $G$, etc.
In this context, the exponential growth rate is a central object connecting geometry, group theory, dynamical systems, etc.

Growth spectrum.
In this article we are interested in the (normal) subgroup growth spectrum of $G$, i.e. the set

\begin{equation*}
\spec{G, X} =  \set{\omega(N,X)}{N \vartriangleleft G}.
\end{equation*}

Note that $\spec{G, X}$ is contained in $\intval 0{\omega_G}$.
In particular, $\omega_N = 0$ ($\omega_N = \omega_G$) if $N$ is finite (has finite index in $G$).
A natural question, which has received much attention, is to understand more precisely the extremal values of this set.
This problem is rather well understood if $G$ is a group acting properly, co-compactly by isometries on a Gromov hyperbolic metric space $X$.
For instance, any infinite normal subgroup  $N \vartriangleleft G$ satisfies

\begin{equation*}
\frac 12 \omega(G,X) < \omega(N,X) \leq \omega(G,X).
\end{equation*}

see Matsuzaki-Yabuki-Jaerisch [37].
Moreover, the second inequality is an equality if and only if $G/N$ is amenable.
Similarly, if $G$ has Kazhdan property (T), then $\omega_N$ cannot be arbitrarily close to $\omega_G$, unless $N$ has finite index in $G$.
See Coulon-Dougall-Schapira-Tapie [18] and the references therein.

Contracting elements.
In the past decades, there have been many efforts to investigate the marks of negative curvature in groups, beyond the context of hyperbolic spaces.
One notion which has emerged is the one of contracting element.
Roughly speaking a subset $Y \subset X$ is contracting, if any ball disjoint from $Y$ has a projection onto $Y$, whose diameter is uniformly bounded [6].

In the literature, this property is sometimes called strong contraction to distinguish it from a weaker version involving a certain system of non-geodesic paths, see for instance [1].
Since we will work with this single notion, we simply call it contraction.

An element $g \in G$ is contracting if the orbit map  $\Z \to G$ sending $n$ to $g^nx$ is a quasi-isometric embedding with contracting image.
Contracting elements can be thought of as hyperbolic directions in the space $X$.
Here are a few examples of group actions with contracting elements.

* If $X$ is a hyperbolic space (in the sense of Gromov) endowed with a proper action of $G$, then every loxodromic element in $G$ is contracting [27].
This is typically the case if $G$ is the fundamental group of a manifold $M$ whose sectional curvature is negative and bounded away from zero, and $X$ the universal cover of $M$.
Any metric quasi-isometric to this one will also work.

* Assume that $G$ is hyperbolic relative to $\{P_1, \dots, P_m\}$.
Suppose that $G$ acts properly, co-compactly on $X$ (e.g. $X$ is the Cayley graph of $G$ with respect to a finite generating set of $G$).
Any infinite order element of $G$ which is not conjugated in some $P_i$ is contracting  [47, 24].

* If $X$ is a CAT(0) space endowed with a proper, co-compact action of $G$, then any rank one element of $G$ is contracting [6].
Recall that the universal cover of any closed, compact manifold with non-positive sectional curvature is CAT(0) [7].

* Let $\Sigma$ be a closed, compact surface.
Assume that $G$ is the mapping class group of $\Sigma$ and $X$ the Teichmüller space of $\Sigma$ endowed with the Teichmüller metric.
Then any pseudo-Anosov element is contracting [38].

Groups with a contracting element are known to be acylindrically hyperbolic, see Sisto [48].
Acylindrical hyperbolicity is a powerful tool for studying the structure of a given group.
Nevertheless it is rather useless here as the exponential growth rate $\omega(G,X)$ heavily depends on the metric space $X$.
The typical spaces $X$ we are interested in are indeed not hyperbolic.

Main results.
The goal of this article is to investigate the extremal values of the subgroup growth spectrum in the context of group actions admitting a contracting element.
Some of our results refine existing statements in the literature.
In particular, we answer most of the questions raised by Arzhantseva and Cashen in [2].
Our main contribution though is the method that we use: we extend to this context the construction of Patterson-Sullivan measures (see below).

When it comes to counting problems, the behavior of the Poincaré series of $G$ at the critical exponent plays a major role.
This motivates the following definition.
The action of $G$ on $X$ is divergent (convergent) if the Poincaré series $\mathcal P_G(s)$ of $G$ diverges (converges) at $s = \omega_G$.
Our first statement deals with the bottom of the subgroup growth spectrum.

[see <ref> and <ref>]

Let $X$ be a proper geodesic metric space.
Let $G$ be a group acting properly, by isometries on $X$ with a contracting element.
Let $N$ be an infinite normal subgroup of $G$.

\begin{equation*}
\omega(N,X) + \frac 12 \omega(G/N, X/N) \geq \omega(G, X).
\end{equation*}

Assume in addition that $G$ is not virtually cyclic and the action of $G$ is divergent.

\begin{equation*}
\omega(N,X) > \frac 12 \omega(G,X).
\end{equation*}

The first inequality was proved by Matsuzaki and Jaerisch when $G$ is a finitely generated free group acting on its Cayley graph with respect to a free basis [29].
Their method involves fine estimates of the Cheeger constant and the spectral radius of the random walk in $G/N$.
To the best of our knowledge this result is new, even if $G$ is a hyperbolic group.

The second inequality is well known in the context of hyperbolic spaces, see Roblin [43] and Matsuzaki-Yabuki-Jaerisch [37].
For groups acting with a contracting element, it was proved by Arzhantseva and Cashen under the stronger assumption that $G$ has pure exponential growth, that is when the map

\begin{equation*}
\ell \mapsto \card{\set{g \in G}{\dist o{go} \leq \ell}} e^{-\omega_G \ell}
\end{equation*}

is bounded from above and away from zero [2].
Note that even if $X$ is Gromov hyperbolic, there are groups $G$ acting on $X$, which are divergent but do not have pure exponential growth.

The next two results focus on the top of the subgroup growth spectrum.
Let $Q$ be a discrete group.
The left action of $Q$ on itself induces an action of $Q$ on $\ell^\infty(Q)$.
The group $Q$ is amenable if there exists a $Q$-invariant mean $\ell^\infty(Q) \to \R$.

[see <ref>]

Let $X$ be a proper, geodesic, metric space.
Let $G$ be a group acting properly, by isometries on $X$ with a contracting element.
Let $N$ be a normal subgroup of $G$.
If $G/N$ is amenable, then $\omega(N,X) = \omega(G,X)$.

This type of results has a long history.
Assume that $G$ is the fundamental group of a compact hyperbolic manifold and $X$ the universal cover of $M$.
Let $N$ be a normal subgroup of $G$.
Brooks proved that $\omega_N = \omega_G$ if and only if $G/N$ is amenable [8] – Brooks' result is actually stated in terms of the bottom spectra of certain Laplace's operators, but they can be related to the growth rates of the groups via Sullivan's formula [49].
A similar statement was obtained independently by Grigorchuk and Cohen when $G$ is a free group acting on its Cayley graph $X$ with respect to a free basis [26, 12].
The “easy direction” stated above was generalized by Roblin to the settings of CAT($-1$) spaces [43].

Recall that two subgroups $H_1$ and $H_2$ of $G$ are commensurable if $H_1 \cap H_2$ has finite index in both $H_1$ and $H_2$.
A subgroup $H \subset G$ is commensurated, if $H$ and $gHg^{-1}$ are commensurable, for every $g \in G$.
The class of commensurated subgroups contains all normal subgroups and finite index subgroups of $G$.
More generally any subgroup of $G$ that is commensurable with a normal subgroup of $G$ is commensurated.
But they are numerous other examples, see <ref>.

[see <ref>]

Let $X$ be a proper, geodesic, metric space.
Let $G$ be a group acting properly, by isometries on $X$ with a contracting element.
Let $H$ be a commensurated subgroup of $G$.
If the action of $H$ on $X$ is divergent, then $\omega(H,X) = \omega(G,X)$ and the action of $G$ on $X$ is divergent.

To the best of our knowledge the statements in the literature only cover the case where $H$ is normal.
With this stronger assumption, it was proved by Matsuzaki and Yabuki, if $G$ is a kleinian group, and generalized  by  Matsuzaki, Yabuki and Jaerisch when $X$ is Gromov hyperbolic [36, 37].

Patterson-Sullivan theory.
Assume that $G$ is the fundamental group of a closed riemannian manifold $M$ with negative sectional curvature.
In this context dynamical systems –
first and foremost the study of the geodesic flow on the unit tangent bundle of $M$ – provide efficient tools to tackle counting problems.
For instance, using the dynamics of the geodesic flow, Margulis proved that the number $c(\ell)$ of simple closed geodesics on $M$ of length at most $\ell$ behaves like

\begin{equation*}
c(\ell) \simop_{\ell \to \infty} \frac{e^{\omega_G \ell}}{\omega_G \ell}.
\end{equation*}

See [35].
Fix a base point $o \in X$.
Denote by $X = \tilde M$ the universal cover of $M$ and $\partial X$ its visual boundary.
In this topic, measures on the boundary play a prominent role.
Recall that a $G$-invariant, $\omega_G$-conformal density is a collection $\nu = (\nu_x)_{x \in X}$ of non-zero finite measures on $X \cup \partial X$, all in the same measure class, satisfying the following properties: for all $g \in G$, for all $x,y \in X$, we have

* $g_\ast \nu_x = \nu_{gx}$ (invariance),

* $\displaystyle \frac{d\nu_x}{d\nu_y}(\xi) = e^{-\omega_G b_\xi(x,y)}$  $\nu$-almost everywhere (conformality),

where $b_\xi$ stands for the Buseman cocycle at $\xi \in \partial X$.
In particular $\omega_G$ can be interpreted as the dimension of the measure $\nu_o$.
Patterson's construction provides examples of such densities which are supported on $\partial X$.
These measures are designed so that the action of $G$ on $(\partial X, \nu_o)$ captures many properties of the geodesic flow on $M$.
The theory can be generalized for groups acting on a Gromov hyperbolic space, see for instance Coornaert [13] and Bader-Furman [3].
For such groups, Theorem <ref>, <ref> and <ref> can be proved using invariant conformal densities.

In the past years, growth problems in groups with a contracting element have been investigated by various people, see for instance [50, 1, 21, 51, 52, 2, 33].
Since no Patterson-Sullivan theory existed in this context, each time the authors developed ad hoc methods.
Actually they often make a point of avoiding “fairly sophisticated” results about Patterson-Sullivan “machinery”.
We adopt here an opposite point of view.
For us, these results witnessed the fact that a Patterson-Sullivan theory should exist.
Building this “missing” theory is the purpose of this work.
If the ambient space $X$ is CAT(0) this task has been achieved by Link [34] extending the work of Knieper [30, 31].
Our approach does not require any CAT(0) assumption though.
Our goal is to stress that this construction is particularly robust and requires very little hypotheses, beside the existence of a contracting element.
Once the basic properties of invariant conformal densities have been established, they provide a unified framework for solving various growth problems.
We believed that these tools can be used for many other applications inspired by non-positive curvature.

We would like to understand the behavior of certain densities supported on the “boundary at infinity” of $X$.
Thus, the first task is to build an appropriate compactification of $X$ to carry these measures.
There have been many attempts to build an analogue of the Gromov boundary for groups with a contracting element: the contracting and Morse boundaries [11, 39, 16], the sublinearly Morse boundary [41, 42], etc.
However these boundaries are sometimes “too small” (for instance the Morse boundary cannot be used as a topological model of the Poisson boundary) and often not compact.
This can be a difficulty to build Patterson-Sullivan measures.
Instead, we choose to work with the horocompactification.
In short, it is the “smallest” compactification $\bar X$ of $X$ such that the map

\begin{equation*}
\begin{array}{ccc}
X \times X \times X & \to & \R \\
(x,y,z) & \mapsto & \dist xz - \dist yz
\end{array}
\end{equation*}

extends continuously to a map $X \times X \times \bar X \to \R$.
The horoboundary of $X$ is $\partial X = \bar X \setminus X$.
A point in the horoboundary is a cocycle $c \colon X \times X \to \R$, playing the role of a Buseman cocycle.
Hence this choice is natural to give a rigorous sense to conformal densities.
If $X$ is CAT(0), then the horoboundary coincides with the visual boundary.
In general, this boundary is slightly too large though for invariant conformal densities to behave as expected.
Let us illustrate this fact with an example.

Consider a group $G$ acting properly, co-compactly, by isometries on a CAT(0) space $X_0$.
Build a new space $X = X_0 \times \intval 01$ endowed with the $L^1$-metric.
Let $G$ act trivially on $\intval 01$ and consider the diagonal action of $G$ on $X$.
This action is still proper and co-compact and $\omega(G, X) = \omega(G, X_0)$.
The horoboundary of $X$ is homeomorphic to $\partial X = \partial X_0 \times \intval 01$.
To carry the analogy with negatively curved manifold, we would like that if $\mu = (\mu_x)$ is a $G$-invariant, $\omega_G$-conformal density supported on $\partial X$ then the action of $G$ on $(\partial X, \mu_o)$ is ergodic.
However, in this example we can choose a $G$-invariant, $\omega_G$-conformal density $\nu = (\nu_x)$ on $\partial X_0$ and form the average $\mu = (\nu^0 + \nu^1)/2$, where $\nu^i$ is a copy of $\nu$ supported on $\partial X_0 \times \{i\}$.
Then the action of $G$ on $(\partial X, \mu_o)$ is not ergodic.

This issue already arises if $X$ is Gromov hyperbolic.
In this context, it can be fixed by passing to the reduced horoboundary.
Endow the horoboundary $\partial X$ with the equivalence relation $\sim$ defined as follows: two cocycles $c, c' \in \partial X$ are equivalent, if $\norm[\infty]{c-c'} < \infty$.
The reduced horoboundary is the quotient $\partial X / \sim$.
If $X$ is hyperbolic, then it coincides with the Gromov boundary.
Moreover the projection $\pi \colon \partial X \onto \partial X/\!\! \sim$ is very well understood, see Coornaert-Papadopoulos [15].
Pushing forward in $\partial X/\!\! \sim$, the densities built in $\partial X$ provides well behaved measures.

However, in general the reduced horoboundary $\partial X /\!\!  \sim$ is a rather nasty topological space.
For instance, if $X = \R^2$ is endowed with the taxicab metric, then $\partial X /\!\!  \sim$ is not even Hausdorff.
To bypass this difficulty we adopt a measure theoretic point of view.
Denote by $\mathfrak R$ the $\sigma$-algebra that consists of all Borel sets which are saturated for the equivalence relation $\sim$.
We make an abuse of vocabulary and call the measurable space $(\partial X, \mathfrak R)$ the reduced horoboundary.
When restricted to the reduced horoboundary, the invariant, conformal densities are well behaved.
For instance, we prove the following partial form of the Hopf-Tsuji-Sullivan dichotomy (we refer the reader to <ref> for the definition of the radial limit set).

[see <ref> and <ref>]

Let $X$ be a proper geodesic metric space and $o \in X$.
Let $G$ be a group acting properly, by isometries on $X$ with a contracting element.
Suppose that $G$ is not virtually cyclic.
Let $\omega \in \R_+$.
Let $\mu = (\mu_x)$ be the restriction to the reduced horoboundary $(\partial X, \mathfrak R)$ of  a $G$-invariant, $\omega$-conformal density.
The following are equivalent.

* The action of $G$ on $X$ is divergent (and thus $\omega = \omega_G$).

* $\mu_o$ gives positive measure to the radial limit set.

* $\mu_o$ gives full measure to the radial limit set.

In a forthcoming work, see [20], we plan to complete the Hopf-Tsuji-Sullivan dichotomy by investigating the ergodicity of the geodesic flow in this context and its consequences for growth problems.

If the action of $G$ on $X$ is divergent, we prove that invariant, conformal densities are ergodic and essentially unique, when restricted to the reduced horoboundary.

[see <ref>]

Let $X$ be a proper, geodesic, metric space and $o \in X$.
Let $G$ be a non virtually cyclic group acting properly, by isometries on $X$ with a contracting element.
Assume that the action of $G$ on $X$ is divergent.
Let $\mu = (\mu_x)$ be the restriction to the reduced horoboundary $(\partial X, \mathfrak R)$ of  a $G$-invariant, $\omega_G$-conformal density.

* $\mu_o$ is ergodic;

* $\mu_o$ is non-atomic;

* $\mu$ is almost unique in the following sense:
there is $C \in \R_+^*$, such that if $\mu' = (\mu'_x)$ is the restriction to the reduced horoboundary of another $G$-invariant, $\omega_G$-conformal density, then for every $x \in X$, we have $\mu'_x \leq C \mu_x$.

Finally, we complete <ref> as follows.

[see <ref>]

Let $X$ be a proper, geodesic, metric space.
Let $G$ be a group acting properly, by isometries on $X$ with a contracting element.
Suppose that $G$ is not virtually cyclic.
Let $H$ be a commensurated subgroup of $G$.
If the action of $H$ on $X$ is divergent, then any $H$-invariant, $\omega_H$-conformal density is $G$-almost invariant when restricted to the reduced horoboundary $(\partial X, \mathfrak R)$.

Other applications can be found in Sections <ref> and <ref>.
In this article we focused on growth problems.
Nevertheless, we believe that the tools we introduced can be used for other purposes, e.g. to generalize the “no proper conjugation” property of divergent subgroups exhibited by Matsuzaki, Yabuki and Jaerisch [37].

Strongly positively recurrent actions.
As we mentioned before, divergent actions play an important role in counting problems.
Any proper and co-compact action is divergent.
In particular, if $G$ acts properly on $X$ with a quasi-convex orbit, then its action is divergent.
This framework has been generalized independently by Schapira-Tapie [45] and Yang [51] under the names strongly positively recurrent action (SPR) and statistically convex co-compact action (SCC) respectively – the idea also implicitly appears in the work of Arzhantseva-Cashen-Tao [1].
The notion has an independent dynamical origin as well, see for instance [28, 44].
Roughly speaking the idea is to ask that the elements of $g \in G$ which “violate” the quasi-convexity of $G$ are statistically very rare.
It was proved by Yang that such actions are divergent.
In <ref> we provide an alternative proof of this fact in the spirit of Schapira-Tapie [45].

Since obtaining the results in this article, we have learned that Wenyuan Yang independently investigated conformal measures in the same context [53].
The techniques used by Wenyuan Yang are slightly different.
For instance, his proof of <ref> (partial form of the Hopf-Tsuji-Sullivan dichotomy) relies on projection complexes introduced by Bestvina, Bromberg, and Fujiwara [5].
In contrast, we tried to use whenever possible low-tech arguments (both of geometric and measure theoretic nature).

The author is grateful to the Centre Henri Lebesgue ANR-11-LABX-0020-01 for creating an attractive mathematical environment.
He acknowledges support from the Agence Nationale de la Recherche under Grant Dagger ANR-16-CE40-0006-01 and GoFR ANR-22-CE40-0004.
He thanks the Institut Henri Poincaré (UAR 839 CNRS-Sorbonne Université) for its hospitality and support (through LabEx CARMIN, ANR-10-LABX-59-01) during the trimester program Groups Acting on Fractals, Hyperbolicity and Self-Similarity in Spring 2022.
The author warmly thanks Françoise Dal'bo, Ilya Gekhtman, Andrea Sambusetti, Barbara Schapira, Samuel Tapie, and Wenyuan Yang for related discussions.
He also thanks the referee for the helpful comments.

§ GROUPS WITH A CONTRACTING ELEMENT

Notations and vocabulary.
In this article $(X,d)$ is a proper, metric space.
A geodesic is a path $\gamma \colon I \to X$ (where $I \subset \R$ is an interval) such that

\begin{equation*}
\dist{\gamma(t)}{\gamma(t')} = \abs{t' - t},\quad \forall t,t' \in I.
\end{equation*}

From now on, we assume that $X$ is geodesic, that is any two points are joined by a (non necessarily unique) geodesic.
Note that we do require $X$ to be geodesically complete.

For every $x \in X$ and $r \in \R_+$, we denote by $B(x,r)$ the open ball of radius $r$ centered at $x$.
Let $Y$ be a closed subset of $X$.
Given $x \in X$, a point $y \in Y$ is a (nearest point) projection of $x$ on $Y$ if $\dist xy = d(x,Y)$.
The projection of a subset $Z\subset X$ onto $Y$ is

\begin{equation*}
\pi_Y(Z) = \set{y \in Y}{ y \ \text{is the projection of a point}\ z \in Z}.
\end{equation*}

Let $I \subset \R$ be a closed interval and  $\gamma \colon I \to X$ a continuous path intersecting $Y$.
The entry point and exit point of $\gamma$ in $Y$ are the points $\gamma(t)$ and $\gamma(t')$ where

\begin{equation*}
t = \inf \set{s \in I}{\gamma(s) \in Y}
\quad \text{and} \quad
t' = \sup \set{s \in I}{\gamma(s) \in Y} .
\end{equation*}

If $I$ is bounded, such points always exist (the subset $Y$ is closed).
Given $d\in \R_+$, we denote by $\mathcal N_d(Y)$ the $d$-neighborhood of $Y$, that is the set of points $x \in X$ such that $d(x,Y) \leq d$.
The distance between two subsets $Y, Y'$ of $X$ is

\begin{equation*}
d(Y,Y') = \inf_{(y,y') \in Y \times Y'} \dist y{y'}.
\end{equation*}

Contracting set.

[Contracting set]
Let $\alpha \in \R^*_+$.
A closed subset $Y \subset X$ is $\alpha$-contracting if for any geodesic $\gamma$ with $d(\gamma, Y)\geq \alpha$, we have $\diam \pi_Y(\gamma) \leq \alpha$.
The set $Y$ is contracting if $Y$ is $\alpha$-contracting for some $\alpha \in \R^*_+$.

The next statements are direct consequences of the definition.
Their proofs are left to the reader.

Let $Y$ be an $\alpha$-contracting subset.
If $\gamma$ is a geodesic joining two points of $\mathcal N_\alpha(Y)$, then $\gamma$ lies in the $5\alpha/2$-neighborhood of $Y$.

Let $Y$ be an $\alpha$-contracting subset.
Let $x,y\in X$ and $\gamma$ be a geodesic from $x$ to $y$.
Let $p$ and $q$ be respective projections of $x$ and $y$ onto $Y$.
If $d(x,Y) < \alpha$ or $\dist pq > \alpha$, then the following holds:

* $d(\gamma, Y) < \alpha$;

* the entry point (exit point) of $\gamma$ in $\mathcal N_\alpha(Y)$ is $2\alpha$-closed to $p$ ($q$);

* $\dist xy \geq \dist xp + \dist pq +  \dist qy - 8\alpha$.

It follows from the above statement that the nearest point projection onto $Y$ is large-scale $1$-Lipschitz.
Actually, refining the above argument one can prove that for every subset $Z \subset X$, we have

\begin{equation*}
\diam (\pi_Y(Z)) \leq \diam(Z) + 4\alpha.
\end{equation*}

For every $\alpha, d \in \R_+$, there exists $\beta \in \R_+$ with the following property.
Let $Y$ and $Z$ be two closed subsets of $X$.
Assume that the Hausdorff distance between them is at most $d$.
If $Y$ is $\alpha$-contracting, then $Z$ is $\beta$-contracting.

Contracting element.
Consider now a group $G$ acting properly, by isometries on $X$.

[Contracting element]
Let $y \in X$.
An element $g \in G$ is contracting, for its action on $X$, if the orbit map  $\Z \to G$ sending $n$ to $g^ny$ is a quasi-isometric embedding with contracting image.

Note that the definition does not depend on the point $y$ (see <ref>).
The next statement is a reformulation of Yang's Lemma 3.3 (and its proof) in [52].

Let $g \in G$ be a contracting element.
For every $d \in \R_+$ and $z \in X$, there is $\alpha \in \R^*_+$ with the following property.
Let $p,q \in \Z$ with $p \leq q$.
Let $x,y \in X$ such that $\dist x{g^pz} \leq d$ and $\dist y{g^qz} \leq d$.
Let $\gamma$ be a geodesic joining $x$ to $y$.
Then any subpath of $\gamma$ is $\alpha$-contracting.
Moreover for every integer $n \in \intvald pq$, the point $g^nz$ is $\alpha$-close to $\gamma$.

Let $g \in G$ be a contracting element and $A$ be the $\group g$-orbit of a point $y \in X$.
Define $E(g)$ as the set of elements $u \in G$ such that the Hausdorff distance between $A$ and $uA$ is finite.
It follows from the definition that $E(g)$ is a subgroup of $G$ that does not depend on $y$.
It is the maximal virtually cyclic subgroup of $G$ containing $\group g$.
Moreover $E(g)$ is almost-malnormal, that is $uE(g)u^{-1} \cap E(g)$ is finite, for every $u \in G \setminus E(g)$, see Yang <cit.>.

Assume that $G$ is not virtually cyclic and contains a contracting element.
Let $H \subset G$ be a commensurated subgroup.
If $H$ is infinite, then $H$ is not virtually cyclic and contains a contracting element.

We only sketch the proof.
For more details we refer the reader to Arzhantseva-Cashen-Tao <cit.> where similar arguments are given.
We first claim that for every contracting element $g \in G$, the group $H$ is not contained in $E(g)$.
Assume on the contrary that $H \subset E(g)$.
Since $G$ is not virtually cyclic, there is $u \in G \setminus E(g)$.
By malnormality, $uHu^{-1} \cap H$ is finite and cannot have finite index in $H$, which contradicts the fact that $H$ is commensurated.

We now fix once and for all a contracting element $g \in G$.
We denote by $A$ an orbit of $\group g$.
It is $\alpha$-contracting for some $\alpha \in \R^*_+$.
Moreover there is $C \in \R_+$ such that for every $u \in G \setminus E(g)$, we have $\diam(\pi_A(uA)) \leq C$, see Yang <cit.>.
We choose $n \in \N$ such that $\dist o{g^no}$ is very large compare to both $\alpha$ and $C$.

We claim that there is an element $h \in H\setminus E(g)$ such that both $g^nhg^{-n}$ and $g^{-n}hg^n$ belong to $H$.
Since $H$ is commensurated, both $g^{-n}Hg^n \cap H$ and $g^nHg^{-n} \cap H$ have finite index in $H$.

\begin{equation*}
H_0 = \left(g^{-n}Hg^n\right) \cap \left(g^nHg^{-n} \right) \cap H
\end{equation*}

has finite index in $H$.
It follows that $H_0$ is not contained in $E(g)$.
Indeed otherwise $E(g)$ should also contain $H$, contradicting our previous claim.
Any element $h \in H_0 \setminus E(g)$ satisfies the conclusions of our second claim.

Consider now the element

\begin{equation*}
f = \left(g^nhg^{-n}\right)\left(g^{-n}h g^n\right)^{-1}
= g^n \left(hg^{-2n} h^{-1}\right) g^n.
\end{equation*}

Note that $f$ belongs to $H$.
We claim that $f$ is contracting.
For simplicity we let $g_0 = g^n$ and $g_1= hg^{-2n}h^{-1}$ so that $f = g_0g_1g_0$.
They respectively act “by translation” on the $\alpha$-contracting sets $A_0 = A$ and $A_1 = hA$.
Fix a point $x \in \pi_A(hA)$.
Since $\pi_A(hA)$ has diameter at most $C$, any geodesic $\gamma \colon \intval 0T \to X$ from $x$ to $fx$ fellow-travels for a long times with $A_0=A$, $g_0A_1=g_0hA$ and $g_0g_1A_0 = fA$ (see <ref>).

$x$ [ ] at 40 25
$g_0x$ [ ] at 88 70
[ ] at 159 116
[ ] at 237 116
$g_0g_1x$ [ ] at 310 80
$g_0g_1g_0x = fx$ [ ] at 320 28
$A_0 = A$ [ ] at 20 82
$g_0A_1 = g_0hA$ [ ] at 205 164
$g_0g_1A_0 = fA$ [ ] at 410 70

The geodesic from $x$ to $fx$. The gray shapes are “axis” of conjugates of $g$.

Note that the construction has been designed so that the end of $\gamma$ and the beginning of $f \gamma$ both fellow-travel with $fA$ in the “same direction”.
Consequently, the concatenation of $\gamma$ and $f\gamma$ cannot backtrack much.
We extend $\gamma$ to a bi-infinite $\group f$-invariant path, still denoted by $\gamma \colon \R \to X$, which is characterized as follows: $\gamma(t + kT) = f^k \gamma(t)$ for every $t \in \R$ and $k \in \Z$.
One proves that $\gamma$ is a quasi-geodesic which  fellow-travels for a long time with $f^kA$, for every $k \in \Z$ (see <ref>).
This suffices to show that $f$ is contracting.

$x$ [ ] at 42 23
$g_0x$ [ ] at 78 16
$fg_0^{-1}x$ [ ] at 140 15
$fx$ [ ] at 177 23
$fg_0x$ [ ] at 214 18
$f^2g_0^{-1}x$ [ ] at 270 16
$f^2x$ [ ] at 316 25
$A$ [ ] at 43 -5
$fA$ [ ] at 178 -5
$f^2A$ [ ] at 316 -5

The “axis” of $f$.

We are left to prove that $H$ is not virtually cyclic.
If $H$ was virtually cyclic, it would be contained in $E(f)$.
This contradicts our first claim.

§ COMPACTIFICATION OF $X$

§.§ Horocompactification

Let $C(X)$ be the set of all real valued, continuous functions on $X$ endowed with the topology of uniform convergence on every compact subset.
We denote by $C^*(X)$ the quotient of $C(X)$ by the subspace consisting of all constant functions, and endowed with the quotient topology.
Given a base point $o \in X$, one can think of $C^*(X)$ as the set of all continuous functions that vanish at $o$.
Alternatively $C^*(X)$ is the set of continuous cocycles $c \colon X \times X \to \R$.
By cocycle we mean that

\begin{equation*}
c(x,z) = c(x,y) + c(y,z), \quad \forall x,y,z \in X.
\end{equation*}

For example, given $z \in X$, we define the cocycle $b_z \colon X \times X \to \R$ by

\begin{equation*}
b_z(x,y) = \dist xz - \dist yz, \quad \forall x,y \in X.
\end{equation*}

Since $X$ is geodesic, the map

\begin{equation*}
\begin{array}{rccc}
\iota \colon & X & \to & C^*(X) \\
& z & \mapsto & b_z
\end{array}
\end{equation*}

is a homeomorphism from $X$ onto its image.

The horocompactification $\bar X$ of $X$ it the closure of $\iota(X)$ in $C^*(X)$.
The horoboundary of $X$ is the set $\partial X = \bar X \setminus \iota(X)$.

From now on, we identify $X$ with its image under the map $\iota \colon X \to C^*(X)$.
By construction, every cocycle $c \in \bar X$ is $1$-Lipschitz, or equivalently $\abs{c(x,x')} \leq \dist x{x'}$, for every $x,x' \in X$.
It is a consequence of the Azerla-Ascoli theorem, that the horocompactification $\bar X$ is indeed a compact set.
We denote by $\mathfrak B$ the Borel $\sigma$-algebra on $\bar  X$.
In the remainder of the article we make an abuse of notations, and write $(\partial X, \mathfrak B)$ to denote the horoboundary endowed with the $\sigma$-algebra $\mathfrak B$ restricted to $\partial X$.

Let $c \in \bar X$ and $\epsilon \in \R_+$.
An $\epsilon$-quasi-gradient arc for $c$ is a path $\gamma \colon I \to X$ parametrized by arc length such that

\begin{equation*}
(t - s) - \epsilon \leq c(\gamma(s),\gamma(t)) \leq t - s, \quad \forall s,t \in I.
\end{equation*}

A gradient arc for $c$ is a $0$-quasi-gradient arc for $c$.
If $I = \R_+$, we call $\gamma$ a (quasi-)gradient ray.

The following observations follow from the definition and/or the triangle inequality.

* Since cocycles in $\bar X$ are $1$-Lipschitz, a gradient arc is always geodesic.

* Conversely, let $x,y \in X$ and $\epsilon \in \R_+$ such that $c(x,y) \geq \dist xy - \epsilon$.
Any geodesic from $x$ to $y$ is an $\epsilon$-quasi-gradient arc for $c$.

* Let $\gamma_1 \colon \intval {a_1}{b_1} \to X$ and $\gamma_2 \colon \intval {a_2}{b_2} \to X$ be two paths such that $\gamma_i$ is an $\epsilon_i$-quasi-gradient arc for $c$.
If $\gamma_1(b_1) = \gamma_2(a_2)$ then the concatenation of $\gamma_1$ and $\gamma_2$ is an $(\epsilon_1 + \epsilon_2)$-quasi-gradient arc for $c$.

The existence of gradient rays is given by the next statement.

Let $c \in \partial X$.
For every $x \in X$, there exists a gradient ray $\gamma \colon \R_+ \to X$ for $c$ such that $\gamma(0) = x$.

Let $(z_n)$ be a sequence of points of $X$ converging to $c$.
For every $n \in \N$, we let $b_n = \iota(z_n)$ and denote by $\gamma_n \colon \intval 0{\ell_n} \to X$ a geodesic from $x$ to $z_n$.
Since $X$ is proper, $(\gamma_n)$ converges, up to passing to a subsequence, to a geodesic ray $\gamma \colon \R_+ \to X$ starting at $x$.
As $(b_n)$ converges uniformly on every compact subset to $c$, we check that $\gamma$ is a gradient ray for $c$.

Given $c \in \partial X$ and a gradient ray $\gamma \colon \R_+ \to X$ for $c$, we think of $\gamma$ as a geodesic from $\gamma(0)$ to $c$.
The next definition is designed to handle simultaneously the cocycles corresponding to points in $X$ and the ones in $\partial X$.

Let $x \in X$ and $c \in \bar X$.
A gradient arc from $x$ to $c$ is

* any geodesic from $x$ to $z$, if $c = \iota(z)$ for some $z \in X$,

* any gradient ray for $c$ starting at $x$, if $c$ belongs to $\partial X$.

§.§ Reduced horoboundary

Given a cocycle $c \in C^*(X)$ we write $\norm[\infty]c$ for its uniform norm, i.e.

\begin{equation*}
\norm[\infty]c = \sup_{x,x' \in X} \abs{c(x,x')}.
\end{equation*}

Note that $\norm[\infty]c$ can be infinite.
If $K \subset X$ is compact, then $\norm[K]c$ is the uniform norm of $c$ restricted to $K \times K$.
We endow $\bar X$ with a binary relation: two cocycles $c,c' \in \bar X$ are equivalent, and we write $c \sim c'$, if one of the following holds:

* either $c$ and $c'$ lie in the image of $\iota \colon X \to C^*(X)$ and $c = c'$,

* or $c,c' \in \partial X$ and $\norm[\infty] {c-c'} < \infty$.

Given a subset $B \subset \bar X$, the saturation of $B$, denoted by $B^+$, is the union of all equivalence classes intersecting $B$.
We say that $B$ is saturated if it is a union of equivalence classes, or equivalently if $B^+ = B$.
Note that the collection of saturated subsets is closed under complement as well as (uncountable) union and intersection.
The reduced algebra, denoted by $\mathfrak R$, is the sub-$\sigma$-algebra of $\mathfrak B$ which consists of all saturated Borel subsets.

The reduced horocompactification and reduced horoboundary of $X$ are respectively the measurable spaces $(\bar X, \mathfrak R)$ and $(\partial X, \mathfrak R)$.

If $F \subset \bar X$ is closed, then $F^+$ belongs to $\mathfrak R$.

It suffices to prove that $F^+$ is a Borel subset.
Without loss of generality we can assume that $F$ is contained in $\partial X$.
Given $D \in \R_+$ and a compact subset $K \subset X$, we write

\begin{equation*}
F_{K,D} = \set{c \in \partial X}{ \exists b \in F, \ \norm[K]{c-b} \leq D}.
\end{equation*}

Since $F$ is compact, $F_{K, D}$ is closed.
Using again the fact that $F$ is compact we observe that

\begin{equation*}
F^+ = \bigcup_{D \in \R_+} \bigcap_{K \subset X} F_{K,D}
\end{equation*}

where $K$ runs over all compact subsets of $X$.
Hence the result.

§.§ Boundary at infinity of a contracting set

The goal of this section is to understand how cocycles at infinity interact with contracting subsets of $X$.

Let $Y$ be a closed subset of $X$.
Let $c \in \bar X$.
A projection of $c$ on $Y$ is a point $q \in Y$ such that for every $y \in Y$, we have $c(q,y) \leq 0$.

Given $z \in X$, the projection of $b = \iota(z)$ on $Y$ coincides with the definition of the nearest point projection.

Let $Y$ be a closed subset of $X$.
Let $c \in \bar X$.
Assume that $c$ admits a projection $q$ on $Y$ and denote by $\gamma \colon I \to \R_+$ a gradient arc for $c$ starting at $q$.
Then for every $t \in I$, the point $q$ is a projection of $\gamma(t)$ on $Y$.

Let $t\in I$.
Note that

\begin{equation*}
\dist q{\gamma(t)}
\leq c(q, \gamma(t))
\leq c(q,y) + c(y,\gamma(t))
\leq c(q,y) + \dist y{\gamma(t)}.
\end{equation*}

The first inequality holds since $\gamma$ is a gradient line for $c$ while the last one follows from the fact that $c$ is $1$-Lipschitz.
However $q$ being a projection of $c$ on $Y$, we have $c(q,y) \leq 0$.
Consequently $\dist q{\gamma(t)} \leq \dist y{\gamma(t)}$, for every $y \in Y$, which completes the proof.

If $c$ is a point in $\partial X$, a projection of $c$ on $Y$ may exist or not.
This leads to the following definition.

Let $Y$ be a closed subset of $X$.
The boundary at infinity of $Y$, denoted by $\partial ^+ Y$, is the set of all cocycles $c \in \partial X$ for which there is no projection of $c$ on $Y$.

We give several equivalent characterizations of the boundary at infinity of a contracting subset.

Let $\alpha \in \R_+$.
Let $Y$ be an $\alpha$-contracting subset of $X$.
Let $(z_n)$ be a sequence of points in $X$ which converges to a cocycle $c \in \partial X$.
For every $n \in \N$, denote by $q_n$ a projection of $z_n$ onto $Y$.
Let $\gamma \colon \R_+ \to X$ a gradient ray for $c$.
Define $T \in \R_+ \cup \{\infty\}$ by

\begin{equation*}
T = \sup \set{t \in \R_+}{d(\gamma(t),Y) \leq \alpha}
\end{equation*}

with the convention that $T = 0$, whenever $\gamma$ does not intersect $\mathcal N_\alpha(Y)$.
The following are equivalent.

$c \notin \partial^+Y$.

For every $x \in X$, the map $Y \to \R$ sending $y$ to $c(x,y)$ is bounded from above.

The ray $\gamma$ does not stay in a neighborhood of $Y$.

The projection $\pi_Y(\gamma)$ is bounded.

$T < \infty$.

The sequence $(q_n)$ is bounded.

Moreover, in this situation,

* the diameter of the set $Q = \pi_Y\left(\restriction \gamma {[T, \infty)}\right)$ is at most $\alpha$.

* any accumulation point $q^*$ of $(q_n)$ is a projection of $c$ on $Y$ which lies in the $\alpha$-neighborhood of $Q$.

It follows from <ref> that $\partial^+Y$ is saturated (as the notation suggested).

The equivalences <ref>$\iff$<ref> and <ref>$\iff$<ref> are standard properties of contracting sets, which only use the fact that $\gamma$ is a geodesic.
Note that $c(x,y) = c(x, x') + c(x',y)$, for every $x,x' \in X$ and $y \in Y$.
The implication <ref> $\Rightarrow$ <ref> follows from this observation.
The proof of <ref> $\Rightarrow$ <ref> is by contraposition.
Suppose that there exists $d \in \R_+$ such that $\gamma$ lies $\mathcal N_d(Y)$.
Let $x = \gamma(0)$.
Let $t \in \R_+$.
We denote by $q_t$ a projection of $\gamma(t)$ onto $Y$.
Using the fact that $c$ is $1$-Lipschitz, we get

\begin{equation*}
c(x, q_t)
\geq c(\gamma(0), \gamma(t)) - \dist{\gamma(t)}{q_t}
\geq t  - d.
\end{equation*}

This inequality holds for every $t \in \R_+$, hence the map $Y \to \R$, sending $y$ to $c(x,y)$ is not bounded from above.

We now focus on <ref>
$\Rightarrow$ <ref>.
For every $n \in \N$, we let $b_n = \iota(z_n)$.
Assume that $T < \infty$.
Let $Q$ be the projection onto $Y$ of $\gamma$ restricted to $[T, \infty)$.
Since $Y$ is contracting, the diameter of $Q$ is at most $\alpha$.
Let $q \in Q$ be a projection of $\gamma(T)$ on $Y$.
We claim that there is $N \in \N$, such that for every $n \geq N$, the point $q_n$ stays at a distance at most $\alpha$ of $Q$.
Assume on the contrary that it is not the case.
Up to passing to a subsequence, $d(q_n, Q) > \alpha$, for every $n \in \N$.
Using <ref>, we observe that for every $t \geq T$, for every $n \in \N$,

\begin{equation*}
b_n(\gamma(t), q) \geq   \dist q{\gamma(t)} - 4\alpha.
\end{equation*}

After passing to the limit we get $c(\gamma(t), q) \geq  \dist q{\gamma(t)} - 4\alpha$, for every $t \geq T$.
In particular, $t \mapsto c(\gamma(t), q)$ diverges to infinity as $t$ tends to infinity, which contradicts the fact that $\gamma$ is a gradient line for $c$.
This completes the proof of our claim and thus implies <ref>.

We finish the proof with <ref> $\Rightarrow$ <ref>.
Assume now that $(q_n)$ is bounded.
Let $q^*$ be an accumulation point of $(q_n)$.
As $Y$ is closed $q^*$ belongs to $Y$.
Observe that for every $y \in Y$, for every $n \in \N$, we have

\begin{equation*}
b_n(q^*,y) \leq b_n(q_n,y) + \dist {q^*}{q_n}  \leq  \dist {q^*}{q_n}.
\end{equation*}

By construction $b_n$ converges to $c$ on every compact subset, hence $c(q^*, y) \leq 0$ for every $y \in Y$.
Thus $q^*$ is a projection of $c$ onto $Y$.
In particular, $c \notin \partial^+Y$.

The next statement is a variation on the previous one, sharpening the estimates.

Let $\alpha \in \R^*_+$.
Let $Y$ be an $\alpha$-contracting set.
Let $c \in \bar X \setminus \partial^+Y$ and $q$ a projection of $c$ on $Y$.
Fix a sequence $(z_n)$ of points in $X$ converging to $c$.
Denote by $q^*$ an accumulation point of $(q_n)$ where $q_n$ stands for a projection of $z_n$ on $Y$.
Then $\dist q{q^*} \leq \alpha$.

Consider a gradient line $\gamma\colon\R_+ \to X$ from $q$ to $c$.
According to <ref>, $q$ is a projection of any point $\gamma(t)$ on $Y$.
Assume that contrary to our claim $\dist q{q^*} > \alpha$.
Reasoning as in the proof of <ref>, we observe that  $c(\gamma(t), q) \geq  \dist q{\gamma(t)} - 4\alpha$, for every $t \in \R_+$, which contradicts the fact that $\gamma$ is a gradient line for $c$.

The two statements extend <ref> for a gradient ray joining a point $x \in X$ to a cocycle $c \in \partial X$.
We distinguish two cases depending whether $c$ belongs to $\partial^+ Y$ or not.

Let $\alpha \in \R^*_+$.
Let $Y$ be an $\alpha$-contracting set.
Let $x \in X$ and  $c \in \bar X \setminus \partial^+Y$.
Let $\gamma$ be a gradient arc from $x$ to $c$.
Let $p$ and $q$ be respective projections of $x$ and $c$ on $Y$.
If $d(x,Y)< \alpha$ or $\dist pq > 4\alpha$, then the following holds:

* $d(\gamma, Y) < \alpha$;

* the entry point (exit point) of $\gamma$ in $\mathcal N_\alpha(Y)$ is $2\alpha$-closed ($5\alpha$-closed) to $p$ ($q$);

* $c(x,q) \geq \dist xp + \dist pq - 14\alpha$.

If $c$ is the cocycle associated to a point $z \in X$, then the statement is just a particular case of <ref>.
Assume now that $c$ belongs to $\partial X \setminus \partial^+ Y$.
Fix a sequence $(z_n)$ of points in $X$ converging to $c$.
Denote by $q^*$ an accumulation point of $(q_n)$ where $q_n$ stands for a projection of $z_n$ on $Y$.
Denote by $\gamma(T)$ the exit point of $\gamma$ from $\mathcal N_\alpha(Y)$, and $Q$ the projection onto $Y$ of $\gamma$ restricted to $[T, \infty)$.
According to <ref>, such an exit point exists.
Moreover, $Q$ has diameter at most $\alpha$, and $q^*$ is a projection of $c$ on $Y$ lying in the $\alpha$-neighborhood of $Q$.

Suppose that $d(x,Y) < \alpha$ or $d(p, q) > 4\alpha$.
According to <ref>, $\dist p{q^*} > 3 \alpha$, hence, $d(p, Q) > \alpha$.
It follows from the definition of contracting sets that  $d(\gamma, Y) < \alpha$.
Let $p'$ ($q'$) be a projection of the entry (exit) point of $\gamma$ in $\mathcal N_\alpha(Y)$ – note that if $d(x, Y) < \alpha$, then $x$ is the entry point of $\gamma$ in $\mathcal N_\alpha(Y)$ so we can choose $p' = p$.
Using again the contraction of $Y$, we see that $\dist p{p'} \leq \alpha$.
Moreover $q'$ belongs to $Q$, thus $\dist {q'}{q^*} \leq 2\alpha$.
In addition,

\begin{equation*}
\dist x{\gamma(T)}
\geq \dist x{p'} + \dist {p'}{q'} + \dist {q'}{\gamma(T)} - 4\alpha.
\end{equation*}

The path $\gamma$ is a gradient line, thus $c(x, \gamma(T))= \dist x{\gamma(T)}$, hence

\begin{equation*}
c(x, \gamma(T))
\geq \dist x{p'} + \dist {p'}{q'} + \dist {q'}{\gamma(T)} - 4\alpha.
\end{equation*}

Combined with the fact that $c$ is $1$-Lipschitz, we get

\begin{equation}
\label{eqn: proj cocycle contracting set}
\geq c(x, \gamma(T)) - \dist {q'}{\gamma(T)}
\geq \dist x{p'} + \dist {p'}{q'} - 4\alpha.
\end{equation}

We observed that $\dist p{p'} \leq \alpha$, while $\dist {q'}q \leq \dist {q'}{q^*} + \dist{q^*}q \leq 4\alpha$.
The conclusion follows from (<ref>) and the triangle inequality.

Let $\alpha \in \R^*_+$.
Let $Y$ be an $\alpha$-contracting set and $c \in \partial^+Y$.
Let $x \in X$ and $p$ be a projection of $x$ onto $Y$.
Let $\gamma \colon \R_+\to X$ be a gradient ray from $x$ to $c$.
Then the following holds:

* $d(\gamma, Y) < \alpha$;

* the entry point of $\gamma$ in $\mathcal N_\alpha(Y)$ is $2\alpha$-closed to $p$;

* $c(x,p) \geq \dist xp - 4\alpha$.

According to <ref> <ref> the set $\pi_Y(\gamma)$ is unbounded.
Thus there exists $s \in \R_+$ and a projection $q$ of $\gamma(s)$ onto $Y$ such that $\dist pq > \alpha$.
It follows that $d(\gamma, Y) < \alpha$.
Let $\gamma(t)$ be the entry point of $\gamma$ in $\mathcal N_\alpha(Y)$.
Using again the contraction of $Y$, we get $\dist p{\gamma(t)} \leq 2\alpha$.
Since $\gamma$ is a gradient line we have $c(x,\gamma(t)) = \dist x{\gamma(t)}$.
Combined with the triangle inequality and the fact that cocycles are $1$-Lipschitz we get $c(x,p) \geq \dist xp - 4\alpha$.

Assume that $G$ is not virtually cyclic.
Let $g$ be a contracting element and $A$ an orbit of $\group g$.
There exists $u \in G$ such that $\partial^+A \cap \partial ^+ (uA) = \emptyset$.

Consider an element $u \in G$ such that $\partial^+A\cap \partial^+(uA)$ is not empty.
Let $c$ be a cocycle in this intersection and $\gamma \colon \R_+ \to X$ a gradient ray for $c$.
By <ref> <ref> there exist $d, T \in \R_+$ such that $\gamma$ restricted to $[T, \infty)$ is contained in $\mathcal N_d(A) \cap \mathcal N_d(uA)$.
In particular, the diameter of this intersection is infinite.
It follows that $u \in E(g)$, see Yang <cit.>.
Recall that $E(g)$, unlike $G$, is virtually cyclic.
Thus there exists $u \in G\setminus E(g)$.
It follows from the above discussion that $\partial^+A \cap\partial ^+(uA) = \emptyset$.

§ CONFORMAL DENSITIES

As previously, $(X,d)$ is a proper, geodesic, metric space, while  $G$ is a group acting properly, by isometries on $X$.

§.§ Definition and existence

If $\mu$ is a finite measure on $\bar X$, we denote by $\norm \mu$ its total mass.

Let $\omega \in \R_+$.
Let $\mathfrak A$ be a $G$-invariant sub-$\sigma$-algebra of the Borel $\sigma$-algebra $\mathfrak B$.
A density on $(\bar X, \mathfrak A)$ is a collection $\nu = (\nu_x)$ of positive finite measures on $(\bar X, \mathfrak A)$ indexed by $X$ such that  $\nu_x \ll \nu_y$, for every $x,y \in X$, and normalized by $\norm{\nu_o} = 1$.
Such a density is

* $G$-invariant, if $g_\ast \nu_x = \nu_{gx}$, for every $g \in G$ and $x \in X$.

* $\omega$-conformal, if for every $x,y \in X$,

\begin{equation*}
\frac{d\nu_x}{d\nu_y} (c) = e^{-\omega c(x,y)}, \quad \nu_y\text{-a.e.}
\end{equation*}

* $\omega$-quasi-conformal, if there is $C \in \R_+^*$ such that for every $x,y \in X$,

\begin{equation*}
\frac 1Ce^{-\omega c(x,y)} \leq \frac{d\nu_x}{d\nu_y} (c) \leq C e^{-\omega c(x,y)} , \quad \nu_y\text{-a.e.}
\end{equation*}

Let $\nu = (\nu_x)$ be a density on $(\bar X, \mathfrak A)$.
We make an abuse of vocabulary and say that a property on $(\bar X, \mathfrak A)$ holds $\nu$-almost everywhere if it holds $\nu_x$-almost everywhere for some (hence every) $x \in X$.

Let $\nu = (\nu_x)$ be an $\omega$-conformal density on $(\bar X, \mathfrak A)$.
Recall that every cocycle in $\bar X$ is $1$-Lipschitz.
It follows that

\begin{equation}
\label{eqn: harnack}
\mu_x(A) \leq e^{\omega \dist xy} \mu_y(A), \quad \forall x,y \in X, \ \forall A \in \mathfrak A.
\end{equation}

This observation will be useful many times later.

In practice, we will consider only two $\sigma$-algebras on $\bar X$: the Borel $\sigma$-algebra $\mathfrak B$ and the reduced $\sigma$-algebra $\mathfrak R$ (see <ref>).
If $\nu$ is a conformal density on $(\bar X, \mathfrak B)$, then its restriction to the reduced $\sigma$-algebra $\mathfrak R$ is not necessarily quasi-conformal.
We will see later that this pathology can be avoided if the action of $G$ on $X$ is divergent.

We denote by $\mathcal D(\omega)$ the set of all $\omega$-conformal densities on the horocompactification $(\bar X, \mathfrak B)$.
We endow $\mathcal D(\omega)$ with the following topology: a sequence $\nu^n = (\nu^n_x)$ of densities converges to $\nu = (\nu_x)$ if for every $x \in X$, the measure $\nu^n_x$ converges to $\nu_x$ for the weak-* topology.
Let $\mathcal P(\bar X)$ be the set of all Borel probability measures on $\bar X$ (endowed wit the weak-* topology).
An $\omega$-conformal density $\nu \in \mathcal D(\omega)$ is entirely determined by the measure $\nu_o$.
More precisely the map

\begin{equation}
\label{eqn: density only depends on base point}
\begin{array}{ccc}
\mathcal D(\omega) &\to & \mathcal P(\bar X) \\
\nu & \mapsto & \nu_o
\end{array}
\end{equation}

is a homeomorphism.
We denote by $\mathcal D(G,\omega)$ the convex closed subspace of $\mathcal D(\omega)$ consisting of all $G$-invariant, $\omega$-conformal densities on $(\bar X, \mathfrak B)$.
The densities $\nu = (\nu_x)$ in $\mathcal D(G, \omega)$ for which the action of $G$ on $(\bar X, \mathfrak B, \nu_o)$ is ergodic are exactly the extremal points of $\mathcal D(G, \omega)$.

Patterson's construction.
We now prove the existence of invariant conformal densities supported on the horoboundary.
Actually we focus on a slightly more general settings that will be useful for our applications.
A map $\chi \colon G \to \R$ is a quasi-morphism if there exists $C \in \R_+$ such that for every $g,g' \in G$, we have

\begin{equation}
\label{eqn: quasi-morphism}
\abs{\chi(g) + \chi(g') - \chi(gg') } \leq C.
\end{equation}

To such a quasi-morphism $\chi$, we associate a twisted Poincaré series defined as

\begin{equation*}
\mathcal P_\chi(s) = \sum_{g \in G} e^{\chi(g)} e^{-s\dist o{go}},
\end{equation*}

and write $\omega_\chi$ for its critical exponent.
It follows from (<ref>) that

\begin{equation*}
e^{-2C} \mathcal P_\chi(s) \leq \mathcal P_{-\chi}(s) \leq e^{2C} \mathcal P_\chi(s), \quad \forall s \in \R_+.
\end{equation*}

Hence $\omega_{-\chi} = \omega_\chi$.
Note also that

\begin{equation*}
\frac12 \left[\mathcal P_\chi(s) + \mathcal P_{-\chi}(s)\right]
\geq \sum_{g \in G} \cosh \left( \chi(g)\right) e^{-s\dist o{go}}
\geq \mathcal P_G(s).
\end{equation*}

Thus $\omega_\chi \geq \omega_G$.

Let $H$ be a subgroup of $G$.
Let $\chi \colon G \to \R$ be a quasi-morphism such that $\chi(hg) = \chi(g)$, for all $h \in H$ and $g \in G$.
There is an $H$-invariant, $\omega_\chi$-conformal density $\nu = (\nu_x)$ on $(\bar X, \mathfrak B)$ with the following properties:

* $\nu$ is supported on $\partial X$;

* there is $C \in \R_+^*$ such that for every $g \in G$, and $x \in X$, we have

\begin{equation*}
\frac 1C \nu_x \leq e^{-\chi(g)} {g^{-1}}_\ast \nu_{gx} \leq C \nu_x.
\end{equation*}

If $H=G$ and $\chi$ is the trivial morphism, then the proposition says that there exists a $G$-invariant, $\omega_G$-conformal density supported on $\partial X$.

The proof follows Patterson's strategy [40].
As Burger and Mozes already observed, this construction can be carried without difficulty in the horocompactification of $X$ [9].
We only review here its main steps.
Note that the twisted Poincaré series $\mathcal P_\chi(s)$ may converge at the critical exponent $s = \omega_\chi$.
Using Patterson's idea, one produces a “slowly growing” function $\theta \colon \R_+ \to \R_+$ with the following properties – see Roblin <cit.>.

For every $\varepsilon > 0$, there exists $t_0 \geq 0$,
such that for every $t \geq t_0$ and $u \geq 0$, we have $ \theta(t + u ) \leq e^{\varepsilon u} \theta(t)$.

The weighted twisted Poincaré series, defined by

\begin{equation}
\label{eqn: weighted patterson series}
\mathcal Q(s)
\sum_{g \in G} \theta(\dist o{g o})e^{\chi(g)}e^{-s\dist{g o}o}
\end{equation}

is divergent whenever $s \leq \omega_\chi$, and convergent otherwise. In particular, $\mathcal Q(s)$ diverges to infinity as $s$ approaches $\omega_\chi$ (from above).

For every $x \in X$ and $s > \omega_\chi$, we define a measure on $\bar X$ by

\begin{equation}
\label{res: patterson approx}
\nu_x^s = \frac 1{\mathcal Q(s)}\sum_{g \in G} \theta(\dist x{go}) e^{\chi(g)}e^{-s\dist x{go}} {\rm Dirac}(go).
\end{equation}

Since $\bar X$ is compact, the space of probability measures on $\bar X$ is compact for the weak-* topology.
Consequently there exists a sequence of real numbers $(s_n)$ converging to $\omega_\chi$ from above, and such that for every $x \in X$, the measure $\nu_x^{s_n}$ converges to a measure on $\bar X$, that we denote by $\nu_x$.
Note that $\nu ^s= (\nu^s_x)$ is $H$-invariant, for every $s > \omega_\chi$.
Moreover, since $\chi$ is a quasi-morphism, there exists $C \in \R_+^*$ such that for every $s > \omega_\chi$, $g \in G$ and $x \in X$, we have

\begin{equation*}
\frac 1C \nu^s_x \leq e^{-\chi(g)} {g^{-1}}_\ast \nu^s_{gx} \leq C \nu^s_x.
\end{equation*}

Hence the same properties hold for $\nu$.
The horocompactification is precisely designed so that  the map

\begin{equation*}
\begin{array}{ccc}
X \times X \times X & \to & \R \\
(x,y,z) & \mapsto & \dist xz - \dist yz
\end{array}
\end{equation*}

extends continuously to a map $X \times X \times \bar X \to \R$.
Taking advantage of this fact, one checks that $\nu$ is $\omega_\chi$-conformal.
Since $\mathcal Q(s)$ diverges when $s$ approaches $\omega_\chi$, the density $\nu$ is supported on $\partial X$.

§.§ Group action on the space of density

Group action.
The action of $G$ on $X$ induces a right action of $G$ on the set of densities.
Let $\mathfrak A$ be a $G$-invariant sub-$\sigma$-algebra of $\mathfrak B$.
Given a density $\nu = (\nu_x)$ on $(\bar X, \mathfrak A)$ and $g \in G$, we define a new density $\nu^g$ as follows

\begin{equation*}
\nu^g_x = \frac 1{\norm{\nu_{go}}} {g^{-1}}_\ast \nu_{gx}, \quad \forall x \in X.
\end{equation*}

We make the following observations.

* If $\nu$ is $\omega$-conformal, then the same holds for $\nu^g$.

* If $\nu$ is $H$-invariant, for some subgroup $H \subset G$, then $\nu^g$ is $H^g$-invariant, where $H^g = g^{-1}Hg$.

In particular, if $N$ is a normal subgroup of $G$, then the map

\begin{equation*}
\begin{array}{ccc}
\mathcal D(N,\omega) \times G & \to & \mathcal D(N,\omega) \\
(\nu, g) & \mapsto & \nu^g
\end{array}
\end{equation*}

defines a right action of $G$ on $\mathcal D(N,\omega)$ which is trivial when restricted to $N$.

Fixed point properties.
For our study, we need to distinguish several fixed point properties for the action of $G$ on the space of densities.
Recall that a density $\nu = (\nu_x)$ is

$G$-invariant, if $g_\ast \nu_x = \nu_{gx}$ for every $g \in G$ and $x \in X$.

We say that $\nu$ is

$G$-almost invariant, if there exists $C \in \R_+^*$ such that for every $g \in G$ and $x \in X$,

\begin{equation*}
\frac 1C \nu_{gx} \leq g_\ast \nu_x \leq C \nu_{gx},
\end{equation*}

fixed by $G$, if $\nu^g = \nu$, for every $g \in G$,

almost-fixed by $G$, if there exists $C \in \R_+^*$ such that for every $g \in G$ and $x \in X$,

\begin{equation*}
\frac 1C \nu_x\leq \nu_x^g \leq C \nu_x,
\end{equation*}

$G$-quasi-invariant, if $g_\ast \nu_o \ll \nu_o$, for every $g \in G$.

If we want to emphasize the constant $C$ in <ref>, we will say that $\nu$ is $C$-almost-fixed by $G$.
These properties are related as follows.

(m) [matrix of math nodes, row sep=0.5em, column sep=2.5em, text height=1.5ex, text depth=0.25ex]

<ref>          <ref>    <ref>.

[-Straight Barb[length=4pt,width=5pt], double] (m-2-1) – (m-1-2) ;
[-Straight Barb[length=4pt,width=5pt], double] (m-2-1) – (m-3-2) ;
[-Straight Barb[length=4pt,width=5pt], double] (m-1-2) – (m-2-3) ;
[-Straight Barb[length=4pt,width=5pt], double] (m-3-2) – (m-2-3) ;
[-Straight Barb[length=4pt,width=5pt], double] (m-2-3) – (m-2-4) ;

The reverse implications do not hold in general.
The next statement highlights the role of quasi-morphisms in our study.

Let  $\nu = (\nu_x)$ be a density.
Assume that $\nu$ is fixed (almost-fixed) by $G$.
Then the map $\chi \colon G \to \R$ sending $g$ to $\ln \norm{\nu_{go}}$ is a morphism (quasi-morphism).

Assume that $\nu$ is almost-fixed by $G$ (if $\nu$ is fixed by $G$ the proof works in the exact same way).
There exists $C \in \R_+^*$ such that for every $g \in G$ and $x \in X$, we have

\begin{equation*}
\frac 1C \nu_x\leq \nu_x^g \leq C \nu_x.
\end{equation*}

Let $g,g' \in G$.
Comparing the total masses of the above measures for $x = g'o$, we get

\begin{equation*}
\frac 1C\norm{\nu_{g'o}}
\leq \frac{\norm{\nu_{gg'o}}}{\norm{\nu_{go}}}
\leq C \norm{\nu_{g'o}}. \qedhere
\end{equation*}

Let $H$ be a subgroup of $G$.
Let $\mu = (\mu_x)$ be an $H$-invariant, $\omega$-quasi-conformal density on the reduced horocompactification $(\bar X, \mathfrak R)$.
Assume that for every $g \in G$, the action of $H^g \cap H$ on $(\bar X, \mathfrak R, \mu_o)$ is ergodic.
If $\mu$ is $G$-quasi-invariant, then $\mu$ is almost-fixed by $G$.

Since $\mu$ is $G$-quasi-invariant, we can define the following map

\begin{equation*}
\begin{array}{rccc}
F \colon & G \times \bar X & \to & \R_+ \\
& (g,c) & \mapsto & \displaystyle \frac{d \left({g^{-1}}_\ast \mu_{go}\right)}{d\mu_o}(c).
\end{array}
\end{equation*}

There exists $C \in \R_+^*$ such that for every $g_1, g_2 \in G$, we have

\begin{equation*}
\frac 1C\frac{F(g_1g_2,c)}{F(g_2,c) } \leq F(g_1,g_2c) \leq C \frac{F(g_1g_2,c)}{F(g_2,c) }, \quad \mu\text{-a.e.}
\end{equation*}

Let $g_1, g_2 \in G$.
The computation gives

\begin{equation}
\label{eqn: ergodic quasi-inv induces almost-fixed}
=  \frac{d \left({g_1^{-1}}_\ast \mu_{g_1g_2o}\right)}{d\mu_{g_2o}}(g_2c) \ \frac{d \left({g_2^{-1}}_\ast \mu_{g_2o}\right)}{d\mu_o}(c), \quad \mu\text{-a.e.}	\end{equation}

Since $\mu$ is $\omega$-quasi-conformal, there exists $C \in \R_+$ such that for every $x ,y\in X$, for every $g \in G$, we have

\begin{equation*}
\frac 1C \frac {d\mu_x}{d\mu_y}
\leq \frac{d\mu_{gx}}{d\mu_{gy}} \circ g
\leq C  \frac {d\mu_x}{d\mu_y}.
\end{equation*}

\begin{equation*}
\frac 1C \frac{d \left({g^{-1}}_\ast \mu_{gy}\right)}{d\mu_y}
\leq \frac{d \left({g^{-1}}_\ast \mu_{gx}\right)}{d\mu_x}
\leq C \frac{d \left({g^{-1}}_\ast \mu_{gy}\right)}{d\mu_y}.
\end{equation*}

It follows that the first factor in the right-hand side of (<ref>) is $F(g_1,g_2c)$  – up to a multiplicative error that does not depend on $g_1$ or $g_2$ – while the second factor is exactly $F(g_2,c)$.
This completes the proof of our claim.

Let $g \in G$ and set $H_0 = H^g \cap H$.
Let $h \in H_0$.
According to our claim we have

\begin{equation*}
\frac 1C\frac{F(gh,c)}{F(h,c) } \leq F(g,hc) \leq C \frac{F(gh,c)}{F(h,c) }, \quad \mu\text{-a.e.}
\end{equation*}

Recall that $\mu$ is $H$-invariant, thus

\begin{equation*}
F(h,c) = 1,
\quad \text{and} \quad
F(gh,c) = F(ghg^{-1}g,c) = F(g,c), \quad \mu\text{-a.e.}
\end{equation*}

Our previous inequalities becomes

\begin{equation*}
\frac 1C F(g,c) \leq F(g,hc) \leq C F(g,c), \quad \mu\text{-a.e.}
\end{equation*}

We now define an auxiliary function $F_g \colon \bar X \to \R_+$ by

\begin{equation*}
F_g(c) = \inf_{h \in H_0} F(g,hc).
\end{equation*}

By construction $F_g$ is $H_0$-invariant.
Since the action of $H_0$ on $(\bar X, \mathfrak R, \mu_o)$ is ergodic, $F_g$ is constant.
From now on, we denote by $F_g$ its essential value.
It follows from our previous observation that

\begin{equation*}
F_g \leq F(g,c) \leq C F_g,
\quad \mu\text{-a.e.}
\end{equation*}

Coming back to the definition of $F$ this means that
\begin{equation*}
F_g \leq  \frac{d \left({g^{-1}}_\ast \mu_{go}\right)}{d\mu_o} \leq C F_g, \quad \mu\text{-a.e.}
\end{equation*}

Integrating these inequalities, we see that $F_g \leq \norm{\mu_{go}} \leq C F_g$ .

\begin{equation*}
\frac 1C \norm{\mu_{go}} \leq  \frac{d \left({g^{-1}}_\ast \mu_{go}\right)}{d\mu_o} \leq C \norm{\mu_{go}}.
\end{equation*}

Recall that $C$ does not depend on $g$.
We have proved that there exists $C \in \R_+^*$ such that for every $g \in G$,

\begin{equation*}
\frac 1C \mu_o \leq \mu^g_o \leq C \mu_o.
\end{equation*}

Using the quasi-conformality of $\mu$, we conclude that $\mu$ is almost-fixed by $G$.

The same argument shows that if $\mu$ is $\omega$-conformal (instead of $\omega$-quasi-conformal) then $\mu$ is fixed by $G$.

§.§ The Shadow Principle

Given $x,y \in X$ and $c \in \bar X$, we define the following Gromov product

\begin{equation*}
\label{eqn: gromov product}
\gro xcy = \frac 12 \left[\dist  xy + c(y,x)\right].
\end{equation*}

If $c = \iota(z)$ for some $z \in X$, then the above formula coincides with the usual definition of the Gromov product.

Since cocycles in $\bar X$ are $1$-Lipschitz, we always have $0 \leq \gro xcy \leq \dist xy$.
We also observe that

\begin{equation}
\label{eqn: gromov product - lip}
\abs{ \gro xcy - \gro{x'}c{y'}} \leq \dist x{x'} + \dist y{y'}, \quad \forall x,x',y,y' \in X.
\end{equation}

Let $x,y \in X$.
Let $r \in \R_+$.
The $r$-shadow of $y$ seen from $x$, is the set

\begin{equation*}
\mathcal O_x(y,r) = \set{ c \in \bar X}{ \gro xcy \leq r}.
\end{equation*}

By construction, $\mathcal O_x(y,r)$ is a closed subset of $\bar X$.
It follows from (<ref>) that for every $x,x',y,y' \in X$ and $r \in \R_+$,

\begin{equation}
\label{eqn: inclusion shadows triangle inequality}
\mathcal O_x(y,r) \subset \mathcal O_{x'}(y',r'), \quad \text{where} \quad r' = r + \dist x{x'} + \dist y{y'}.
\end{equation}

A more intuitive definition of shadows could have been the following: a cocycle $c \in \bar X$ belongs to $\mathcal O_x(y,r)$ if some gradient arc from $x$ to $c$ passes at a distance at most $r$ from $y$.
Nevertheless, unlike our approach, this definition is very sensitive to the change of point $x$.

Following Roblin with small variations we define the Shadow Principle [43].

Let $\omega \in \R_+$ and $(\epsilon, r_0) \in \R_+^* \times \R_+$.
Let $\nu = (\nu_x)$ be an $\omega$-conformal density on $(\bar X, \mathfrak B)$.
We say that $(G, \nu)$ satisfies the Shadow Principle with parameters $(\epsilon, r_0)$ if for every $g \in G$ and $r \geq r_0$, we have

\begin{equation}
\label{eqn: shadow principle}
\nu_o\left( \mathcal O_o(go,r)\right)
\geq \epsilon\norm{\nu_{go}}e^{-\omega \dist o{go}}.
\end{equation}

We say that $(G, \nu)$ satisfies the Shadow Principle, if there are $(\epsilon, r_0) \in \R_+^* \times \R_+$ such that $(G, \nu)$ satisfies the Shadow Principle with parameters $(\epsilon, r_0)$.

If the inequality (<ref>) holds for $r = r_0$, then it automatically holds for every $r \geq r_0$.
We will see later that a similar upper bound is always satisfied without any additional assumption.

Our next task is to adapt Sullivan's celebrated Shadow Lemma (<ref>).
It states that $(G,\nu)$ satisfies the Shadow Principle whenever $\nu$ is an $\omega$-conformal density which is $N$-invariant for some infinite normal subgroup $N \vartriangleleft G$.

Assume that $G$ is not virtually cyclic and contains a contracting element.
Let $\mathcal D_0$ be a closed subset of $G$-quasi-invariant densities on $\bar X$.
There exists $(\epsilon, r_0)  \in \R_+^*\times \R_+$ with the following property.
For every $r \geq r_0$, for every density $\nu = (\nu_x)$ in $\mathcal D_0$, for every $z \in X$, we have

\begin{equation*}
\nu_o\left(\mathcal O_z(o,r)\right) \geq \epsilon.
\end{equation*}

Assume that our claim fails.
We can find a sequence $(r_n)$ diverging to infinity, a sequence $(z_n)$ of points in $X$, and a sequence $\nu^n = (\nu^n_x)$ of densities in $\mathcal D_0$ such that

\begin{equation*}
\nu^n_o\left(\mathcal O_{z_n}(o,r_n)\right)
\end{equation*}

converges to zero.
Since $\mathcal D_0$ is closed, up to passing to a subsequence, we may assume that $\nu^n$ converges to a density $\nu$ in $\mathcal D_0$.
Note that for every $x,y \in X$ and $r \in \R_+$, we have $\mathcal O_x(y,r) = \bar X$, whenever $\dist xy \leq r$.
Hence $\dist o{z_n}$ necessarily diverges to infinity.
Up to passing again to a subsequence, we can assume that $z_n$ converges to $b \in \partial X$.

By assumption $G$ is not virtually cyclic and contains a contracting element
According to <ref>, there exists a contracting element $h \in G$, such that $b \notin \partial ^+A$, where $A = \group h o$ is $\alpha$-contracting for some $\alpha \in \R_+^*$.
For every $n \in \N$, we write $p_n$ for a projection of $z_n$ onto $A$.
Up to passing to a subsequence $(p_n)$ converges to a point $p \in A$ which is a projection of $b$ onto $A$ (<ref>).
We introduce the following closed subset of $\bar X$

\begin{equation*}
F = \set{c \in \bar X}{ c(p,y) \leq 4\alpha, \ \forall y \in Y}.
\end{equation*}

We are going to prove that $\nu_o(F) = 1$.

Let $c \in \bar X \setminus F$.
Suppose first that $c \notin \partial^+A$.
Let $q$ be a projection of $c$ onto $A$.
Since $c$ is $1$-Lipschitz, we have

\begin{equation*}
c(p,y) \leq \dist pq + c(q,y) \leq \dist pq, \quad \forall y \in A.
\end{equation*}

As $c$ does not belong to $F$, necessarily $\dist pq > 4\alpha$.
In particular, there exists $N_0 \in \N$, such that for every $n \geq N_0$, we have $\dist {p_n}q > 4\alpha$.
According to <ref>, $\gro {z_n}c{p_n} \leq 2\alpha$.
Note that the conclusion still holds if $c \in \partial^+A$.
This is indeed a consequence of <ref>.
Hence in all cases, we have

\begin{equation*}
\gro{z_n}co \leq \gro{z_n}c{p_n} + \dist o{p_n} \leq 2\alpha + \dist o{p_n}.
\end{equation*}

Recall that $(p_n)$ is bounded.
Consequently, there is $N_1 \geq N_0$ such that for every $n \geq N_1$, the set $\bar X \setminus F$ is contained in $\mathcal O_{z_n}(o,r_n)$.
In particular, $\nu^n_o (\bar X \setminus F)$ converges to zero.
Since $\bar X \setminus F$ is an open subset of $\bar X$, we deduce that $\nu_o(\bar X \setminus F) = 0$, i.e. $\nu_o(F) = 1$.

The group $\group h$ has unbounded orbits.
Thus there is $g \in \group h$ such that $\dist{gp}p > 36\alpha$.
We claim that $F \cap gF$ is empty.
Let $c \in F$.
Remember that $c$ does not belong to $\partial^+A$ by <ref> <ref>.
Choose a projection $q$ of $c$ onto $A$.
It follows from <ref> that

\begin{equation*}
4\alpha \geq c(p,q) \geq \dist pq - 14\alpha.
\end{equation*}

Hence $\dist pq \leq 18\alpha$.
Suppose now that contrary to our claim $c$ also belongs to $gF$.
In particular, $g^{-1}q$ is a projection of $g^{-1}c$ onto $A$.
Following the same argument as above, we get $\dist p{g^{-1}q} \leq 18\alpha$.
Consequently $\dist p{gp} \leq 36\alpha$, a contradiction.
Observe that

\begin{equation*}
\nu_o(gF) = {g^{-1}}_*\nu_o(F) = \norm{\nu_{go}} \nu^g_{g^{-1}o}(F).
\end{equation*}

As an element of $\mathcal D_0$, the density $\nu$ is $G$-quasi-invariant, hence $\nu_o \ll \nu^g_{g^{-1}o}$.
We prove that $\nu_o(F) > 0$, thus $\nu_o(gF) > 0$.

\begin{equation*}
\nu_o( F \cup gF)  = \nu_o( F) + \nu_o(g F) > 1.
\end{equation*}

This contradicts the fact that $\nu_o$ is a probability measure.

Assume that $G$ is not virtually cyclic and has a contracting element.
Let $N$ be an infinite normal subgroup of $G$.
Let $\omega \in \R_+$.
There exists $(\epsilon, r_0)  \in \R_+^* \times \R_+$ such that for every $r \geq r_0$, for every $N$-invariant, $\omega$-conformal density $\nu = (\nu_x)$, for every $g \in G$, we have

\begin{equation}
\label{eqn: shadow lemma}
\epsilon \norm{\nu_{go}}e^{-\omega \dist o{go}}
\leq \nu_o\left( \mathcal O_o(go,r)\right)
\leq e^{2\omega r}  \norm{\nu_{go}}e^{-\omega \dist o{go}}.
\end{equation}

In particular, $(G,\nu)$ satisfies the Shadow Principle with parameters $(\epsilon, r_0)$.

Let $\nu = (\nu_x)$ be an $\omega$-conformal density.
A classical computation shows that for every $g \in G$ and $r \in \R_+$, we have

\begin{equation*}
\nu_o\left(\mathcal O_o(go,r)\right)
= \norm{\nu_{go}}\int \mathbb 1_{\mathcal O_{g^{-1}o}(o, r)}(c) e^{-\omega c(g^{-1}o,o)} d\nu^g_o(c).
\end{equation*}

Shadows have been designed so that for every $c \in \mathcal O_{g^{-1}o}(o, r)$, we have

\begin{equation*}
\dist o{go} -2 r \leq c(g^{-1}o,o) \leq \dist o{go}.
\end{equation*}

\begin{equation*}
\nu^g_o\left(\mathcal O_{g^{-1}o}(o,r)\right)
\leq \frac { e^{\omega\dist {go}o}}{\norm{\nu_{go}}} \nu_o\left(\mathcal O_o(go,r)\right)
\leq e^{2\omega r}\nu^g_o\left(\mathcal O_{g^{-1}o}(o,r)\right).
\end{equation*}

The upper bound in (<ref>) follows from the fact that $\nu^g_o$ is a probability measure.
Let us focus on the lower bound.
To that end, we assume now that $\nu$ is $N$-invariant.
As $N$ is a normal subgroup of $G$, the density $\nu^g$ is $N$-invariant as well.
Since $G$ contains a contracting element, and is not virtually cyclic, the same holds for $N$ (<ref>).
The result now follows from <ref> applied with the group $N$ and the closed set $\mathcal D_0 = \mathcal D(N,\omega)$.

Note that the upper bound in (<ref>) was proved without assuming any invariance for $\nu$.
Moreover it works for any $r \in \R_+$.

Here is another variation of the Shadow Lemma.

Assume that $G$ is not virtually cyclic and contains a contracting element.
Let $\omega \in \R_+$ and $\nu = (\nu_x)$ be an $\omega$-conformal density.
If $\nu$ is almost fixed by $G$, then $(G,\nu)$ satisfies the Shadow Principle.

Let $g \in G$ and $r \in \R_+$.
Reasoning as in the proof of <ref>, we see that

\begin{equation*}
\nu_o\left(\mathcal O_o(go,r)\right)
\geq \norm{\nu_{go}} e^{-\omega \dist o{go}} \nu^g_o\left(\mathcal O_{g^{-1}o}(o,r)\right).
\end{equation*}

Since $\nu$ is almost-fixed by $G$, there is $\epsilon \in \R_+^*$, which does not depend on $g$ or $r$, such that $\nu^g_o \geq \epsilon \nu_o$.

\begin{equation*}
\nu_o\left(\mathcal O_o(go,r)\right)
\geq \epsilon\norm{\nu_{go}} e^{-\omega \dist o{go}} \nu_o\left(\mathcal O_{g^{-1}o}(o,r)\right).
\end{equation*}

The result now follows from <ref> applied with $\mathcal D_0 = \{ \nu\}$.

§.§ Contracting tails

In order to take full advantage of the Shadow Lemma, we consider a particular kind of shadows, namely shadows of the form $\mathcal O_x(y,r)$ where $x$ is joined to $y$ by a “geodesic” whose tail is contracting.
The next definition quantifies the “contraction strength” of this tail.

Let $\alpha \in \R_+^*$ and  $L \in \R_+$.
Let $x,y \in X$.
The pair $(x,y)$ has an $(\alpha, L)$-contracting tail, if there exists an $\alpha$-contracting geodesic $\tau$ ending at $y$ and a projection $p$ of $x$ on $\tau$ satisfying $\dist py \geq L$.
The path $\tau$ is called a (contracting) tail of $(x,y)$ — the contracting strength and the length of the tail should be clear from the context.

Given $\alpha \in \R^*_+$ and $L \in \R_+$, we denote by $\mathcal T(\alpha, L)$ the set of all elements $g \in G$ such that the pair $(o,go)$ has an $(\alpha, L)$-contracting tail.

The next statement is essentially a reformulation of <ref>.
It states that if $(x,y)$ has a sufficiently long contracting tail, then the shadow of $y$ seen from $x$ behaves according to our intuition coming from hyperbolic geometry.

Let $\alpha \in \R_+^*$ and $r, L \in \R_+$ with $L > r + 13\alpha$.
Let $x,y \in X$.
Assume that $(x,y)$ has an $(\alpha, L)$-contracting tail, say $\tau$.
Let $p$ be a projection of $x$ on $\tau$.
Let $c\in \mathcal O_x(y,r)$.
Let $\gamma$ be a gradient arc from $x$ to $c$.
Let $q$ be a projection of $c$ onto $\tau$.
Then the following holds.

$\dist yq \leq r+7\alpha$.

* $d(\gamma, \tau) < \alpha$.

* The entry point of $\gamma$ in $\mathcal N_\alpha(\tau)$ is $2\alpha$-close to $p$.

* The exit point of $\gamma$ from $\mathcal N_\alpha(\tau)$ is $5\alpha$-close to $q$.

Since $\tau$ is an contracting tail, there is a projection $p'$ of $x$ on $\tau$ such that $\dist {p'}y \geq L$.
The computation shows that

\begin{equation*}
\dist yq + \gro xc{p'} + \gro {p'}cq
= \gro xcy + \gro xy{p'} + \gro ycq + \gro {p'}yq,
\end{equation*}

(it suffices to expand the definition of the Gromov products).
On the one hand, we observed that $\gro xc{p'} \geq 0$ and $\gro {p'}cq \geq 0$.
On the other hand, since $\tau$ is contracting, we have $\gro xy{p'} \leq 2\alpha$ (<ref>) and $\gro ycq \leq 5\alpha$ (<ref>).
It follows that

\begin{equation*}
\dist yq
\leq r + 7 \alpha + \gro {p'}yq.
\end{equation*}

We now discuss the relative positions of $p'$, $q$ and $y$ on $\tau$.
If $p'$ lies between $q$ and $y$, then $\gro {p'}yq = \dist {p'}q$ while $\dist yq = \dist y{p'} + \dist {p'}q$.
It forces $L\leq \dist y{p'} \leq r + 7 \alpha$, which contradicts our assumption.
Thus $q$ lies between $p'$ and $y$ so that $\gro {p'}yq = 0$.
Consequently $\dist yq \leq r + 7 \alpha$, which completes the proof of <ref>.

By assumption, $p$ is another projection of $x$ on $\tau$, which is $\alpha$-contracting, thus $\dist p{p'} \leq 2 \alpha$.

\begin{equation*}
\dist pq \geq \dist {p'}y -  \dist p{p'} - \dist yq \geq L - (r+9\alpha)  > 4\alpha.
\end{equation*}

According to <ref>, $d(\gamma,\tau) < \alpha$.
Moreover the entry (exit) point of $\gamma$ in $\mathcal N_\alpha(\tau)$ is  $2\alpha$-close to $p$ ($5\alpha$-closed to $q$).
Hence the result.

Let $\alpha \in \R_+^*$ and $r, L \in \R_+$ with $L > r + 13\alpha$.
Let $x, y_1, y_2 \in X$ such that $(x,y_i)$ has an $(\alpha, L)$-contracting tail, say $\tau_i$.
Let $p_i$ be a projection of $x$ on $\tau_i$.
If $\mathcal O_x(y_1, r) \cap \mathcal O_x(y_2,r)$ is non empty, then

* $\dist {y_1}{y_2} \leq \abs{\dist x{y_1} - \dist x{y_2}} + 4r + 44\alpha$, and

* $\dist {p_1}{p_2} \leq \abs{\dist x{p_1} - \dist x{p_2}} +  8\alpha$.

Let $c$ be a cocycle in the intersection $\mathcal O_x(y_1, r) \cap \mathcal O_x(y_2,r)$.
Let $\gamma$ be a gradient arc from $x$ to $c$.
According to <ref> the points $y_1$ and $y_2$ lie in the $(r+11\alpha)$-neighborhood of $\gamma$, while $p_1$ and $p_2$ are $2\alpha$-closed to $\gamma$.
The result follows.

In order to state the next lemma, we need a notion of spheres in $G$ (for the metric induced by $X$).
This is the purpose of the following notation.
Given $\ell, a \in \R_+$, we let

\begin{align*}
S(\ell, a) & = \set{g \in G}{\ell - a \leq \dist o{go} < \ell + a},
\end{align*}

Assume that $G$ has a contracting element.
There is $\alpha \in \R^*_+$ such that for every $L \in \R_+$, for every $g \in G$, there exist $u \in S(L,\alpha)$ and a geodesic $\tau \colon \intval 0T \to X$ from $o$ to $guo$ with the following properties

* The path $\gamma$ restricted to $\intval{T-L}T$ is $\alpha$-contracting.
In particular, $gu \in \mathcal T(\alpha, L)$.

* The point $go$ is $\alpha$-close to $\tau(T-L)$.

Let $h \in G$ be a contracting element.
Denote by $A$ the $\group h$-orbit of $o$.
It is $\beta$-contracting for some $\beta \in \R^*_+$.
Let $\alpha_0 \in \R^*_+$ be the parameter given by <ref> applied with the element $h$, the point $z = o$ and $d = 2\beta$.
Up to increasing the value of $\alpha_0$ we can assume that $\alpha_0 \geq  2 \beta + \dist o{ho}$.

Let $L \in \R_+$.
Since $h$ is contracting, the orbit map $\Z \to X$ sending $n$ to $h^no$ is a quasi-isometric embedding.
Thus there is $N \in \N$, such that for every $n \geq N$, we have $\dist o{h^no} \geq L+ 2\beta$.
We choose $N$ minimal with the above property.
In particular,

\begin{equation*}
L + 2\beta
\leq \dist o{h^No} \leq \dist o{h^{N-1}o} + \dist o{ho}
< L + 2\beta + \dist o{ho}.
\end{equation*}

Hence $h^N$ and $h^{-N}$ belong to $S(L, \alpha_0)$.

Let $g \in G$.
There is $k \in \Z$, such that $h^ko$ is a projection of $g^{-1}o$ onto $A$.
If $k \leq 0$ ($k \geq 0$), we choose $u = h^N$ ($u = h^{-N}$).
We now prove that $u$ satisfies the announced properties.
We suppose that $k \geq 0$.
The other case works in the exact same way.
Let $\gamma \colon \intval 0T \to X$ be a geodesic from $g^{-1}o$ to $h^{-N}o$.
According to <ref>, $h^ko$ is $2\beta$-close to the entry point $\gamma(t)$ of $\gamma$ in $\mathcal N_\beta(A)$.
By our choice of $N$, we have

\begin{equation*}
T - t \geq \dist{\gamma(t)}{\gamma(T)} \geq \dist {h^ko}{h^{-N}o} - 2 \beta \geq \dist o{h^{N+k}o} - 2\beta \geq L.
\end{equation*}

In particular, $t \leq T-L\leq T$.
It follows from <ref> and our choice of $\alpha_0$ that

* $\gamma$ restricted to $\intval {T-L}T$ is  $\alpha_0$-contracting;

* there is $s \in \intval {T-L}T$ such that $\dist {\gamma(s)}o \leq \alpha_0$.

Using the triangle inequality we observe that

\begin{equation*}
\abs{\dist{\gamma(s)}{h^{-N}o} - \dist o{h^{-N}o}} \leq \dist{\gamma(s)}o \leq \alpha_0.
\end{equation*}

On the one hand $\dist {\gamma(s)}{h^{-N}o} = T - s$.
On the other hand $h^{-N} \in S(L, \alpha_0)$.
Thus $\abs{(T-L) - s} \leq 2\alpha_0$.
Consequently the triangle inequality yields

\begin{equation*}
\dist o{\gamma(T-L)}
\leq \dist o{\gamma(s)} + \dist{\gamma(s)}{\gamma(T-L)} \leq 3\alpha_0.
\end{equation*}

Observe now that the path $\tau = g\gamma$ satisfies the conclusion of the lemma with the parameter $\alpha = 3\alpha_0$.

Given $\alpha, r, L, \ell \in \R_+$, we consider the following set

\begin{equation*}
A_\ell(\alpha,r, L) = \bigcup_{ g \in S(\ell, r) \cap \mathcal T(\alpha, L) } \mathcal O_o(go, r).
\end{equation*}

Observe that for a fixed $\ell \in \R$, the set $A_\ell(\alpha,r,L)$ is a non-decreasing function of $r$ ($\alpha$) and a non-increasing function of $L$.

Assume that $G$ contains a contracting element.
There is $\alpha \in \R^*_+$, such that for every $\omega, a \in \R_+$ and $(\epsilon, r_0) \in \R_+^* \times \R_+$, there exist $r_1, C \in \R_+^*$, with the following property.
Let $\nu = (\nu_x)$ be an $\omega$-conformal density.
If $(G,\nu)$ satisfies the Shadow Principle with parameters $(\epsilon, r_0)$, then for every $r \geq r_1$, $L > r + 13\alpha$, and $\ell \in \R_+$,

\begin{equation*}
\sum_{g \in S(\ell, a)}  \norm{\nu_{go}}  e^{-\omega \dist o{go}}
\leq C e^{2\omega L}\nu_o \left( A_{\ell+L}(\alpha, r,L) \right).
\end{equation*}

We choose for $\alpha$ the parameter given by <ref>.
Let $\omega, a \in \R_+$ and $(\epsilon, r_0) \in \R_+^* \times \R_+$.
The action of $G$ on $X$ is proper, so there is $M \in \N$ such that

\begin{equation*}
\card{\set{g \in G}{\dist o{go} \leq 2a + 12\alpha}} \leq M.
\end{equation*}

In addition, we set

\begin{equation*}
r_1 = \max\{r_0, a + 3\alpha\}.
\end{equation*}

Let $\nu = (\nu_x)$ be an $\omega$-conformal density such that $(G, \nu)$ satisfies the Shadow Principle with parameters $(\epsilon, r_0)$.
Let $r \geq r_1$,  $L > r + 13\alpha$, and $\ell \in \R_+$.
According to <ref>, for every $g \in S(\ell, a)$, there is $u_g \in S(L,\alpha)$ such that $gu_g$ belongs to $\mathcal T(\alpha, L)$.
Moreover there is a geodesic $\tau_g \colon \intval 0{T_g} \to X$ joining $o$ to $gu_go$, whose restriction to $\intval {T_g-L}{T_g}$ is $\alpha$-contracting, and such that $go$ is $\alpha$-close to $z_g = \tau_g(T_g-L)$.
In particular,

\begin{equation*}
\abs{\dist o{go}  + \dist o{u_go} - \dist o{gu_go}}
\leq 2\alpha.
\end{equation*}

Since $g \in S(\ell, a)$ and $u_g \in S(L,\alpha)$, we get $gu_g \in S(\ell + L, a + 3\alpha)$.
Hence $\mathcal O_o(gu_go,r)$ is contained in $A_{\ell+L}(\alpha, r, L)$.

Consider now $g,g' \in S(\ell, a)$ such that $\mathcal O_o(gu_go,r) \cap \mathcal O_o(g'u_{g'}o,r)$ is non empty.
By construction, $go$ and $g'o$ are $\alpha$-close to $z_g$ and $z_{g'}$.
Note that $z_g = \tau_g(T_g -L)$ is the (unique) projection of $o$ on $\tau_g$ restricted to $\intval{T_g -L}{T_g}$, which is $\alpha$-contracting.
A similar statement holds for $\tau_{g'}$.
Using <ref> and the triangle inequality, we observe that

\begin{align*}
\dist {go}{g'o}
\leq \dist {z_g}{z_{g'}} + 2\alpha
& \leq \abs{ \dist o{z_g} - \dist o{z_{g'}}} + 10\alpha \\
& \leq \abs{ \dist o{go} - \dist o{g'o}} + 12\alpha \\
& \leq 2a + 12\alpha.
\end{align*}

It follows from our choice of $M$ that any cocycle $c \in \bar X$ belongs to a most $M$ shadows of the form $\mathcal O_o(gu_go,r)$, where $g \in S(\ell,a)$.

\begin{equation}
\label{eqn: pre upper bounds}
\sum_{g \in S(\ell, a)} \nu_o\left(\mathcal O_o(gu_go,r) \right)
\leq M\nu_o\left(A_{\ell+L}(\alpha, r, L)\right).
\end{equation}

Recall that $(G, \nu)$ satisfies the Shadow Principle with parameters $(\epsilon, r_0)$.
Hence for every $g \in S(\ell, a)$, we have

\begin{equation*}
\nu_o\left(\mathcal O_o(gu_go,r)\right)
\geq \epsilon \norm{\nu_{gu_go}}e^{-\omega\dist o{gu_go}}
\geq \epsilon e^{-2\omega (L+\alpha)}\norm{\nu_{go}}e^{-\omega\dist o{go}}.
\end{equation*}

The second inequality follows from (<ref>).
Consequently (<ref>) becomes

\begin{equation*}
\sum_{g \in S(\ell, a)}  \norm{\nu_{go}}  e^{-\omega \dist o{go}}
\leq  \left(\frac{M e^{2\omega\alpha}}\epsilon\right) e^{2\omega L}\nu_o\left(A_{\ell+L}(\alpha, r, L)\right). \qedhere
\end{equation*}

Assume that $G$ contains a contracting element.
For every $\omega, a \in \R_+$ and $(\epsilon, r_0) \in \R_+^* \times \R_+$, there exists $C \in \R_+^*$, with the following property.
Let $\nu = (\nu_x)$ be an $\omega$-conformal density.
Assume that $(G, \nu)$ satisfies the Shadow Principle with parameters $(\epsilon, r_0)$.
For every $\ell \in \R_+$,

\begin{equation*}
\sum_{g \in S(\ell, a)}  \norm{\nu_{go}}  e^{-\omega \dist o{go}}
\leq C \nu_o \left( \bar X \setminus B(o, \ell) \right).
\end{equation*}

Denote by $\alpha, r_1, C$ the parameters given by <ref> applied with $\omega$, $a$, and $(\epsilon,r_0)$.
We choose $r \geq r_1$ and $L > \max \{ 2r, r + 15\alpha\}$.
Let $\nu = (\nu_x)$ be an $\omega$-conformal density such that $(G, \nu)$ satisfies the Shadow Principle with parameters $(\epsilon, r_0)$.
By <ref>, we have

\begin{equation*}
\sum_{g \in S(\ell, a)}  \norm{\nu_{go}}  e^{-\omega \dist o{go}}
\leq Ce^{2\omega L}\nu_o \left( A_{\ell+L}( \alpha, r, L) \right), \quad \forall \ell \in \R_+.
\end{equation*}

Consider now $x,y,z \in X$.
By the very definition of shadows, if $z \in \mathcal O_x(y,r)$, then

\begin{equation*}
\dist xy - \dist xz \leq \gro xzy \leq r.
\end{equation*}

Hence for every $\ell \in \R_+$, the set $A_{\ell+L}(\alpha, r, L)$ lies in $\bar X \setminus B(o, \ell + L - 2r)$.
The result follows from the fact that $L \geq 2r$.

§.§ First applications

For our first applications, we assume that $G$ is a group acting properly, by isometries on $X$ with a contracting element.
In addition, we suppose that $G$ is not virtually cyclic.

There exists $C \in \R_+$ such that for every $\ell \in \R_+$,

\begin{equation*}
\card{ \set{g \in G}{ \dist o{go} \leq \ell}} \leq C e^{\omega_G\ell}.
\end{equation*}

An alternative proof of this fact can be found in Yang [51].

According to <ref> there exists a $G$-invariant, $\omega_G$-conformal density $\nu = (\nu_x)$.
By <ref>, the pair $(G, \nu)$ satisfies the Shadow Principle for some parameters $(\epsilon, r_0) \in \R_+^* \times \R_+$.
Fix $a \in \R_+^*$.
Applying <ref>, there exists $C \in \R_+$ such that for every $\ell \in \R_+$,

\begin{equation*}
\sum_{g \in S(\ell, a)} \norm{\nu_{go}}  e^{-\omega_G \dist o{go}}
\leq C \nu_o \left( \bar X \setminus B(o, \ell) \right) \leq C.
\end{equation*}

However, since $\nu$ is $G$-invariant, $\norm{\nu_{go}} = \norm{\nu_o} = 1$, for every $g \in G$.
Hence we get

\begin{equation*}
\card{S(\ell, a)} \leq Ce^{\omega_G a}e^{\omega_G \ell}, \quad \forall \ell \in \R_+.
\end{equation*}

The result follows by summing this inequality over $\ell \in a \N$.

Before stating our next application, we define the radial limit set for the action of $G$ on $X$.

Let $r \in \R_+$ and $ \in X$.
The set $\Lambda_{\rm rad}(G, x,r)$ consists of all cocycles $c \in \partial X$ with the following property:  for every $T \geq 0$, there exists $g \in G$ with $\dist x{go} \geq T$ such that $c \in \mathcal O_x(go,r)$.
The radial limit set of $G$ is the union

\begin{equation*}
\Lambda_{\rm rad}(G)
= \bigcup_{ r \geq 0}
G \Lambda_{\rm rad}(G, o, r).
\end{equation*}

Note that $\Lambda_{\rm rad}(G, x,r)$ is a non-decreasing function of $r$.
The radial limit set $\Lambda_{\rm rad}(G)$ is $G$-invariant.
It follows from (<ref>) that

\begin{equation*}
\Lambda_{\rm rad}(G) =
\bigcup_{r \geq 0}
\Lambda_{\rm rad}(G, o, r).
\end{equation*}

The radial limit set is saturated.

Let $x,y \in X$ and $r \in \R_+$.
Let $c, c' \in \partial X$ such that $c \sim c'$.
Note that if $c$ belongs to $\mathcal O_x(y,r)$ then $c'$ belongs to $\mathcal O_x(y,r')$ where $r' = r + \norm[\infty]{c- c'}$.
The result follows from this observation.

Let $\omega \in \R_+$ and $\nu = (\nu_x)$ be an $\omega$-conformal density.
If $(G, \nu)$ satisfies the Shadow Principle then the series

\begin{equation}
\label{eqn: series roblin}
\sum_{g \in G} \norm{\nu_{go}}e^{-s\dist o{go}}
\end{equation}

converges whenever $s > \omega$.
If, in addition, $\nu_o$ gives positive measure the radial limit set $\Lambda_{\rm rad}(G)$, then the series diverges at $s = \omega$.
In particular its critical exponent is exactly $\omega$.

Fix $a \in \R_+^*$.
According to <ref> there is $C \in \R_+^*$ such that for every $\ell \in \R_+$,

\begin{equation*}
\sum_{g \in S(\ell, a)}\norm{\nu_{go}}  e^{-\omega \dist o{go}}
\leq C \nu_o \left( \bar X \setminus B(o, \ell) \right)
\leq C.
\end{equation*}

Hence the series (<ref>) converges whenever $s > \omega$.
The second part of the proposition is proved by contraposition.
Suppose that the series (<ref>) converges at $s = \omega$.
Let $r \in \R_+$.
For simplicity we write  $\Lambda = \Lambda_{\rm rad}(G, o, r)$.
Observe that for every $T \in \R_+$,

\begin{equation*}
\Lambda \subset  \bigcup_{\substack{g \in G \\ \dist o{go} \geq T}} \mathcal O_o(go, r).
\end{equation*}

It follows from <ref>, that

\begin{equation*}
\nu_o(\Lambda)
\leq \sum_{\substack{g \in G \\ \dist o{go} \geq T}} \nu_o\left(\mathcal O_o(go, r)\right)
\leq e^{2\omega r} \sum_{\substack{g \in G \\ \dist o{go} \geq T}} \norm{\nu_{go}}e^{-\omega \dist o{go}}.
\end{equation*}

The right-hand side of the inequality is the remainder of the series we are interested in.
Since this series converges at $s = \omega$, we get $\nu_o(\Lambda) = 0$.
We observed in <ref> that

\begin{equation*}
\Lambda_{\rm rad}(G) =  \bigcup_{r \geq 0} \Lambda_{\rm rad}(G, o, r).
\end{equation*}

Hence $\nu_o(\Lambda_{\rm rad}(G)) = 0$.

Note that the proof of the second assertion (the series diverges at $s = \omega$, whenever $\nu_o$ gives positive measure to the radial limit set) only uses the upper estimate from the Shadow lemma.
Hence it holds even if $G$ is virtually cyclic or does not contain a contracting element.

Let $\omega \in \R_+$.
Let $\nu = (\nu_x)$ be a $G$-invariant, $\omega$-conformal density.
Then $\omega \geq \omega_G$.
Moreover if $\nu_o$ gives positive measure to the radial limit set $\Lambda_{\rm rad} (G)$, then $\omega = \omega_G$ and the action of $G$ on $X$ is divergent.

If the action of $G$ on $X$ is proper an co-compact, one checks that the radial limit set is actually $\partial X$.
Hence if $\nu = (\nu_x)$ is a $G$-invariant, $\omega$-conformal density supported on $\partial X$, then $\omega = \omega_G$.

By <ref>, the pair $(G, \nu)$ satisfies the Shadow Principle.
Since the density $\nu$ is $G$-invariant, the series (<ref>) which appears in <ref> is exactly the Poincaré series of $G$.
The result follows.

Let $\omega \in \R_+$.
Let $\nu = (\nu_x)$ be an $\omega$-conformal density and $\mu = (\mu_x)$ its restriction to the reduced horocompactification $(\bar X, \mathfrak R)$.
Assume that $(G, \nu)$ satisfies the Shadow Principle.
If $\mu$ is almost-fixed by $G$, then $\omega \geq \omega_G$.

According to <ref> the map $\chi \colon G \to \R_+$ sending $g$ to $\ln\norm{\nu_{go}}$ is a quasi-morphism.
Note that the exponent of the series

\begin{equation*}
\sum_{g \in G} \norm{\nu_{go}} e^{-s\dist o{go}}
= \sum_{g \in G} e^{\chi(g)} e^{-s\dist o{go}}
\end{equation*}

is exactly $\omega_\chi$.
We observed earlier that $\omega_\chi \geq \omega_G$.
The result follows from <ref>.

Let $N$ be a normal subgroup of $G$ such that $G/N$ is amenable.
Then $\omega_N = \omega_G$.

The proof follows Roblin's argument [43].
Assume first that $N$ is finite.
This forces $G$ to be itself amenable.
However since $G$ contains a contracting element, $G$ has to be virtually cyclic (otherwise $G$ would contain a free subgroup).
Hence $\omega_G = 0$ and the result holds.
Suppose now that $N$ is infinite.
For simplicity we let $Q = G/N$.
Denote by $\nu = (\nu_x)$ an $N$-invariant, $\omega_N$-conformal density on $\bar X$.
We are going to “average” the $G$-orbit of $\nu$ to produce another $N$-invariant, $\omega_N$-conformal density.
The construction goes as follows.
Choose a $Q$-invariant mean $M \colon \ell^\infty(Q) \to \R$.
Let $x \in X$ and $f \in C(\bar X)$.
Consider the function

\begin{equation*}
\begin{array}{rccc}
\psi_{x,f} \colon & G & \to & \R \\
& u & \mapsto & \displaystyle \int f d\nu_x^u
\end{array}
\end{equation*}

According to (<ref>), we have $\norm {\nu_x^u} \leq e^{\omega_N \dist ox}$, for every $u \in G$.
Consequently, $\psi_{x,f}$ is a bounded function.
Since $\nu$ is $N$-invariant, we also observe that $\psi_{x,f}$ induces a map bounded map $Q \to \R$, that we still denote by $\psi_{x,f}$.
Using Riesz representation theorem we define a new measure $\mu_x$ by imposing

\begin{equation*}
\int f d\mu_x = M \left(\psi_{x,f}\right),\quad \forall f\in C(\bar X).
\end{equation*}

One checks that $\mu = (\mu_x)$ is still $\omega_N$-conformal.
Since $N$ is a normal subgroup, it is also $N$-invariant.
In particular, it satisfies the Shadow Principle (<ref>).
According to <ref> the series

\begin{equation}
\label{eqn: amenability roblin}
\sum_{g \in G} \norm{\mu_{go}}e^{-s\dist o{go}}
\end{equation}

converges whenever $s > \omega_N$. Consider now the map

\begin{equation*}
\begin{array}{rccc}
\chi \colon & G & \to & \R \\
& g & \mapsto & M \left(\ln \psi_{go,\mathbb 1}\right).
\end{array}
\end{equation*}

Note that

\begin{equation*}
\psi_{g_1g_2o,\mathbb 1}(u) = \psi_{g_2o,\mathbb 1}(ug_1) \psi_{g_1o,\mathbb 1}(u), \quad \forall u,g_1,g_2 \in G.
\end{equation*}

Since $M$ is $Q$-invariant, $\chi$ is a homomorphism (factoring through the projection $G \onto Q$).
The exponential is a convex function, hence the Jensen inequality yields

\begin{equation*}
e^{\chi(g)} \leq \norm {\mu_{go}}, \quad \forall g \in G.
\end{equation*}

\begin{equation*}
\sum_{g \in G} \norm{\mu_{go}}e^{-s\dist o{go}}
\geq \sum_{g \in G} e^{\chi(g)}e^{-s\dist o{go}}
\end{equation*}

The critical exponent of the sum on the right hand side is $\omega_\chi$.
Combined with the above discussion it yields $\omega_N \geq \omega_\chi$.
We observed previously that for a homomorphism $\chi$, we have $\omega_\chi \geq \omega_G$, thus $\omega_N \geq \omega_G$.
The other inequality follows from the fact that $N \subset G$.

In the proof, we have used the fact that $N$ is normal in a crucial way to ensure that $\mu$ is $N$-invariant and therefore satisfies the Shadow Principle.
It would be nice to generalize this strategy to the case of a co-amenable subgroup $H$, i.e. when there is a $G$-invariant mean on the coset space $H\backslash G$.
We know from the Patterson-Sullivan construction that $H$ fixes a point $\nu$ in $\mathcal D(\omega_H)$.
Since $G$ acts on this compact space, it would be tempting to use the fixed point characterization of co-amenability due to Eymard <cit.> and conclude that $G$ should fix a point $\mu$ in $\mathcal D(\omega_H)$.
Even though $\mu$ may not be $H$-invariant that would be enough to ensure that it satisfies the Shadow Principle.
One has to be careful though that the action of $G$ on $\mathcal D(\omega_H)$ is not affine.
Hence the fixed point criterion does not apply here.
In this context, one can easily prove that the “average” density $\mu$ we have built is $G$-quasi-invariant.
Unfortunately, this information is not sufficient to prove that it satisfies the Shadow Principle.

Let $N$ be an infinite, normal subgroup of $G$.

\begin{equation*}
\omega(N,X) + \frac 12 \omega(G/N, X/N) \geq \omega(G, X).
\end{equation*}

Let $Q = G/N$.
Denote by $\pi \colon G \onto Q$ and $\zeta \colon X \to X/N$ the canonical projections.
For simplicity we write $\omega_N$ and $\omega_G$ for the growth rates of $N$ and $G$ acting on $X$, while $\omega_Q$ stands for the growth rate of $Q$ acting on $X/N$.
We denote by $\mathcal H$ the Hilbert space $\mathcal H = \ell^2(Q)$.
Given $s,t \in \R_+$ we consider the following maps

\begin{equation*}
\begin{array}{rccccrccc}
\phi_s \colon & Q & \to & \R
& \quad \text{and} \quad \
& \psi_t \colon & Q & \to & \R \\
& q & \mapsto &\displaystyle  \sum_{g \in \pi^{-1}(q)} e^{-s\dist o{go}}
& q & \mapsto & \displaystyle e^{-t\dist {\zeta(o)}{q\zeta(o)}}
\end{array}
\end{equation*}

One checks that $\psi_t \in \mathcal H$ ($\psi_t \notin \mathcal H$) whenever $2t >  \omega_Q$ ($2t <  \omega_Q$).
Similarly $\phi_s \notin \mathcal H$, whenever $s < \omega_N$.
We now prove that the converse essentially holds true.

If $s > \omega_N$, then $\phi_s \in \mathcal H$.

Let $s > \omega_N$.
Consider the $N$-invariant, $s$-conformal density $\nu^s = (\nu^s_x)$ on $\bar X$ defined by

\begin{equation*}
\nu^s_x = \frac 1 {\mathcal P_N(s)} \sum_{h \in N} e^{-s \dist x{ho}} {\rm Dirac}(ho).
\end{equation*}

The computation shows that

\begin{align*}
\norm{\phi_s}^2
= \sum_{\substack{g_1,g_2 \in G\\ \pi(g_1)=\pi(g_2)}} e^{-s \left[\dist o{g_1o} + \dist o{g_2o}\right]}
& = \sum_{g \in G}e^{-s \dist o{go}} \left(\sum_{h \in N} e^{-s\dist{go}{ho}}\right) \\
& = \mathcal P_N(s)  \sum_{g \in G} \norm{\nu^s_{go}} e^{-s \dist o{go}} .
\end{align*}

Fix $a \in \R_+^*$.
According to <ref>, $\nu^s$ satisfies the Shadow Principle.
By <ref> there exists $C \in\R_+^*$, such that for every $\ell \in \R_+$,

\begin{equation*}
\sum_{g \in S(\ell, a)} \norm{\nu^s_{go}}  e^{-s \dist o{go}}
\leq C\nu^s_o\left(\bar X \setminus B(o, \ell)\right).
\end{equation*}

Unfolding the definition of $\nu^s$, we get

\begin{equation*}
\mathcal P_N(s)\sum_{g \in S(\ell, a)}  \norm{\nu^s_{go}}  e^{-s \dist o{go}}
\leq C \sum_{\substack{h \in N \\ \dist o{ho} \geq \ell}} e^{-s \dist o{ho}}.
\end{equation*}

It follows that

\begin{equation*}
\norm{\phi_s}^2
\leq C \sum_{k \in \N} \sum_{\substack{h \in N \\ \dist o{ho} \geq ka}} e^{-s \dist o{ho}}
\leq C \sum_{h \in N} \sum_{\substack{k \in \N \\ ka \leq \dist o{ho}}}  e^{-s \dist o{ho}}.
\end{equation*}

Up to replacing $C$ by a larger constant, we have

\begin{equation}
\label{eqn: improved lower bound growth normal sbgp}
\norm{\phi_s}^2
\leq C \sum_{h \in N} \left[1 + \dist o{ho}\right] e^{-s \dist o{ho}}.
\end{equation}

Note that the series

\begin{equation*}
-\sum_{h \in N} \dist o{ho} e^{-s\dist o{ho}}
\end{equation*}

is the derivative of the Poincaré series of $N$, hence it converges.
Consequently the right-hand side in (<ref>) converges.
Thus $\norm{\phi_s}$ is finite, which completes the proof of our claim.

Let $s, t \in \R_+$ with $s > \omega_N$ and $2t > \omega_Q$.
The scalar product of $\phi_s$ and $\psi_t$ can be computed as follows:

\begin{align*}
\left( \phi_s, \psi_t\right)
& = \sum_{q \in Q} e^{-t \dist {\zeta(o)}{q\zeta(o)}} \left(\sum_{g \in \pi^{-1}(q)} e^{-s\dist o{go}}\right) \\
& = \sum_{g \in G} e^{-s \dist o{go}}e^{-t \dist{\zeta(o)}{\pi(g)\zeta(o)}}.
\end{align*}

The projection $\zeta \colon X \onto X/N$ is $1$-Lipschitz.
Combined with the Cauchy-Schwartz inequality, it gives

\begin{equation*}
\mathcal P_G(s+t)
\leq \left( \phi_s, \psi_t\right)
\leq \norm{\phi_s}\norm{\psi_t} < \infty.
\end{equation*}

Consequently $s + t \geq \omega_G$.
This inequality holds for every $s, t \in \R_+$ with $s > \omega_N$ and $2t > \omega_Q$, whence the result.

We keep the notations of the proof.
<ref> is sharp.

* Indeed, assume that $X$ is a Cayley graph of $G$.
If $Q$ has subexponential growth, then $\omega_Q = 0$ and $Q$ is amenable, so that $\omega_N = \omega_G$ by <ref>.

* At the other end of the spectrum, assume that $G = \free r$ is the free group of rank $r$ acting on its Cayley graph $X$ with respect to a free basis.
Let $g \in G \setminus\{1\}$.
For every $k \in \N$, denote by $N_k$ the normal closure of $g^k$.

\begin{equation*}
\lim_{k \to \infty}\omega_{N_k} = \frac 12 \omega_G
\quad \text{and} \quad
\lim_{k \to \infty}{\omega_{G/N_k}} = \omega_G.
\end{equation*}

The first limit is due to Grigorchuk [25], see also Champetier [10].
The second limit was proved by Shukhov [46], see also Coulon [17].

§ DIVERGENT ACTIONS

§.§ Contracting tails, continued

We continue here our study of shadows of elements $g \in \mathcal T(\alpha, L)$ having a contracting tail.
Before we recall the proof of the following classical result.

Let $C \in \R_+$.
Let $x,y \in X$ and  $\gamma \colon \intval ab \to X$ be a continuous path from $x$ to $y$.
Let $\nu \colon I \to X$ be a geodesic.
Denote by $p = \nu(c)$ and $q = \nu(d)$ respective projections of $x$ and $y$ on $\nu$ and suppose that $c \leq d$.
Assume in addition that $\gamma$ lies in the $C$-neighborhood of $\nu$.
Then $\nu$ restricted to $\intval cd$ lies in the $2C$-neighborhood of $\gamma$.

Fix a point $r = \nu(t_0)$ on $\nu$ where $t_0 \in \intval cd$.
Consider the set $J$ consisting of all $s \in \intval ab$ such that $\gamma(s)$ admits a projection on $\nu$ of the form $\nu(t)$ with $t \leq t_0$.
Note that $J$ is non empty as it contains $a$.
Now set $s_0 = \sup J$.
Let $\epsilon > 0$.
Since $\gamma$ is continuous, there is $\eta > 0$ such that for every $s \in \intval ab$, we have $\dist {\gamma(s)}{\gamma(s_0)} \leq \epsilon$ provided $\abs{s-s_0} \leq \eta$.
By definition of $J$, there is $s_1 \in (s_0-\eta, s_0]$ such that $\gamma(s_1)$ admits a projection on $\nu$ of the form $\nu(t_1)$ with $t_1 \leq t_0$.
On the other hand, we claim that there is $s_2 \in [s_0, s_0 + \eta)$ such that $\gamma(s_2)$ admits a projection on $\nu$ of the form $\nu(t_2)$ with $t_2 \geq t_0$.
Indeed if $s_0 = b$, we can simply take $s_2 = b$.
Otherwise it follows from the definition of $J$.
By assumption $\dist{\nu(t_i)}{\gamma(s_i)} \leq C$.
Hence the triangle inequality yields

\begin{equation*}
\dist {\nu(t_1)}{\nu(t_2)}
\leq \dist {\gamma(s_1)}{\gamma(s_2)} + 2C
\leq 2C + 2\epsilon.
\end{equation*}

and then

\begin{equation*}
d(r, \gamma)
\leq \frac 12 \dist {\nu(t_1)}{\nu(t_2)} + \max \left\{ \dist{\nu(t_1)}{\gamma(s_1)}, \dist{\nu(t_2)}{\gamma(s_2)}\right\}
\leq 2C + \epsilon.
\end{equation*}

This inequality holds for every $\epsilon > 0$, whence the result.

Let $\alpha \in \R_+^*$ and $r, L \in \R_+$ with $L > r + 16\alpha$.
Let $x,y_1, y_2 \in X$ with $\dist x{y_1} \leq \dist x{y_2}$.
Assume that the pairs $(x,y_1)$ and $(x,y_2)$ have an $(\alpha, L)$-contracting tail.
If $\mathcal O_x(y_1,r) \cap \mathcal O_x(y_2,r)$ is not empty, then $\mathcal O_x(y_2,r)$ is contained in $\mathcal O_x(y_1,r + 42\alpha)$.

By assumption there is an $\alpha$-contracting geodesic $\tau_i$ ending at $y_i$, and a projection $p_i$ of $x$ on $\tau_i$ satisfying $\dist{p_i}{y_i} \geq L$.
In addition, we write $z_i$ for the point on $\tau_i$ satisfying $\dist {y_i}{z_i} = r + 13\alpha$.
We split the proof in several claims.

Let $c \in \mathcal O_x(y_i,r)$ and $\gamma \colon I \to X$ a gradient line from $x$ to $c$.
Then $z_i$ lies in the $5\alpha$-neighborhood of $\gamma$.
In particular $c \in \mathcal O(z_i, 5\alpha)$.

According to <ref>, $\gamma$ intersects $\mathcal N_\alpha(\tau_i)$.
Denote by $\gamma(s_i)$ and $\gamma(t_i)$ the entry and exit point of $\gamma$ in $\mathcal N_\alpha(\tau_i)$.
We know, again from <ref>, that $\dist{\gamma(s_i)}{p_i} \leq 2 \alpha$ and $\dist{\gamma(t_i)}{y_i} \leq r + 12\alpha$.
In particular, if $m_i$ and $n_i$ stands for projections of $\gamma(s_i)$ and $\gamma(t_i)$ on $\tau_i$ respectively then $m_i$, $z_i$, $n_i$ and $y_i$ are aligned in this order along $\tau_i$.
The path $\gamma$ restricted to $\intval{s_i}{t_i}$ lies in the $5\alpha/2$-neighborhood of $\tau$ (<ref>).
It follows that $z_i$ is $5\alpha$-close to $\gamma$ (<ref>).

We have $\gro x{z_2}{z_1} \leq 24\alpha$.

According to our assumption $\mathcal O_x(y_1,r) \cap \mathcal O_x(y_2,r)$ is not empty.
Let $\gamma \colon I \to X$ be a gradient arc from $x$ to a cocycle $c$ in this intersection.
It follows form our previous claim that there is $u_i \in \R_+$ such that $\gamma(u_i)$ is $5\alpha$-close to $z_i$.
Hence the triangle inequality yields

\begin{equation}
\label{eqn: pre-vitali}
\gro x{z_2}{z_1} \leq \gro x{\gamma(u_2)}{\gamma(u_1)} + 10\alpha.
\end{equation}

Observe that

\begin{equation*}
\dist  x{z_1} + \dist{z_1}{y_1} - 4 \alpha
\leq \dist x{y_1}
\leq \dist x{y_2}
\leq \dist x{z_2} + \dist {z_2}{y_2}.
\end{equation*}

Indeed, the first inequality follows from <ref> applied to $\tau_1$, the second one is part of our assumptions, while the last one is just the triangle inequality.
By construction $\dist {y_1}{z_1} = \dist {y_2}{z_2}$.
Combined with the triangle inequality it yields

\begin{equation*}
\dist x{\gamma(u_1)} \leq \dist x{\gamma(u_2)} + 14\alpha.
\end{equation*}

Therefore $\gro x{\gamma(u_2)}{\gamma(u_1)} \leq 14\alpha$, which combined with (<ref>) completes the proof of our second claim.

$\mathcal O_x(y_2,r)$ is contained in $\mathcal O_x(y_1,r + 42\alpha)$.

Consider $c \in \mathcal O_x(y_2,r)$.
According to our first claim $\gro xc{z_2} \leq 5\alpha$.
Since cocyles are $1$-Lipschitz, we get in combination with the previous claim

\begin{equation*}
\gro xc{z_1} \leq \gro xc{z_2} + \gro x{z_2}{z_1} \leq 29 \alpha.
\end{equation*}

However $\dist {y_1}{z_1} = r + 13\alpha$.
It follows from (<ref>) that $\gro xc{y_1} \leq r + 42\alpha$.

The next statement will be used later to estimate the measures of various saturated sets using a Vitali type argument.

Let $\alpha \in \R_+^*$ and $r, L \in \R_+$ with $L > r + 16\alpha$.
Let $S$ be a subset of $\mathcal T(\alpha, L)$.
There is a subset $S^* \subset S$ with the following properties

The collection $\left(\mathcal O_o(go,r)\right)_{g \in S^*}$ is pairwise disjoint.

$\displaystyle \bigcup_{g \in S} \mathcal O_o(go,r) \subset  \bigcup_{g \in S^*} \mathcal O_o(go,r+42\alpha)$.

Since the action of $G$ on $X$ is proper, we can index the elements $g_0, g_1, g_2 \dots$ of $S$ such that $\dist o{g_io} \leq \dist o{g_{i+1}o}$, for every $i \in \N$.
We build by induction a sequence $(i_n)$ as follows.
Set $i_0 = 0$.
Let $n \in \N$ for which $i_n$ has been defined.
We search for the minimal index $j > i_n$ such that

\begin{equation*}
\left( \bigcup_{k = 0}^n \mathcal O_o(g_{i_k}o,r)\right) \cap  \mathcal O_o(g_jo,r)  = \emptyset.
\end{equation*}

If such an index exists, we let $i_{n+1} = j$.
Otherwise we let $i_{n+1} = i_n$, in other words, the sequence $(i_n)$ eventually stabilizes.
Finally we set

\begin{equation*}
S^* = \set{g_{i_n}}{n \in \N}.
\end{equation*}

Note that <ref> directly follows from the construction.
Let $g \in S$.
If $g$ does not belong to $S^*$, it means that there is $h \in S^*$, with $\dist o{ho} \leq \dist o{go}$ such that $\mathcal O_o(go,r) \cap \mathcal O_o(ho,r)$ is non-empty.
Hence by <ref>, the shadow $\mathcal O_o(go,r)$ is contained in $\mathcal O_o(ho,r+42\alpha)$.
This completes the proof of <ref>.

Let $\alpha \in \R_+^*$ and $r, L \in \R_+$ with $L > r + 13\alpha$.
Let $x,y \in X$ such that $(x,y)$ has an $(\alpha, L)$-contracting tail.
Let $K$ be a closed ball of radius $R$ centered at $x$.
If $\dist xy > R + r +  13\alpha$, then for every $c,c' \in \mathcal O_x(y,r)$, we have $\norm[K]{c-c'} \leq 20\alpha$.

Denote by $\tau$ a contracting tail of $(x,y)$ and let $p$ be a projection of $x$ on $\tau$ such that $\dist py \geq L$.
Fix $x' \in K$ and denote by $p'$ a projection of $x'$ onto $\tau$.
Let $q$ be a projection of $c$ on $\tau$, so that $\dist yq \leq r + 7\alpha$  (<ref>).
We claim that $\dist {p'}q > 4\alpha$.
If $\dist p{p'} \leq \alpha$, then this is just a consequence of the triangle inequality.
Thus we can assume that $\dist p{p'} > \alpha$.
According to <ref> we have $\dist x{p'} \leq \dist x{x'} + 2 \alpha$.

\begin{align*}
\dist {p'}q
\geq \dist xy - \dist x{p'} - \dist qy
& \geq \dist xy - \dist x{x'} - \dist qy -2 \alpha \\
& \geq \dist xy - (R + r + 9\alpha)
> 4 \alpha.
\end{align*}

It follows from <ref> that

\begin{equation*}
\dist {x'}q - 10\alpha \leq c(x',q) \leq \dist {x'}q.
\end{equation*}

This estimate holds for every $x' \in K$.
Consider now $x_1,x_2 \in K$.
Since $c$ is a cocycle, $c(x_1,x_2) = c(x_1,q) + c(q,x_2)$, thus

\begin{equation*}
\abs{c(x_1,x_2) - \left[\dist {x_1}q - \dist {x_2}q\right]}\leq 10\alpha.
\end{equation*}

The same argument holds for $c'$, thus $\norm[K]{c-c'} \leq 20\alpha$.

§.§ The contracting limit set

Contracting limit set.
We now introduce a variation of the the radial limit set that keeps track of the elements fellow-traveling with some contracting geodesics.
Let $\alpha,r,L \in \R_+$ and $x \in X$.
The set $\Lambda_{\rm ctg}(G, x, \alpha, r, L)$ consists of all cocycles $c \in \partial X$ with the following property:  for every $T \geq 0$, there exists $g \in G$ such that

* $\dist x{go} \geq T$,

* $(x,go)$ has an $(\alpha, L)$-contracting tail,

* $c \in \mathcal O_x(go,r)$.

We also let

\begin{equation*}
\Lambda_{\rm ctg} (G, x, \alpha,r) = \bigcap_{ L \in \R_+} \Lambda_{\rm ctg} (G, x, \alpha, r, L).
\end{equation*}

Observe that the set $\Lambda_{\rm ctg}(G, x, \alpha, r, L)$ is a non-decreasing (non-increasing) function of $\alpha$ and $r$, ($L$).

The contracting limit set of $G$ is

\begin{equation*}
\Lambda_{\rm ctg}(G) =
\bigcup_{\alpha, r \in \R_+} G \Lambda_{\rm ctg}(G, o,\alpha, r).
\end{equation*}

It follows from the definition that the contracting limit set is $G$-invariant and contained in the radial limit set.

Let $\alpha \in \R^*_+$.
For every $r \in \R_+$ and $L > r + 13\alpha$ we have

\begin{equation*}
G \Lambda_{\rm ctg}(G, o,\alpha, r, L) \subset \Lambda_{\rm ctg}(G, o, \alpha,r+12\alpha, L- \alpha).
\end{equation*}

In particular, $G \Lambda_{\rm ctg}(G, o,\alpha, r) \subset \Lambda_{\rm ctg}(G, o, \alpha,r+12\alpha)$.
Moreover, the contracting limit set can also be described as

\begin{equation*}
\Lambda_{\rm ctg}(G) =
\bigcup_{\alpha, r \in \R_+}
\Lambda_{\rm ctg}(G, o, \alpha, r).
\end{equation*}

Let $h \in G$ and $c$ be a cocycle in

\begin{equation*}
h\Lambda_{\rm ctg}(G, o,\alpha, r, L) = \Lambda_{\rm ctg}(G, ho,\alpha, r, L).
\end{equation*}

For simplicity we let $b = h^{-1}c$.
For every $n \in \N$, we can find an element $g_n \in \mathcal T(\alpha, L)$ such that $\dist o{g_no} \geq n$ and $b \in \mathcal O_o(g_no, r)$.
Denote by $\tau_n$ the contracting tail for the pair $(o,g_no)$ and $p_n$ a projection of $o$ on $\tau_n$ satisfying $\dist {p_n}{g_no} \geq L$.
Up to passing to a subsequence, we can assume that $\dist o{g_no} \geq \dist o{h^{-1}o} + L + \alpha$, for every $n \in \N$.

Let $n \in \N$.
We claim that $hg_n$ belongs to $\mathcal T(\alpha, L - \alpha)$.
Let $p'_n$ be a projection of $h^{-1}o$ on $\tau_n$.
It suffices to prove that $\dist {p'_n}{g_no} \geq L-\alpha$.
Indeed, after translating the figure by $h$, it tells us that $h\tau_n$ is a contracting tail for the pair $(o, hg_no)$.
We distinguish two cases.
Assume first that $\dist {p_n}{p'_n} > \alpha$.
It follows from <ref> that $\dist o{p'_n} \leq \dist o{h^{-1}o} + 2\alpha$.
The triangle inequality yields

\begin{equation*}
\dist{p'_n}{g_no} \geq \dist o{g_no} - \dist o{p'_n}
\geq \dist o{g_no} - \dist o{h^{-1}o} - 2\alpha
\geq L - \alpha
\end{equation*}

Assume now that $\dist{p_n}{p'_n} \leq \alpha$.
The triangle inequality yields

\begin{equation*}
\dist{p'_n}{g_no} \geq \dist {p_n}{g_no} - \dist {p_n}{p'_n}
\geq L - \alpha,
\end{equation*}

which completes the proof of our claim.

We now prove that $c \in \mathcal O_o(hg_no, r +18\alpha)$.
Let $q_n$ be a projection of $b$ onto $\tau_n$.
According to <ref>, we have $\dist {q_n}{g_no} \leq r + 7\alpha$.
Combining the above discussion with the triangle inequality we have

\begin{equation*}
\dist {p'_n}{q_n}
\geq \dist {p'_n}{g_no} - \dist {q_n}{g_no}
\geq L - (r + 8\alpha) > 4\alpha.
\end{equation*}

On the one hand, by <ref>, we have

\begin{equation*}
b(h^{-1}o, q_n) \geq \dist {h^{-1}o}{q_n} - 10\alpha,
\end{equation*}

i.e. $\gro{h^{-1}o}b{q_n} \leq 5\alpha$.

\begin{equation*}
\gro {h^{-1}o}b{g_no} \leq \gro {h^{-1}o}b{q_n} + \dist {q_n}{g_no}  \leq r + 12\alpha.
\end{equation*}

Recall that $b = h^{-1}c$.
The above inequality implies that $c$ belongs to the shadow $\mathcal O_o(hg_no, r + 12\alpha)$, which completes the proof of our claim.
Note that $\dist o{hg_no}$ diverges to infinity, hence $c \in  \Lambda_{\rm ctg}(G, o, \alpha, r+12\alpha,L-\alpha)$.

The points in the contracting limit set have a specific behavior with respect to the equivalence relation $\sim$  used to define the reduced horoboundary (see <ref>).
Proceeding as for the radial limit set (see <ref>) one checks that the contracting limit set is saturated.
The next statement precises this fact.

Let $\alpha \in \R_+^*$ and $r, L \in \R_+$ with $L > r + 29\alpha$.
Let $c,c' \in \partial X$ such that $c \sim c'$.
Assume that $c$ belongs to $\Lambda_{\rm ctg}(G,o, \alpha, r, L)$.
Then $\norm[\infty] {c-c'} \leq 20\alpha$ and $c'$ belongs to $\Lambda_{\rm ctg}(G,o, \alpha, r + 16\alpha, L)$.
In particular, the saturation of $\Lambda_{\rm ctg}(G,o, \alpha, r)$ is contained in $\Lambda_{\rm ctg}(G,o, \alpha, r + 16\alpha)$.

Since $c$ belongs to $\Lambda_{\rm ctg}(G,o,  \alpha, r, L)$, there exists a sequence of elements $g_n \in \mathcal T(\alpha, L)$ such that $\dist o{g_no}$ diverges to infinity and $c \in \mathcal O_o(g_no,r)$ for every $n \in \N$.
For each $n \in \N$,  the pair $(o,g_no)$ has an $(\alpha, L)$-contracting tail, say $\tau_n$.
Let $q_n$ ($q'_n$) be a projection of $c$ ($c'$) onto $\tau_n$.
We break the proof in several steps.

For every $n \in \N$, we have $\dist {q_n}{q'_n} \leq 4\alpha$.

Fix $n \in \N$.
Let $\gamma \colon \R_+ \to X$ be a gradient ray for $c$ starting at $q_n$.
Let $t \in \R_+$.
According to <ref>, $q_n$ is a projection of $\gamma(t)$ on $\tau_n$.
Assume that contrary to our claim $\dist {q'_n}{q_n} > 4\alpha$.
It follows from <ref> applied with $c'$ that

\begin{equation*}
c'(\gamma(t), q'_n) \geq \dist {q'_n}{\gamma(t)} - 10\alpha \geq - 10\alpha.
\end{equation*}

On the other hand, since $\gamma$ is a gradient ray for $c$, we know that

\begin{equation*}
c(\gamma(t), q'_n)  = c(\gamma(t), q_n) + c(q_n,q'_n)  = - t + c(q_n,q'_n).
\end{equation*}

These two estimates hold for every $t \geq T$, thus contradicting the fact that $\norm[\infty]{c - c'} < \infty$.
It completes the proof of our first claim.

For every $n \in \N$, the cocycle $c'$ belongs to $\mathcal O_o(g_no, r+ 16\alpha)$.
In particular, $c' \in \Lambda_{\rm ctg}(G,o, \alpha, r+16\alpha, L)$.

Let $n \in \N$.
Since $(o, g_no)$ has a contracting tail, there is a projection $p_n$ of $o$ on $\tau_n$ such that $\dist{p_n}{g_no} \geq L$.
According to <ref>, $\dist {q_n}{g_no} \leq r + 7\alpha$.
Our previous claim, combined with the triangle inequality, gives

\begin{equation*}
\dist {p_n}{q'_n}
\geq \dist {p_n}{g_no} - \dist{q'_n}{q_n} - \dist{q_n}{g_no}
\geq L - (r + 11\alpha)   > 4\alpha.
\end{equation*}

Hence by <ref>, we have $\gro o{c'}{q'_n} \leq 5\alpha$, which combined with  (<ref>) yields

\begin{equation*}
\gro o{c'}{g_no} \leq \gro o{c'}{q'_n} + \dist {q'_n}{q_n} + \dist {q_n}{g_no}  \leq r + 16\alpha.
\end{equation*}

This completes the proof of our second claim.

$\norm[\infty]{c -c'} \leq 20\alpha$.

Consider a closed ball $K$ of radius $R$ centered at $o$.
If $n \in \N$ is sufficiently large, then $\dist o{g_no} > R + r + 29\alpha$.
According to our previous claim $c$ and $c'$ both belong to $\mathcal O_o(g_no, r+ 16\alpha)$.
It follows from <ref> that $\norm[K]{c-c'} \leq 20\alpha$.
This estimate does not depend on $K$, whence the result.

In a CAT(-1) settings, shadows are known to provide a basis of open neighborhoods for the points in the boundary at infinity.
This is no more the case in our context.
However we can still approximate saturated subsets of the contracting limit set using shadows.
This is the purpose of the next lemma.

Let $\alpha \in \R_+^*$ and $r, L \in \R_+$ with $L > r + 13\alpha$.
Let $B$ be a saturated subset of $\Lambda_{\rm ctg}(G,o, \alpha, r, L)$ and $V$ an open subset of $\bar X$ containing $B$.
Let $b \in B$.
There exists $T \in \R_+$ such that for every $g \in \mathcal T(\alpha, L)$ with $\dist o{go} \geq T$, if $b$ belongs to $\mathcal O_o(go,r)$ then $\mathcal O_o(go,r) \subset V$.

Assume on the contrary that our statement fails.
We can find a sequence of elements $g_n \in \mathcal T(\alpha, L)$ such that $\dist o{g_no}$ diverges to infinity, $b$ belongs to $\mathcal O_o(g_no, r)$ and $\mathcal O_o(g_no, r) \setminus V$ is non-empty.
For every $n \in \N$, we write $c_n$ for a cocycle in $\mathcal O_o(g_no, r) \setminus V$.
Up to passing to a subsequence, we can assume that $c_n$ converges to $c \in \bar X$.
As $V$ is open, $c \notin V$.
We claim that $\norm[\infty]{c - b} < \infty$.
Let $K$ be a closed ball of radius $R$ centered at $o$.
As $c_n$ converges to $c$, there is $N \in \N$, such that for every $n \geq N$, we have $\norm[K]{c_n - c} \leq 1$.
By construction $b$ and $c_n$ both belong to $\mathcal O_o(g_no,r)$.
According to <ref>, if $n$ is sufficiently large $\norm[K]{b - c_n} \leq 20\alpha$ so that $\norm[K]{b - c} \leq 20\alpha +1$.
This inequality holds for every compact subset $K \subset X$,  which completes the proof of our claim.
Since $B$ is saturated, $c \in B$.
It contradicts the fact that $B \subset V$.

§.§ Measure of the contracting limit set

From now on, we assume that  $G$ is not virtually cyclic and contains a contracting element.
According to <ref>, there are $(\epsilon, r_0) \in \R_+^* \times \R_+$ such that any $G$-invariant, $\omega_G$-conformal density $\nu = (\nu_x)$ satisfies the Shadow Principle with parameters $(\epsilon, r_0)$.
The goal of this section is to prove that if the action of $G$ on $X$ is divergent, then any such density gives full measure to the contracting limit set.
The proof is an application of the Kochen-Stone theorem, which generalizes the second Borel-Cantelli Lemma.

Let $(\Omega,\mu)$ be a probability space.
Let $(B_n)$ be a sequence of subsets of $\Omega$ such that

\begin{equation*}
\sum_{n \in \N} \mu(B_n) = \infty.
\end{equation*}

Assume that there exists $C \in \R_+^*$ such that for every $N \in \N$,

\begin{equation*}
\sum_{n_1 = 0}^N \sum_{n_2 = 0}^N \mu\left(B_{n_1} \cap B_{n_2} \right) \leq C \left( \sum_{n = 0}^N \mu\left(B_n\right) \right)^2.
\end{equation*}

\begin{equation*}
\mu\left( \bigcap_{N \in \N} \bigcup_{n \geq N} B_n\right) \geq \frac 1C.
\end{equation*}

Recall that for every $\alpha, r, \ell,  L \in \R_+$, the set $A_\ell(\alpha, r,L)$ is defined by

\begin{equation*}
A_\ell(\alpha, r,L) = \bigcup_{g \in S(\ell, r) \cap  \mathcal T(\alpha, L)} \mathcal O_o(go, r).
\end{equation*}

These are the sets with which we will apply the Kochen-Stone theorem.
The aim of the next lemmas is to make sure that the hypotheses of <ref> are satisfied.

Let $a \in \R_+^*$.
There are $\alpha, r_1, C_1\in \R_+^*$, such that for every $r \geq r_1$ and $L > r + 13\alpha$, the following holds.
Let $\nu = (\nu_x)$ be a $G$-invariant, $\omega_G$-conformal density.
For every integer $N \in \N$,

\begin{equation*}
\sum_{\substack{ g \in G \\ \dist o{go} \leq Na - L}}e^{-\omega_G \dist o{go}}
\leq C_1e^{2\omega_GL}
\sum_{n = 0}^N \nu_o(B_n),
\end{equation*}

where $B_n = A_{na}(\alpha,r,L)$.

Recall that $(\epsilon, r_0)$ are the Shadow Principle parameters which have been fixed once and for all at the beginning of <ref>.
We denote by $\alpha, r_1, C \in \R_+$ the parameters given by <ref> applied with $\omega_G$, $a$, and $(\epsilon, r_0)$.
Fix $r \geq r_1$ and $L > r + 13\alpha$.
Let $\nu = (\nu_x)$ be a $G$-invariant, $\omega_G$-conformal density.
Recall that $(G, \nu)$ satisfies the Shadow Principle with parameters $(\epsilon, r_0)$.
It follows from <ref> that for every $\ell \in \R_+$,

\begin{equation*}
\sum_{g \in S(\ell, a)} e^{-\omega _G\dist o{go}}
\leq C e^{2\omega L}\nu_o \left( A_{\ell+L}(\alpha, r,L) \right).
\end{equation*}

Summing this identity we get for every $N \in \N$,

\begin{equation*}
\sum_{n = 0}^N \sum_{g \in S(na, a)} e^{-\omega _G\dist o{go}}
\leq C e^{2\omega L} \sum_{n = 0}^N  \nu_o \left( A_{na+L}(\alpha, r,L) \right).
\end{equation*}

Note that $A_{na+L}(\alpha, r,L)$ is covered by $A_{(m-1)a}(\alpha, r,L)$ and $A_{ma}(\alpha, r,L)$ where $m = n + \lceil L/a \rceil$.

\begin{equation*}
\sum_{\substack{ g \in G \\ \dist o{go} \leq (N+1)a}} e^{-\omega _G\dist o{go}}
\leq 2C e^{2\omega L} \sum_{n = 0}^{N +\lceil L/a \rceil} \nu_o \left( B_n \right),
\end{equation*}

whence the result.

Let $a, \alpha, r \in \R_+^*$.
There are $b, C_2 \in \R_+$ such that for every $L > r + 13\alpha$, the following holds.
Let $\nu = (\nu_x)$ be a $G$-invariant, $\omega_G$-conformal density.
For every $N \in \N$, we have

\begin{equation*}
\sum_{n_1 = 0}^N\sum_{n_2 = 0}^N \nu_o(B _{n_1}\cap B_{n_2})
\leq C_2 \left( \sum_{\substack{ g \in G \\ \dist o{go} \leq Na+ b}}e^{-\omega_G \dist o{go}}\right)^2,
\end{equation*}

where $B_n = A_{na}(\alpha,r,L)$.

Let $L > r + 13\alpha$.
Let $\nu = (\nu_x)$ be a $G$-invariant, $\omega_G$-conformal density.
Let $N \in \N$.
Observe first that

\begin{equation}
\label{eqn: borel-cantelli - upper hyp}
\begin{split}
\sum_{n_1 = 0}^N\sum_{n_2 = 0}^N \nu_o(B _{n_1}\cap B_{n_2})
& \leq 2 \sum_{n_1 = 0}^N\sum_{n_2 = n_1}^N \nu_o(B _{n_1}\cap B_{n_2}) \\
& \leq 2  \sum_{n_1 = 0}^N\sum_{n_2=0}^{N-n_1} \nu_o(B _{n_1}\cap B_{n_1+n_2}).
\end{split}
\end{equation}

Consider now $n_1, n_2 \in \N$ with $0 \leq n_1 \leq n_1 + n_2 \leq N$.
By definition, we have

\begin{equation*}
B_{n_1} \cap B_{n_1+n_2}
= \bigcup_{(g_1,g_2) \in U}
\mathcal O_o(g_1o,r) \cap \mathcal O_o(g_1g_2o,r),
\end{equation*}

where $U$ is the set of pairs $(g_1,g_2) \in G$ with the following properties:

$g_1, g_1g_2 \in \mathcal T(\alpha, L)$,

$g_1 \in S(n_1a,r)$ and $g_1g_2 \in S(n_1a + n_2a,r)$.

Let $(g_1,g_2) \in U$ for which $\mathcal O_o(g_1o,r) \cap \mathcal O_o(g_1g_2o,r)$ is non-empty.
According to <ref>, we have

\begin{equation*}
\dist o{g_2o} \leq \dist{g_1o}{g_1g_2o}
\leq \abs{\dist o{g_1g_2o} - \dist o{g_1o}} + 4r + 44\alpha.
\end{equation*}

In particular,

\begin{equation*}
\dist o{g_1g_2o} \geq \dist o{g_1o} + \dist o{g_2o} - 4r - 44\alpha.
\end{equation*}

Moreover, combined with  <ref>, it shows that $g_2 \in S(n_2a, 6r + 44\alpha)$.
Using <ref>, we estimate the measure of each shadow:

\begin{align*}
\nu_o\left( O_o(g_1o,r) \cap \mathcal O_o(g_1g_2o,r)\right)
& \leq \nu_o\left( \mathcal O_o(g_1g_2o,r)\right) \\
& \leq e^{2\omega_Gr}e^{-\omega_G\dist o{g_1g_2o}} \\
& \leq e^{\omega_G(6r+44\alpha)} e^{-\omega_G\left[\dist o{g_1o} + \dist o{g_2o}\right]}.
\end{align*}

\begin{equation*}
\nu_o\left(B_{n_1} \cap B_{n_1+n_2}\right)
\leq e^{\omega_G(6r+44\alpha)} \sum_{\substack {g_1 \in S(n_1a,r), \\  g_2 \in S(n_2a, 6r + 44\alpha)}} e^{-\omega_G\left[\dist o{g_1o} + \dist o{g_2o}\right]}.
\end{equation*}

Note that for every $d \in \R_+$, an element $g \in G$ belongs to at most $\lceil 2d/a\rceil$ spheres of the form $S(na,d)$ when $n$ runs over $\N$.
Summing the previous inequality over $n_1$ and $n_2$, and using (<ref>) we get

\begin{align*}
\sum_{n_1 = 0}^N\sum_{n_2 = 0}^N \nu_o(B _{n_1}\cap B_{n_2})
& \leq C_2\sum_{\substack {g_1,g_2 \in G \\ \dist o{g_1o},   \dist o{g_2o} \leq Na +b}} e^{-\omega_G\left[\dist o{g_1o} + \dist o{g_2o}\right]} \\
& \leq C_2 \left( \sum_{\substack{ g \in G \\ \dist o{go} \leq Na + b}}e^{-\omega_G \dist o{go}}\right)^2,
\end{align*}

\begin{equation*}
b = a + 6r + 44 \alpha,
\quad \text{and} \quad
C_2 = 2e^{\omega_G(6r+44\alpha)} \left\lceil  \frac {12r + 88\alpha}a\right\rceil ^2
\end{equation*}

only depends on $a$, $\alpha$ and $r$.

Assume that the action of $G$ on $X$ is divergent.
There exists $\alpha, r \in \R_+$ with the following property.
Let $\nu = (\nu_x)$ be a $G$-invariant, $\omega_G$-conformal density.
For every $L > r + 13\alpha$, we have

\begin{equation*}
\nu_0 \left( \Lambda_{\rm ctg}(G, o, \alpha, r, L) \right) > 0.
\end{equation*}

Fix $a \in \R_+^*$.
Let $\alpha, r_1, C_1$ be the parameters given by <ref>.
Fix $r \geq r_1$.
Let $b, C_2$ be the parameters given by <ref> applied with $a$, $\alpha$ and $r$.
Choose $L > r+13\alpha$ and set $a' = \max\{a, b, L\}$.
We write $C$ for the constant given by <ref> applied with $\omega_G$, $a'$ and $(\epsilon, r_0)$.

Let $\nu = (\nu_x)$ be a $G$-invariant, $\omega_G$-conformal density.
For simplicity; for every $n \in \N$,  we let $B_n  = A_{na}(r,\alpha, L)$.
Since the action of $G$ is divergent, <ref> tells us that

\begin{equation*}
\sum_{n \in \N} \nu_o(B_n) = \infty.
\end{equation*}

Recall that $(G,\nu)$ satisfies the Shadow Principle with parameters $(\epsilon, r_0)$.
By <ref>

\begin{equation*}
\sum_{g \in S(\ell, a')} e^{-\omega_G\dist o{go}} \leq C, \quad \forall \ell \in \R_+.
\end{equation*}

Since the Poincaré series of $G$ diverges at $s = \omega_G$, we deduce from  Lemmas <ref> and  <ref> that there exists $C' \in \R_+^*$, such that for every sufficiently large $N \in \N$,

\begin{equation*}
\sum_{n_1 = 0}^N \sum_{n_2 = 0}^N \nu_o\left(B_{n_1} \cap B_{n_2} \right)
\leq C' \left( \sum_{n = 0}^N \nu_o\left(B_n\right) \right)^2.
\end{equation*}

Applying <ref>, we observe that

\begin{equation*}
\nu_0 \left( \Lambda_{\rm ctg}(G, o, \alpha, r, L) \right)
= \nu_o\left( \bigcap_{N \in \N} \bigcup_{n \geq N} B_n\right) \geq \frac 1{C'} .\qedhere
\end{equation*}

Assume that the action of $G$ on $X$ is divergent.
There exist $\alpha, r \in \R_+^*$ with the following property.
If $\nu = (\nu_x)$ is a $G$-invariant, $\omega_G$-conformal density, then $\nu_o$ gives full measure to

\begin{equation*}
\Lambda_{\rm ctg} (G, o, \alpha, r) = \bigcap_{ L \in \R_+} \Lambda_{\rm ctg} (G, o, \alpha, r, L).
\end{equation*}

In particular, $\nu_o$ gives full measure to the contracting limit set $\Lambda_{\rm ctg} (G)$ and thus to the radial limit set $\Lambda_{\rm rad}(G)$.

The corollary is a reminiscence of the fact that contracting elements are “generic” in $G$, see Yang [52].

Let $\alpha, r$ be the parameters given by <ref>.
Let $\nu = (\nu_x)$ be a $G$-invariant, $\omega_G$-conformal density.
Let $L > r + 13\alpha$.
For simplicity we let

\begin{equation*}
B = G \Lambda_{\rm ctg} (G, o, \alpha, r, L).
\end{equation*}

We claim that $\nu_o$ gives full measure to $B$.
Assume on the contrary that the set $A = \bar X \setminus B$ has positive measure.
We define a new density $\nu^*  = (\nu^*_x)$ by

\begin{equation*}
\nu^*_x = \frac 1{\nu_o(A)}  \mathbb 1_A \nu_x.
\end{equation*}

Note that $\nu^*$ is $\omega_G$-conformal.
By construction $B$, and thus $A$, is $G$-invariant.
Hence $\nu^*$ is also $G$-invariant.
It follows from <ref>, that the $\nu^*_o$ gives positive measure to $B$, a contradiction.
According to <ref>, the set $B$ is contained in $\Lambda_{\rm ctg} (G, o, \alpha, r + 12\alpha, L- \alpha)$.
Hence the latter has full measure as well.
This facts holds for every $L > r + 13\alpha$.

\begin{equation*}
\nu_0 \left(\bigcap_{L \in \R_+} \Lambda_{\rm ctg}(G, o,\alpha, r+12\alpha, L) \right) = 1,
\end{equation*}

\begin{equation*}
\nu_0 \left( \Lambda_{\rm ctg}(G, o, \alpha, r + 12\alpha) \right) = 1. \qedhere
\end{equation*}

§.§ Passing to the reduced horoboundary

We now study the restriction to the reduced horoboundary of invariant conformal densities.
We still assume that $G$ is not virtually cyclic and contains a contracting element.
The Shadow Principle parameters $(\epsilon, r_0) \in \R_+^* \times \R_+$ are as in the previous section.

Assume that the action of $G$ on $X$ is divergent.
There is $C \in \R_+^*$ with the following property.
Let $\nu = (\nu_x)$ and $\nu' = (\nu'_x)$ be two $G$-invariant, $\omega_G$-conformal densities.
Denote by $\mu = (\mu_x)$ and $\mu' = (\mu'_x)$ their respective restrictions to the reduced horocompactification $(\bar X, \mathfrak R)$.
Then $\mu_o \leq C\mu'_o$.

Let $\alpha, r \in \R^*_+$ be the parameters given by <ref>.
Without loss of generality we can assume that $r \geq r_0$.
Set $r' = r + 16\alpha$.
For simplicity we write $\Lambda$ and $\Lambda'$ for the set  $\Lambda_{\rm ctg} (G, o, \alpha, r)$ and $\Lambda_{\rm ctg} (G, o, \alpha, r')$ respectively.
Let $\nu = (\nu_x)$ and $\nu' = (\nu'_x)$ be as in the statement.
Let $B \subset \partial X$ be a saturated Borel subset.
We want to compare $\mu_o(B) = \nu_o(B)$ and $\mu'_o(B) = \nu'_o(B)$.
According to <ref>, both $\nu_o$ and $\nu'_o$ give full measures to $\Lambda$.
In addition, the saturated set $(B \cap \Lambda)^+$ is contained in $\Lambda'$, see <ref>
(note that we do not claim that $(B \cap \Lambda)^+$ is measurable; we will use it only as an auxiliary tool to describe sets and will never compute its measure).
Let $V$ be an open subset of $\bar X$ containing $B$.
Choose $L  > r' + 16\alpha$.
Using <ref> with $(B \cap \Lambda)^+$, we build a subset $S \subset \mathcal T(\alpha, L)$ such that

\begin{equation*}
B \cap \Lambda \subset (B \cap \Lambda)^+ \subset \bigcup_{g \in S}\mathcal O_o(go,r') \subset V.
\end{equation*}

According to <ref>, there is a subset $S^*$ of $S$ such that

* the collection $\left( \mathcal O_o(go,r')\right)_{g \in S^*}$ is pairwise disjoint, and

* $B \cap \Lambda$ is covered by $\left( \mathcal O_o(go,r'+42\alpha)\right)_{g \in S^*}$

Recall that $(G,\nu')$ satisfies the Shadow Principle.
Since $\Lambda$ has full measure, we get

\begin{align*}
\nu_o(B)
\leq \nu_o(B \cap \Lambda)
\leq \sum_{g \in S^*} \nu_o(\mathcal O_o(go,r'+42\alpha))
& \leq e^{2\omega_G(r'+42\alpha)} \sum_{g \in S^*} e^{-\omega_G \dist o{go}} \\
& \leq C \sum_{g \in S^*} \nu'_o(\mathcal O_o(go,r)),
\end{align*}

where $C = e^{2\omega_G(r'+42\alpha)}/\epsilon$ does not depend on $\nu$ and $\nu'$.
Hence $\nu_o(B) \leq C \nu'_o(V)$.
This inequality holds for every open subset $V$ containing $B$, thus $\nu_o(B) \leq C\nu'_o(B)$.

Let $\nu = (\nu_x)$ be a $G$-invariant, $\omega_G$-conformal density and $\mu = (\mu_x)$ its restriction to the reduced horocompactification $(\bar X, \mathfrak R)$.
Assume that the action of $G$ on $X$ is divergent.

$\mu_o$ is supported on the contracting limit set

$\mu_o$ is ergodic.

$\mu_o$ is non-atomic, in particular, no equivalence class for $\sim$ has positive measure.

$\mu$ is a $G$-invariant, $\omega_G$-quasi-conformal density.

$\mu$ is almost-unique, that is there is $C \in \R_+^*$ such that  if $\mu' = (\mu'_x)$ is the restriction to the reduced horocompactification of another $G$-invariant, $\omega_G$-conformal density, then for every $x \in X$, we have $\mu'_x \leq C \mu_x$.

Let $\alpha, r \in \R^*_+$ be the parameters given by <ref> and $C \in \R_+$  the one given by <ref>.
Let $\nu = (\nu_x)$ be a $G$-invariant, $\omega_G$-conformal density and $\mu = (\mu_x)$ its restriction to the reduced horocompactifiction $(\bar X, \mathfrak R)$.
It follows from <ref> that $\mu_o$ gives full measure to the contracting limit set.
Let us prove the ergodicity of $\mu_o$.
Let $B$ be a $G$-invariant saturated Borel subset such that $\mu_o(B) > 0$.
Consider the density $\nu^* = (\nu^*_x)$ defined by

\begin{equation*}
\nu^*_x = \frac 1{\nu_o(B)} \mathbb 1_B \nu_x.
\end{equation*}

Since $B$ is $G$-invariant, $\nu^*$ is a $G$-invariant, $\omega_G$-conformal density.
Denote by $\mu^* = (\mu^*_x)$ its restriction to the reduced horocompactifiction.
It follows from <ref> that $\mu_o \leq C \mu^*_o$.
Consequently $\mu_o(\bar X \setminus B) = 0$, that is $\mu_o(B) = 1$.

We now focus on non-atomicity.
Assume on the contrary that $B \in \mathfrak R$ is an atom of $\mu_o$.
For every $g \in G$, we write $\mathcal O^+_o(go,r)$ for the saturation of the shadow $\mathcal O_o(go,r)$.
Note that $\mathcal O^+_o(go,r)$ is mesurable (<ref>).
Consequently the measure of $B \cap \mathcal O^+_o(go,r)$ is either zero or equals $\mu_o(B)$.
By <ref> $\mu_o$ only charges the set $\Lambda = \Lambda_{\rm ctg}(G, o, \alpha,r)$, hence $\Lambda_{\rm rad}(G,o,r)$.
Consequently, for every $n \in \N$, there is $g_n \in G$, with $\dist o{g_no} \geq n$ such that

\begin{equation*}
\mu_o \left(\mathcal O^+_o(g_no,r) \right)