# Patterson-Sullivan measures for relatively Anosov groups

Richard Canary  University of Michigan  ,  Tengren Zhang  National University
of Singapore  and  Andrew Zimmer  University of Wisconsin-Madison

###### Abstract.

We establish existence, uniqueness and ergodicity results for Patterson-
Sullivan measures for relatively Anosov groups. As applications we obtain an
entropy gap theorem and a strict concavity result for entropies associated to
linear functionals.

Canary was partially supported by grant DMS-1906441 from the National Science
Foundation. Zhang was partially supported by the NUS-MOE grant
A-8000458-00-00. Zimmer was partially supported by a Sloan research fellowship
and grants DMS-2105580 and DMS-2104381 from the National Science Foundation.

###### Contents

  1. 1 Introduction
  2. 2 Semisimple Lie groups
  3. 3 Relatively hyperbolic groups
  4. 4 Discrete subgroups of semisimple Lie groups
  5. 5 Multiplicative estimates
  6. 6 Resolution of singularities
  7. 7 Entropy gap for peripheral subgroups
  8. 8 Divergence of the Poincaré series at its critical exponent
  9. 9 Relatively quasiconvex subgroups
  10. 10 Characterizing linear functions with finite critical exponent
  11. A Proof of Proposition 2.3
  12. B Proof of Theorem 4.4

##  1\. Introduction

Anosov subgroups and relatively Anosov subgroups of semisimple Lie groups are
respectively natural generalizations of convex cocompact subgroups and
geometrically finite subgroups of rank one semisimple Lie groups to the higher
rank setting. Patterson-Sullivan measures for Anosov subgroups have been
extensively studied, see [18, 36, 12, 31]. They have also been studied for
relatively Anosov subgroups of the projective general linear group which
preserve a properly convex domain, see [4, 5, 9, 40, 10]. More generally,
Patterson-Sullivan measures can be constructed for transverse subgroups, of
which Anosov subgroups, relatively Anosov subgroups, and discrete subgroups of
rank one semisimple Lie groups are examples, see [15]. For a more detailed
historical discussion of Patterson-Sullivan measures for discrete subgroups of
semisimple Lie groups see [15, Sec. 1.1].

In this paper we study Patterson-Sullivan measures for relatively Anosov
subgroups of semisimple Lie groups. The key new result here is that the
Poincaré series associated to such a Patterson-Sullivan measure always
diverges at its critical exponent if this critical exponent is finite. As a
consequence, we establish uniqueness and ergodicity results for such
Patterson-Sullivan measures. We then derive an entropy gap theorem and a
strict concavity result for the entropy.

Delaying precise definitions until Sections 2, 3 and 4, we informally
introduce the notation necessary to state our main results. For the entire
paper, $\mathsf{G}$ will be a connected semisimple Lie group without compact
factors and with finite center. We fix a Cartan decomposition
$\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ of the Lie algebra of $\mathsf{G}$, a
Cartan subspace ${\mathfrak{a}}\subset{\mathfrak{p}}$, and a Weyl chamber
$\mathfrak{a}^{+}\subset\mathfrak{a}$. Let $\Delta\subset{\mathfrak{a}}^{*}$
be the corresponding system of simple restricted roots, and let
$\kappa:\mathsf{G}\to{\mathfrak{a}}^{+}$ denote the associated Cartan
projection. Given a subset $\theta\subset\Delta$, we let
$\operatorname{\mathsf{P}}_{\theta}\subset\mathsf{G}$ denote the associated
parabolic subgroup and let
$\operatorname{\mathcal{F}}_{\theta}=\mathsf{G}/\operatorname{\mathsf{P}}_{\theta}$
denote the associated flag manifold. We will always assume that $\theta$ is
symmetric.

Suppose $\Gamma\subset\mathsf{G}$ is a non-elementary discrete subgroup which,
as an abstract group, is relatively hyperbolic with respect to a finite
collection $\mathcal{P}$ of subgroups in $\Gamma$. Let
$\partial(\Gamma,\mathcal{P})$ denote its associated Bowditch boundary. Then
$\Gamma$ is _$\operatorname{\mathsf{P}}_{\theta}$ -Anosov relative to
$\mathcal{P}$_ if $\alpha\circ\kappa$ is proper on $\Gamma$ for all
$\alpha\in\theta$, its limit set $\Lambda_{\theta}(\Gamma)$ in
$\operatorname{\mathsf{P}}_{\theta}$ consists of mutually transverse flags and
there exists a continuous $\Gamma$-equivariant map

$\xi:\partial(\Gamma,\mathcal{P})\rightarrow\operatorname{\mathcal{F}}_{\theta}$

which is a homeomorphism onto $\Lambda_{\theta}(\Gamma)$.

The action of $\mathsf{G}$ on $\operatorname{\mathcal{F}}_{\theta}$ preserves
a vector valued cocycle
$B_{\theta}:\mathsf{G}\times\operatorname{\mathcal{F}}\rightarrow{\mathfrak{a}}_{\theta}$,
called the _partial Iwasawa cocycle_ , whose image lies in a subspace
${\mathfrak{a}}_{\theta}\subset{\mathfrak{a}}$ associated to $\theta$. If
$\phi\in{\mathfrak{a}}_{\theta}^{*}$, then the composition $\phi\circ
B_{\theta}$ is a real valued cocycle, which allows us to define Patterson-
Sullivan measures associated to such $\phi$. More precisely, if
$\phi\in{\mathfrak{a}}_{\theta}^{*}$ and $\Gamma\subset\mathsf{G}$ is
$\operatorname{\mathsf{P}}_{\theta}$-Anosov relative to $\mathcal{P}$, then a
_$\phi$ -Patterson-Sullivan measure for $\Gamma$ of dimension $\beta$_ is a
Borel probability measure $\mu$ on $\operatorname{\mathcal{F}}_{\theta}$ where

  1. (1)

${\rm supp}(\mu)\subset\Lambda_{\theta}(\Gamma)$,

  2. (2)

for every $\gamma\in\Gamma$ the measures $\gamma_{*}\mu$, $\mu$ are absolutely
continuous and

$\frac{d\gamma_{*}\mu}{d\mu}(F)=e^{-\beta\phi(B_{\theta}(\gamma^{-1},F))}$

for $\mu$-almost every $F\in\operatorname{\mathcal{F}}_{\theta}$.

In the case when $\mathsf{G}=\mathsf{SO}_{0}(d,1)$ is the group of
orientation-preserving isometries of real hyperbolic $d$-space
$\operatorname{\mathbb{H}}^{d}$, there is a single simple restricted root
$\Delta=\\{\alpha\\}$ and $\operatorname{\mathcal{F}}_{\alpha}$ naturally
identifies with the geodesic boundary of
$\operatorname{\mathbb{H}}^{d}_{\operatorname{\mathbb{R}}}$. Further (up to
scaling), $\alpha\circ B_{\alpha}$ identifies with the usual Busemann cocycle.
Since a discrete subgroup $\Gamma$ of $\mathsf{SO}_{0}(d,1)$ is relatively
$\operatorname{\mathsf{P}}_{\alpha}$-Anosov if and only if it is geometrically
finite, the above definition encompasses the classical notion of Patterson-
Sullivan measures for geometrically finite Kleinian groups.

As in the classical theory, there is an associated Poincaré series and
critical exponent. Let
$\kappa_{\theta}:\mathsf{G}\rightarrow{\mathfrak{a}}_{\theta}$ denote the
_partial Cartan projection_ defined in Section 2. Then given
$\phi\in{\mathfrak{a}}_{\theta}^{*}$, $s>0$ and a discrete group
$\Gamma\subset\mathsf{G}$, the _$\phi$ -Poincaré series_ is

$Q_{\Gamma}^{\phi}(s):=\sum_{\gamma\in\Gamma}e^{-s\phi(\kappa_{\theta}(\gamma))}.$

The _$\phi$ -critical exponent_, denoted
$\delta^{\phi}(\Gamma)\in[0,+\infty]$, is the critical exponent of the above
series, that is $Q_{\Gamma}^{\phi}(s)$ converges when
$s>\delta^{\phi}(\Gamma)$ and diverges when $0<s<\delta^{\phi}(\Gamma)$. Given
$\phi\in{\mathfrak{a}}_{\theta}^{*}$ let
$\bar{\phi}\in{\mathfrak{a}}_{\theta}^{*}$ be the unique functional where
$\bar{\phi}(\kappa_{\theta}(g))=\phi(\kappa_{\theta}(g^{-1}))$ for all
$g\in\mathsf{G}$. Notice that $Q^{\phi}_{\Gamma}=Q^{\bar{\phi}}_{\Gamma}$ and
so $\delta^{\phi}(\Gamma)=\delta^{\bar{\phi}}(\Gamma)$.

The main result of this paper is that if $\Gamma$ is relatively Anosov and its
$\phi$-critical exponent is finite, then its $\phi$-Poincaré series diverges
at its critical exponent.

######  Theorem 1.1 (Theorem 8.1).

If $\Gamma\subset\mathsf{G}$ is a $\operatorname{\mathsf{P}}_{\theta}$-Anosov
subgroup relative to $\mathcal{P}$, $\phi\in{\mathfrak{a}}_{\theta}^{*}$ and
$\delta^{\phi}(\Gamma)<+\infty$, then $Q_{\Gamma}^{\phi}$ diverges at its
critical exponent.

Later, we will discuss some important consequences of Theorem 1.1, and also
provide an outline of its proof (in Sections 1.1 and 1.2 respectively). Notice
that Theorem 1.1 fails in the setting of transverse groups, since the Poincaré
series of any finitely generated, geometrically infinite discrete subgroup of
$\mathsf{SO}(3,1)$ whose limit set is not all of $\partial\mathbb{H}^{3}$
converges at its critical exponent (see [13, Cor. 4.2] and [38, Thm, 2.17,
Cor. 2.18]).

We also provide a characterization of the functionals with finite critical
exponent, surprisingly the only requirement is that
$\phi(\kappa_{\theta}(\cdot))$ converges to infinity along escaping sequences
in the group. We also show that if $\operatorname{\mathsf{P}}_{\theta}$
contains no simple factors of $\mathsf{G}$, then
$\phi(\kappa_{\theta}(\cdot))$ has linear lower and upper bounds in terms of
the distance ${\rm d}_{M}$ on the Riemannian symmetric space $M$ associated to
$\mathsf{G}$.

######  Theorem 1.2 (see Section 10).

Suppose $\Gamma\subset\mathsf{G}$ is a
$\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative to $\mathcal{P}$
and $\phi\in\mathfrak{a}_{\theta}^{*}$. The following are equivalent:

  1. (1)

$\lim_{n\rightarrow\infty}\phi(\kappa_{\theta}(\gamma_{n}))=+\infty$ for every
sequence of distinct elements $\\{\gamma_{n}\\}$ in $\Gamma$.

  2. (2)

$\delta^{\phi}(\Gamma)<+\infty$.

Moreover, if $\operatorname{\mathsf{P}}_{\theta}$ contains no simple factors
of $\mathsf{G}$, then the above conditions are equivalent to:

  1. (3)

If $x_{0}\in M$ , there exist constants $c\geq 1$ and $C>0$ such that

$\frac{1}{c}{\rm
d}_{M}(\gamma(x_{0}),x_{0})-C\leq\phi(\kappa_{\theta}(\gamma))\leq c{\rm
d}_{M}(\gamma(x_{0}),x_{0})+C$

for all $\gamma\in\Gamma$.

We will observe later, see Section 1.3, that in Theorem 1.2 , Condition (3) is
not equivalent to Conditions (1) and (2) without the assumption that
$\operatorname{\mathsf{P}}_{\theta}$ contains no simple factors of
$\mathsf{G}$. Also, in the spirit of Sambarino’s analogous result for Anosov
groups [36], we observe that Theorem 1.2 implies that
$\phi\in\mathfrak{a}_{\theta}^{*}$ has finite critical exponent if and only if
$\phi$ is positive on the $\theta$-Benoist limit cone, see Section 10.

###  1.1. Consequences of Theorem 1.1

We now recall some results from our previous work [15] from which we can
derive several consequences of Theorem 1.1.

A discrete subgroup $\Gamma\subset\mathsf{G}$ is
_$\operatorname{\mathsf{P}}_{\theta}$ -transverse_ if $\alpha\circ\kappa$ is
proper on $\Gamma$ for all $\alpha\in\theta$ and its limit set
$\Lambda_{\theta}(\Gamma)$ in $\operatorname{\mathsf{P}}_{\theta}$ consists of
mutually transverse flags. A $\operatorname{\mathsf{P}}_{\theta}$-transverse
subgroup acts on this limit set as a convergence group, so one can define the
set of conical limit points $\Lambda^{\rm
con}_{\theta}(\Gamma)\subset\Lambda_{\theta}(\Gamma)$ using the standard
convergence group action definition. In this setting, we established in [15]
the following analogue of the Hopf-Tsuji-Sullivan dichotomy.

######  Theorem 1.3 ([15, Thm 1.4]).

Suppose $\Gamma\subset\mathsf{G}$ is a non-elementary
$\operatorname{\mathsf{P}}_{\theta}$-transverse subgroup,
$\phi\in\mathfrak{a}^{*}_{\theta}$ and
$\delta:=\delta^{\phi}(\Gamma)<+\infty$.

  * •

If $Q_{\Gamma}^{\phi}(\delta)=+\infty$, then there exists a unique
$\phi$-Patterson-Sullivan measure $\mu_{\phi}$ for $\Gamma$ of dimension
$\delta$ and there exists a unique $\bar{\phi}$-Patterson-Sullivan measure
$\mu_{\bar{\phi}}$ for $\Gamma$ of dimension $\delta$. Moreover:

    1. (1)

$\mu_{\phi}(\Lambda_{\theta}^{\rm
con}(\Gamma))=1=\mu_{\bar{\phi}}(\Lambda_{\theta}^{\rm con}(\Gamma))$.

    2. (2)

The action of $\Gamma$ on $(\Lambda_{\theta}(\Gamma),\mu_{\phi})$ and
$(\Lambda_{\theta}(\Gamma),\mu_{\bar{\phi}})$ is ergodic.

    3. (3)

The action of $\Gamma$ on
$(\Lambda_{\theta}(\Gamma)^{2},\mu_{\bar{\phi}}\otimes\mu_{\phi})$ is ergodic.

  * •

If $Q_{\Gamma}^{\phi}(\delta)<+\infty$, then $\mu(\Lambda_{\theta}^{\rm
con}(\Gamma))=0$ for any $\phi$-Patterson-Sullivan measure $\mu$ for $\Gamma$.

If $\Gamma$ is a $\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative
to $\mathcal{P}$, then $\Gamma$ is
$\operatorname{\mathsf{P}}_{\theta}$-transverse. Thus, we obtain, as a
consequence of Theorem 1.1 and Theorem 1.3, ergodicity and uniqueness results
for their Patterson-Sullivan measures. These generalize results earlier
obtained for Anosov groups (see [18, 36, 12, 31]).

######  Corollary 1.4.

Suppose $\Gamma\subset\mathsf{G}$ is a
$\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative to
$\mathcal{P}$, $\phi\in{\mathfrak{a}}_{\theta}^{*}$ and
$\delta:=\delta^{\phi}(\Gamma)<+\infty$. Then:

  1. (1)

There is a unique $\phi$-Patterson-Sullivan measure $\mu_{\phi}$ for $\Gamma$
of dimension $\delta$ and a unique $\bar{\phi}$-Patterson-Sullivan measure
$\mu_{\bar{\phi}}$ for $\Gamma$ of dimension $\delta$.

  2. (2)

$\Gamma$ acts ergodically on $(\Lambda_{\theta}(\Gamma),\mu_{\phi})$ and
$(\Lambda_{\theta}(\Gamma),\mu_{\bar{\phi}})$.

  3. (3)

$\Gamma$ acts ergodically on
$(\Lambda_{\theta}(\Gamma)\times\Lambda_{\theta}(\Gamma),\mu_{\phi}\otimes\mu_{\bar{\phi}})$.

In [15], we also established a criterion for when the critical exponent of a
subgroup of a transverse group is strictly less than that of the entire group.

######  Theorem 1.5 ([15, Thm 1.2]).

Suppose $\Gamma\subset\mathsf{G}$ is a non-elementary
$\operatorname{\mathsf{P}}_{\theta}$-transverse subgroup,
$\phi\in\mathfrak{a}_{\theta}^{*}$ and $\delta^{\phi}(\Gamma)<+\infty$. If
$\Gamma_{0}$ is a subgroup of $\Gamma$ such that $Q_{\Gamma_{0}}^{\phi}$
diverges at its critical exponent and $\Lambda_{\theta}(\Gamma_{0})$ is a
proper subset of $\Lambda_{\theta}(\Gamma)$, then
$\delta^{\phi}(\Gamma)>\delta^{\phi}(\Gamma_{0}).$

Suppose that $\Gamma\subset\mathsf{G}$ is
$\operatorname{\mathsf{P}}_{\theta}$-Anosov relative to $\mathcal{P}$. A
subgroup $\Gamma_{0}\subset\Gamma$ is _relatively quasiconvex_ if its action
on its limit set $\Lambda(\Gamma_{0})\subset\partial(\Gamma,\mathcal{P})$ is
geometrically finite. We prove that non-elementary, relatively quasiconvex
subgroups of $\Gamma$ are themselves
$\operatorname{\mathsf{P}}_{\theta}$-Anosov relative to $\mathcal{P}$ (see
Proposition 9.4), and that the limit set of any infinite index quasiconvex
subgroup of $\Gamma$ is a closed, proper subset of the limit set of $\Gamma$
(see Lemma 9.3). Combining these facts with Theorems 1.1 and 1.5, we may then
prove the following result for relatively quasiconvex subgroups of infinite
index.

######  Corollary 1.6 (Corollary 9.1).

Suppose $\Gamma\subset\mathsf{G}$ is a
$\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative to
$\mathcal{P}$, and $\phi\in{\mathfrak{a}}_{\theta}^{*}$ such that
$\delta^{\phi}(\Gamma)<+\infty$. If $\Gamma_{0}$ is infinite index relatively
quasiconvex subgroup of $(\Gamma,\mathcal{P})$, then

$\delta^{\phi}(\Gamma)>\delta^{\phi}(\Gamma_{0}).$

Finally, in [15], we established that the critical exponent is strictly
concave on the space of linear functionals which diverge at their critical
exponent, except when there is agreement of length functionals. More
precisely, given $\phi\in{\mathfrak{a}}_{\theta}^{*}$ and $g\in\mathsf{G}$,
the _$\phi$ -length_ of $g$ is

$\ell^{\phi}(g):=\lim_{n\rightarrow\infty}\frac{1}{n}\phi(\kappa_{\theta}(g^{n})).$

######  Theorem 1.7 ([15, Thm. 1.5]).

Suppose $\Gamma$ is a non-elementary $\mathsf{P}_{\theta}$-transverse subgroup
of $\mathsf{G}$ and $\phi_{1},\phi_{2}\in\mathfrak{a}_{\theta}^{*}$ satisfy
$\delta^{\phi_{1}}(\Gamma)=\delta^{\phi_{2}}(\Gamma)=1$. If
$\phi=\lambda\phi_{1}+(1-\lambda)\phi_{2}$ for some $\lambda\in(0,1)$, then

$\delta:=\delta^{\phi}(\Gamma)\leq 1.$

Moreover, if $Q_{\Gamma}^{\phi}(\delta)=+\infty$, then equality occurs if and
only if $\ell^{\phi_{1}}(\gamma)=\ell^{\phi_{2}}(\gamma)$ for all
$\gamma\in\Gamma$.

Together, Theorem 1.1 and Theorem 1.7 give the following result.

######  Corollary 1.8.

Suppose $\Gamma\subset\mathsf{G}$ is a
$\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative to $\mathcal{P}$
and $\phi_{1},\phi_{2}\in{\mathfrak{a}}_{\theta}^{*}$ satisfy
$\delta^{\phi_{1}}(\Gamma)=\delta^{\phi_{2}}(\Gamma)=1$. If
$\phi=\lambda\phi_{1}+(1-\lambda)\phi_{2}$ for some $\lambda\in(0,1)$, then

$\delta^{\phi}(\Gamma)\leq 1$

with equality if and only if $\ell^{\phi_{1}}(\gamma)=\ell^{\phi_{2}}(\gamma)$
for all $\gamma\in\Gamma$.

In the Zariski dense case, see [15, Cor 1.6], a theorem of Benoist [2] implies
that distinct length functions cannot agree.

######  Corollary 1.9.

Suppose $\Gamma\subset\mathsf{G}$ is a Zariski dense
$\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative to
$\mathcal{P}$, and $\phi_{1},\phi_{2}\in{\mathfrak{a}}_{\theta}^{*}$ are
distinct and satisfy $\delta^{\phi_{1}}(\Gamma)=\delta^{\phi_{2}}(\Gamma)=1$.
If $\phi=\lambda\phi_{1}+(1-\lambda)\phi_{2}$ for some $\lambda\in(0,1)$, then
$\delta^{\phi}(\Gamma)<1$.

###  1.2. Outline of the proof of Theorem 1.1

The strategy of the proof of Theorem 1.1 is inspired by earlier work of
Blayac-Zhu [5] in the context of relatively hyperbolic groups preserving
properly convex domains and Patterson-Sullivan measures defined using the
Busemann functions associated to the Hilbert distance. The key technical
result needed to prove Theorem 1.1 is that if $\Gamma$ is
$\operatorname{\mathsf{P}}_{\theta}$-Anosov relative to $\mathcal{P}$ and
$\delta^{\phi}(\Gamma)<+\infty$, then the $\phi$-Poincaré series of any
peripheral subgroup diverges at its critical exponent.

######  Theorem 1.10 (Theorem 7.1).

Suppose $\Gamma\subset\mathsf{G}$ is a
$\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative to
$\mathcal{P}$, $\phi\in{\mathfrak{a}}_{\theta}^{*}$ and
$\delta^{\phi}(\Gamma)<+\infty$. If $P\in\mathcal{P}$, then $Q_{P}^{\phi}$
diverges at its critical exponent.

Together, Theorems 1.5 and 1.10 imply that
$\delta^{\phi}(P)<\delta^{\phi}(\Gamma)$ for all $P\in\mathcal{P}$. We may
then adapt arguments of Dal’bo-Otal-Piegné [17] to our setting to conclude
Theorem 1.1.

The proof of Theorem 1.10 makes use of Hironaka’s famous result on the
resolution of singularities. More precisely, in Section 7, we relate the
Poincaré series $Q_{P}^{\phi}(s)$ associated to a peripheral subgroup $P$ to
an integral of the form

$\int_{\operatorname{\mathbb{R}}^{n}}\big{(}R_{1}^{\ell_{1}}\cdots
R_{m}^{\ell_{m}}\big{)}^{-s}d\lambda$ (1)

where $\lambda$ is the Lebesgue measure,
$R_{1},\dots,R_{m}:\operatorname{\mathbb{R}}^{n}\rightarrow\operatorname{\mathbb{R}}$
are positive rational functions that are defined everywhere and
$\ell_{1},\dots,\ell_{m}$ are real numbers such that $R_{1}^{\ell_{1}}\cdots
R_{m}^{\ell_{m}}$ is a proper function. In Section 6, we use Hironaka’s
resolution of singularities to show that the integral in Equation (1) diverges
at its critical exponent. This in turn implies that the Poincaré series
associated to any peripheral subgroup diverges at its critical exponent.

In Blayac and Zhu’s setting, the integral in Equation (1) involves a single
rational function (i.e. $m=1$) and one can deduce that it diverges at its
critical exponent from a result of Benoist-Oh [3, Prop. 7.2] (see the proof of
[5, Lem. 8.9]). The case when $m>1$ is more technical and our use of
Hironaka’s resolution of singularities to understand the integral at infinity
is motivated by Benoist and Oh’s arguments.

###  1.3. Conditions (1) and (2) are not equivalent to (3) in Theorem 1.2

We now give an example which shows that Condition (3) in Theorem 1.2 may fail
to be equivalent to Conditions (1) and (2) when
$\operatorname{\mathsf{P}}_{\theta}$ contains a simple factor of $\mathsf{G}$.

######  Example 1.11.

Let
$\mathsf{G}=\mathsf{PSL}(2,\operatorname{\mathbb{R}})\times\mathsf{PSL}(2,\operatorname{\mathbb{R}})$.
Note that

$\mathfrak{a}:=\left\\{{\rm
diag}(a_{1},\dots,a_{4}):a_{1}+a_{2}=0=a_{3}+a_{4}\right\\}$

is a Cartan subspace of $\mathsf{G}$ and
$\Delta:=\\{\alpha_{1},\alpha_{1}^{\prime}\\}\subset\mathfrak{a}^{*}$ is a
system of simple restricted roots, where

$\alpha_{1}({\rm
diag}(a_{1},\dots,a_{4}))=a_{1}-a_{2}\quad\text{and}\quad\alpha_{1}^{\prime}({\rm
diag}(a_{1},\dots,a_{4}))=a_{3}-a_{4}.$

Set $\theta:=\\{\alpha_{1}\\}$, and note that

${\mathfrak{a}}_{\theta}=\\{{\rm
diag}(a,-a,0,0):a\in\operatorname{\mathbb{R}}\\}$

and

$\kappa_{\theta}((g,h))=(\log\sigma_{1}(g),\log\sigma_{2}(g),0,0)$

where $\sigma_{1}(g)\geq\sigma_{2}(g)$ are the singular values of some (any)
lift of $g$ to $\operatorname{\mathsf{SL}}(2,\operatorname{\mathbb{R}})$.
Also,

$\operatorname{\mathsf{P}}_{\theta}=\operatorname{\mathsf{P}}_{1}\times\mathsf{PSL}(2,\operatorname{\mathbb{R}})$

where
$\operatorname{\mathsf{P}}_{1}\subset\mathsf{PSL}(2,\operatorname{\mathbb{R}})$
stabilizes the line $\operatorname{\mathbb{R}}\cdot\begin{pmatrix}1\\\
0\end{pmatrix}$. In particular, $\operatorname{\mathsf{P}}_{\theta}$ contains
a simple factor of $\mathsf{G}$.

Let $S$ be a thrice punctured sphere equipped with a complete hyperbolic
structure such that one of its ends is a cusp while the other two ends are
hyperbolic funnels. Then there is a discrete subgroup
$\Gamma_{0}\subset\mathsf{PSL}(2,\operatorname{\mathbb{R}})$ such that
$S=\Gamma_{0}\backslash\operatorname{\mathbb{H}}^{2}$. Let $h$ be a hyperbolic
element in $\mathsf{PSL}(2,\operatorname{\mathbb{R}})$ whose conjugacy class
$[h]$ corresponds to an oriented geodesic going once around one of the
hyperbolic funnels of $S$, and let $u$ be a unipotent element in
$\mathsf{PSL}(2,\operatorname{\mathbb{R}})$ whose conjugacy class $[u]$
corresponds to an oriented closed curve going once around the cusp of $S$. We
may assume that $\Gamma_{0}=\left\langle u\right\rangle*\left\langle
h\right\rangle$. Then $\Gamma_{0}$ is $\operatorname{\mathsf{P}}_{1}$-Anosov
relative to $\mathcal{P}_{0}:=\\{\left\langle u\right\rangle\\}$. Thus, if we
set $\Gamma:=\left\langle\gamma,\eta\right\rangle\subset\mathsf{G}$, where

$\gamma:=(u,h)\in\mathsf{G}\quad\text{ and
}\quad\eta:=(h,\operatorname{\mathrm{id}})\in\mathsf{G},$

then $\Gamma$ is $\operatorname{\mathsf{P}}_{\theta}$-Anosov relative to
$\mathcal{P}:=\\{\left\langle\gamma\right\rangle\\}$. Furthermore,

$\lim_{n\rightarrow\infty}\alpha_{1}(\kappa_{\theta}(g_{n}))=+\infty$

for every sequence of distinct elements $\\{g_{n}\\}$ in $\Gamma$, so (1) in
Theorem 1.2 holds.

On the other hand, the Riemannian symmetric space $M$ associated to
$\mathsf{G}$ is the product

$M=\operatorname{\mathbb{H}}^{2}_{\operatorname{\mathbb{R}}}\times\operatorname{\mathbb{H}}^{2}_{\operatorname{\mathbb{R}}}$

where $\operatorname{\mathbb{H}}^{2}_{\operatorname{\mathbb{R}}}$ is the real
hyperbolic 2-space. Then for any $p_{0}=(x_{0},y_{0})\in M$ we have

$\liminf_{n\rightarrow\infty}\frac{1}{n}\log{\rm
d}_{M}(\gamma^{n}(p_{0}),p_{0})=\liminf_{n\rightarrow\infty}\frac{1}{n}\log{\rm
d}_{\operatorname{\mathbb{H}}_{\operatorname{\mathbb{R}}}^{2}}(h^{n}(y_{0}),y_{0})>0$

since $h$ is hyperbolic and $u$ is unipotent. However,

$\limsup_{n\rightarrow\infty}\frac{1}{n}\log\phi(\kappa_{\theta}(\gamma^{n}))=\limsup_{n\rightarrow\infty}\frac{1}{n}\log\frac{\sigma_{1}(u^{n})}{\sigma_{2}(u^{n})}=0$

since $u$ is unipotent. So (3) in Theorem 1.2 does not hold.

##  2\. Semisimple Lie groups

In this expository section we introduce some of the notation used throughout
the paper. All the notation introduced is the same as in [15].

As in the introduction (and the rest of the paper), let $\mathsf{G}$ be a
connected semisimple Lie group without compact factors and with finite center.
Fix a Cartan decomposition

${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{p}}$

of the Lie algebra ${\mathfrak{g}}$ of $G$, a Cartan subspace
$\mathfrak{a}\subset\mathfrak{p}$, and a Weyl chamber by
${\mathfrak{a}}^{+}\subset{\mathfrak{a}}$. Let $\Delta\subset\mathfrak{a}^{*}$
be the system of simple restricted roots associated to $\mathfrak{a}^{+}$, and
let $\operatorname{\mathsf{K}}\subset\mathsf{G}$ denote the maximal compact
Lie subgroup whose Lie algebra is ${\mathfrak{k}}$.

###  2.1. Cartan projection

The _Cartan projection_ is the map
$\kappa:\mathsf{G}\rightarrow{\mathfrak{a}}^{+}$ with the defining property
that $\kappa(g)\in{\mathfrak{a}}^{+}$ is the unique element such that

$g=me^{\kappa(g)}\ell$

for some $m,\ell\in\operatorname{\mathsf{K}}$ (in general $m$ and $\ell$ are
not uniquely determined by $g$). Such a decomposition $g=me^{\kappa(g)}\ell$
is called a _$\mathsf{KAK}$ -decomposition_ of $g$, see [22, Chap. IX, Thm.
1.1].

There is a unique involutive linear map
$\iota:{\mathfrak{a}}\rightarrow{\mathfrak{a}}$, called the _opposition
involution_ , such that $\iota(-{\mathfrak{a}}^{+})={\mathfrak{a}}^{+}$ and

$\iota(\kappa(g))=\kappa(g^{-1})$

for all $g\in\mathsf{G}$. The dual of $\iota$ satisfies

$\iota^{*}(\Delta)=\Delta.$

As in the introduction we let $M$ denote the standard Riemannian symmetric
space associated to $\mathsf{G}$, and let ${\rm d}_{M}$ denote the distance
function on $M$ induced by the Riemannian metric. As a $\mathsf{G}$-space,
$M=\mathsf{G}/\operatorname{\mathsf{K}}$, and ${\rm d}_{M}$ satisfies

${\rm
d}_{M}(g\operatorname{\mathsf{K}},h\operatorname{\mathsf{K}})=\left\|\kappa(g^{-1}h)\right\|\quad\text{for
all}\quad g,h\in\mathsf{G},$

where $\left\|\cdot\right\|$ is the norm on $\mathfrak{a}$ induced by the
restriction of the Killing from to $\mathfrak{a}$. The Cartan projection
satisfies the following estimates.

######  Lemma 2.1 (see e.g. [29, Lem. 2.3]).

If $g,h\in\mathsf{G}$, then

$\left\|\kappa(gh)-\kappa(h)\right\|\leq\left\|\kappa(g)\right\|\quad\text{and}\quad\left\|\kappa(gh)-\kappa(g)\right\|\leq\left\|\kappa(h)\right\|.$

###  2.2. Parabolic subgroups and flag manifolds

Given a subset $\theta\subset\Delta$, the _parabolic subgroup associated to
$\theta$_, denoted by $\operatorname{\mathsf{P}}_{\theta}\subset\mathsf{G}$,
is the normalizer of

${\mathfrak{u}}_{\theta}:=\bigoplus_{\alpha\in\Sigma^{+}_{\theta}}{\mathfrak{g}}_{\alpha}$

where $\Sigma^{+}_{\theta}:=\Sigma^{+}-{\rm Span}(\Delta-\theta)$. Then the
_flag manifold associated to $\theta$_ is

$\operatorname{\mathcal{F}}_{\theta}:=\mathsf{G}/\mathsf{P}_{\theta}.$

In this paper will only consider the case when $\theta\subset\Delta$ is
_symmetric_ , that is $\iota^{*}(\theta)=\theta$. In this case, there is a
unique open $\mathsf{G}$-orbit in

$\operatorname{\mathcal{F}}_{\theta}\times\operatorname{\mathcal{F}}_{\theta}$

and we say that two flags $F_{1},F_{2}\in\operatorname{\mathcal{F}}_{\theta}$
are _transverse_ if $(F_{1},F_{2})$ is in this orbit. For a flag
$F\in\operatorname{\mathcal{F}}_{\theta}$, we let

$\mathcal{Z}_{F}\subset\operatorname{\mathcal{F}}_{\theta}$

denote the set of flags that are not transverse to $F$.

Following the notation in [21], we define a map

$U_{\theta}:\mathsf{G}\rightarrow\operatorname{\mathcal{F}}_{\theta}$

by fixing a $\mathsf{KAK}$-decomposition $g=m_{g}e^{\kappa(g)}\ell_{g}$ for
each $g\in\mathsf{G}$ and then letting
$U_{\theta}(g):=m_{g}\operatorname{\mathsf{P}}_{\theta}$. One can show that if
$\alpha(\kappa(g))>0$ for all $\alpha\in\theta$, then $U_{\theta}(g)$ is
independent of the choice of $\mathsf{KAK}$-decomposition, see [22, Chap. IX,
Thm. 1.1], and hence $U_{\theta}$ is continuous on the set

$\left\\{g\in\mathsf{G}:\alpha(\kappa(g))>0\text{ for all
}\alpha\in\theta\right\\}.$

The action of $\mathsf{G}$ on $\operatorname{\mathcal{F}}_{\theta}$ has the
following north-south like dynamics.

######  Proposition 2.2 (see e.g. [15, Prop. 2.3]).

Suppose $\theta\subset\Delta$ is symmetric,
$F^{\pm}\in\operatorname{\mathcal{F}}_{\theta}$ and $\\{g_{n}\\}$ is a
sequence in $\mathsf{G}$. The following are equivalent:

  1. (1)

$U_{\theta}(g_{n})\rightarrow F^{+}$, $U_{\theta}(g_{n}^{-1})\rightarrow
F^{-}$ and $\lim_{n\rightarrow\infty}\alpha(\kappa(g_{n}))=\infty$ for every
$\alpha\in\theta$,

  2. (2)

$g_{n}(F)\to F^{+}$ for all
$F\in\operatorname{\mathcal{F}}_{\theta}-\mathcal{Z}_{F^{-}}$, and this
convergence is uniform on compact subsets of
$\operatorname{\mathcal{F}}_{\theta}-\mathcal{Z}_{F^{-}}$.

  3. (3)

$g_{n}^{-1}(F)=F^{-}$ for all
$F\in\operatorname{\mathcal{F}}_{\theta}-\mathcal{Z}_{F^{+}}$, and this
convergence is uniform on compact subsets of
$\operatorname{\mathcal{F}}_{\theta}-\mathcal{Z}_{F^{+}}$.

  4. (4)

there are open sets
$\mathcal{U}^{\pm}\subset\operatorname{\mathcal{F}}_{\theta}$ such that
$g_{n}(F)\to F^{+}$ for all $F\in\mathcal{U}^{+}$ and $g_{n}^{-1}(F)\to F^{-}$
for all $F\in\mathcal{U}^{-}$.

###  2.3. Fundamental weights and partial Cartan projections

For any $\alpha\in\Delta$, let $\omega_{\alpha}\in{\mathfrak{a}}^{*}$ denote
the associated fundamental weight.

Given a subset $\theta\subset\Delta$, the _partial Cartan subspace associated
to $\theta$_ is

${\mathfrak{a}}_{\theta}:=\\{H\in{\mathfrak{a}}:\alpha(H)=0\text{ for all
}\alpha\in\Delta-\theta\\}.$

Then there is a unique projection

$p_{\theta}:{\mathfrak{a}}\rightarrow{\mathfrak{a}}_{\theta}$

such that $\omega_{\alpha}(X)=\omega_{\alpha}(p_{\theta}(X))$ for all
$\alpha\in\theta$ and $X\in\mathfrak{a}$. The _partial Cartan projection
associated to $\theta$_ is

$\kappa_{\theta}:=p_{\theta}\circ\kappa:\mathsf{G}\to\mathfrak{a}_{\theta}.$

One can show that
$\\{\omega_{\alpha}|_{{\mathfrak{a}}_{\theta}}:\alpha\in\theta\\}$ is a basis
of ${\mathfrak{a}}_{\theta}^{*}$ and hence we will identify

${\mathfrak{a}}_{\theta}^{*}={\rm
Span}\\{\omega_{\alpha}:\alpha\in\theta\\}\subset\mathfrak{a}^{*}.$

Note that $\omega_{\alpha}(\kappa_{\theta}(g))=\omega_{\alpha}(\kappa(g))$ for
all $\alpha\in\theta$ and $g\in\mathsf{G}$ and so

$\phi(\kappa_{\theta}(g))=\phi(\kappa(g))$ (2)

for all $\phi\in{\mathfrak{a}}_{\theta}^{*}$ and $g\in\mathsf{G}$.

###  2.4. The partial Iwasawa cocycle

Let $\operatorname{\mathsf{U}}:=\exp({\mathfrak{u}}_{\Delta})$. The _Iwasawa
decomposition_ states that the map

$(k,a,u)\in\operatorname{\mathsf{K}}\times\exp({\mathfrak{a}})\times\operatorname{\mathsf{U}}\mapsto
kau\in\mathsf{G}$

is a diffeomorphism, see [30, Chap. VI, Prop. 6.46]. Using this, Quint [35]
defined the _Iwasawa cocycle_

$B:\mathsf{G}\times\operatorname{\mathcal{F}}_{\Delta}\rightarrow{\mathfrak{a}}$

with the defining property that
$gk\in\operatorname{\mathsf{K}}\cdot\exp(B(g,F))\cdot\operatorname{\mathsf{U}}$
for all $(g,F)\in\mathsf{G}\times\operatorname{\mathcal{F}}_{\Delta}$, where
$k\in\operatorname{\mathsf{K}}$ is an element such that
$F=k\operatorname{\mathsf{P}}_{\Delta}$.

For any $\theta\subset\Delta$, note that
$\operatorname{\mathsf{P}}_{\Delta}\subset\operatorname{\mathsf{P}}_{\theta}$,
so the identity map on $\mathsf{G}$ induces a surjection
$\Pi_{\theta}:\operatorname{\mathcal{F}}_{\Delta}\to\operatorname{\mathcal{F}}_{\theta}$.
The _partial Iwasawa cocycle_ is the map

$B_{\theta}:\mathsf{G}\times\operatorname{\mathcal{F}}_{\theta}\rightarrow{\mathfrak{a}}_{\theta}$

defined by $B_{\theta}(g,F)=p_{\theta}(B(g,F^{\prime}))$ for some (all)
$F^{\prime}\in\Pi_{\theta}^{-1}(F)$. By [35, Lem. 6.1 and 6.2], this is a
well-defined cocycle, that is

$B_{\theta}(gh,F)=B_{\theta}(g,hF)+B_{\theta}(h,F)$

for all $g,h\in\mathsf{G}$ and $F\in\operatorname{\mathcal{F}}_{\theta}$.

###  2.5. The Linear Case

We now briefly describe the above notations when
$\mathsf{G}=\mathsf{PSL}(d,\operatorname{\mathbb{R}})$. Let
$\mathfrak{sl}(d,\operatorname{\mathbb{R}})=\mathfrak{k}+\mathfrak{p}$ denote
the standard Cartan decomposition of
$\mathfrak{sl}(d,\operatorname{\mathbb{R}})$, that is

$\displaystyle\mathfrak{k}=\\{X\in\mathfrak{sl}(d,\operatorname{\mathbb{R}}):{{}^{t}X}=-X\\}\quad\text{and}\quad\mathfrak{p}=\\{X\in\mathfrak{sl}_{d}(\operatorname{\mathbb{R}}):{{}^{t}X}=X\\}.$

Also let

$\mathfrak{a}:=\\{{\rm
diag}(a_{1},\ldots,a_{d})\in\mathfrak{sl}(d,\operatorname{\mathbb{R}}):\
a_{1}+\cdots+a_{d}=0\\}\subset\mathfrak{p}$

denote the standard Cartan subspace with the standard positive Weyl chamber

$\mathfrak{a}^{+}:=\\{{\rm diag}(a_{1},\ldots,a_{d})\in\mathfrak{a}:\
a_{1}\geq\dots\geq a_{d}\\}\subset\mathfrak{p}.$

Then the opposite involution is given by

$\iota({\rm diag}(a_{1},\ldots,a_{d}))={\rm diag}(-a_{d},\ldots,-a_{1})$

and the Cartan projection is given by

$\kappa(g)=(\log\sigma_{1}(g),\cdots,\log\sigma_{d}(g))$

where $\sigma_{1}(g)\geq\cdots\geq\sigma_{d}(g)$ are the singular values of
any lift of $g$ to $\mathsf{SL}(d,\operatorname{\mathbb{R}})$.

The standard system of simple restricted roots is
$\Delta:=\\{\alpha_{1},\dots,\alpha_{d-1}\\}\subset\mathfrak{a}^{*}$ where

$\alpha_{j}({\rm diag}(a_{1},\ldots,a_{d}))=a_{j}-a_{j+1}$

for all ${\rm diag}(a_{1},\ldots,a_{d})\in\mathfrak{a}$. Then the fundamental
weights are given by

$\omega_{\alpha_{j}}({\rm diag}(a_{1},\ldots,a_{d}))=a_{1}+\dots+a_{j}$

and so $\iota^{*}(\omega_{\alpha_{j}})=\omega_{\alpha_{d-j}}$.

Let $e_{1},\dots,e_{d}$ denote the standard basis of
$\operatorname{\mathbb{R}}^{d}$ and let $F_{0}^{j}:=\left\langle
e_{1},\dots,e_{j}\right\rangle$ be the subspace spanned by
$e_{1},\dots,e_{j}$. Then given
$\theta=\\{\alpha_{j_{1}},\dots,\alpha_{j_{n}}\\}\subset\Delta$ with
$j_{1}<j_{2}<\dots<j_{n}$, the parabolic subgroup
$\operatorname{\mathsf{P}}_{\theta}\subset\mathsf{PSL}(d,\operatorname{\mathbb{R}})$
is the stabilizer of the flag

$(F_{0}^{j_{1}},F_{0}^{j_{2}},\cdots,F^{j_{n}}_{0})$

and $\operatorname{\mathcal{F}}_{\theta}$ is the partial flag manifold

$\mathcal{F}_{\theta}=\left\\{(F^{j_{1}},\dots,F^{j_{n}}):\
\mathrm{dim}\left(F^{j_{i}}\right)=j_{i}\text{ and }F^{j_{1}}\subset
F^{j_{2}}\subset\cdots\subset F^{j_{n}}\right\\}.$

In the case when $\theta$ is symmetric, two flags
$F_{1},F_{2}\in\mathcal{F}_{\theta}$ are transverse if and only if they are
transverse in the usual sense: $F_{1}^{j}\oplus
F_{2}^{d-j}=\operatorname{\mathbb{R}}^{d}$ for all $\alpha_{j}\in\theta$.

In the linear case we often replace subscripts of the form
$\theta=\\{\alpha_{j_{1}},\dots,\alpha_{j_{n}}\\}$ with $j_{1},\cdots,j_{n}$.
So for instance

$\operatorname{\mathcal{F}}_{1,d-1}=\operatorname{\mathcal{F}}_{\\{\alpha_{1},\alpha_{d-1}\\}}$

denotes the line/hyperplane partial flag manifold.

###  2.6. Properties of unipotent subgroups

Recall that a $d\times d$ real matrix $A$ is _unipotent_ if 1 is the only
eigenvalue of $A$ (over $\operatorname{\mathbb{C}}$) and a subgroup
$\mathsf{U}\subset\mathsf{GL}(d,\operatorname{\mathbb{R}})$ is called
_unipotent_ if every element in $\mathsf{U}$ is unipotent. When $\mathsf{G}$
has trivial center, a subgroup $\mathsf{U}\subset\mathsf{G}$ is _unipotent_ if
${\rm Ad}(\mathsf{U})\subset\operatorname{\mathsf{SL}}(\mathfrak{g})$ is
unipotent. Given a Lie subgroup $\mathsf{H}\subset\mathsf{G}$, the _unipotent
radical_ of $\mathsf{H}$ is the maximal connected normal unipotent subgroup of
$\mathsf{H}$.

Unipotent subgroups have the following well-known properties.

######  Proposition 2.3.

Suppose $Z(\mathsf{G})$ is trivial and $\mathfrak{u}$ is the Lie algebra of a
connected unipotent subgroup $\mathsf{U}\subset\mathsf{G}$. Then:

  1. (1)

The exponential map induces a diffeomorphism
$\mathfrak{u}\rightarrow\mathsf{U}$.

  2. (2)

Let $\lambda$ be a measure on $\mathfrak{u}$ which is obtained by pulling back
the Lebesgue measure on $\operatorname{\mathbb{R}}^{\dim\mathfrak{u}}$ via a
linear isomorphism (notice that any two measures obtained this way are scalar
multiplies of each other). Then the push-forward $\mu:=\exp_{*}\lambda$ is a
Haar measure on $\mathsf{U}$.

  3. (3)

If
$\Phi:\mathsf{G}\rightarrow\operatorname{\mathsf{SL}}(d,\operatorname{\mathbb{R}})$
is a representation, then
$\Phi(\mathsf{U})\subset\operatorname{\mathsf{SL}}(d,\operatorname{\mathbb{R}})$
is unipotent.

  4. (4)

There exists $C>0$ such that

$\left\|\kappa(\exp(Y))\right\|\leq C+C\log(1+\left\|Y\right\|)$

for all $Y\in\mathfrak{u}$.

  5. (5)

For any $\alpha\in\Delta$ there exist
$M_{\alpha}\in\operatorname{\mathbb{N}}$, $C_{\alpha}>1$ and a positive
everywhere defined rational function
$R_{\alpha}:\mathfrak{u}\rightarrow\operatorname{\mathbb{R}}$ where

$\frac{1}{C_{\alpha}}R_{\alpha}(Y)^{1/M_{\alpha}}\leq
e^{\omega_{\alpha}(\kappa(\exp(Y)))}\leq
C_{\alpha}R_{\alpha}(Y)^{1/M_{\alpha}}$

for all $Y\in\mathfrak{u}$.

We provide a proof of Proposition 2.3 in Appendix A.

##  3\. Relatively hyperbolic groups

In this expository section we recall one of the many equivalent definitions of
a relatively hyperbolic group, for more background and other definitions see
[8, 19, 33, 39].

###  3.1. Relatively hyperbolic groups

Recall that the action, by homeomorphisms, of a group $\Gamma$ on a compact
metrizable space $M$ is said to be a (discrete) convergence group action if
whenever $\\{\gamma_{n}\\}$ is a sequence of distinct elements in $\Gamma$,
then there are points $x,y\in M$ and a subsequence $\\{\gamma_{n_{j}}\\}$ so
that $\gamma_{n_{j}}(z)$ converges to $x$ for all $z\in M-\\{y\\}$ and the
convergence is uniform on compact subsets of $M-\\{y\\}$.

Suppose that $\Gamma$ acts on $M$ as a convergence group, then:

  * •

A point $x\in M$ is a _conical limit point_ if there exist $a,b\in M$ distinct
and a sequence $\\{\gamma_{n}\\}$ in $\Gamma$ such that $\gamma_{n}(x)\to a$
and $\gamma_{n}(y)\to b$ for all $y\in M-\\{x\\}$.

  * •

An element $\gamma\in\Gamma$ is _parabolic_ if it has infinite order and fixes
exactly one point in $M$.

  * •

A infinite order subgroup $H\subset\Gamma$ is _parabolic_ if it fixes some
point of $M$ and each infinite order element in $H$ is parabolic. The fixed
point of a parabolic subgroup is called a _parabolic point_.

  * •

A parabolic point $x\in M$ is _bounded_ if the quotient ${\rm
Stab}_{\Gamma}(x)\backslash(M-\\{x\\})$ is compact.

Finally, $\Gamma$ is called a _geometrically finite convergence group_ if
every point in $M$ is either a conical limit point or a bounded parabolic
point.

######  Definition 3.1.

Given a finitely generated group $\Gamma$ and a collection $\mathcal{P}$ of
finitely generated infinite subgroups, we say that $\Gamma$ is _hyperbolic
relative to $\mathcal{P}$_, or that $(\Gamma,\mathcal{P})$ is _relatively
hyperbolic_ , if $\Gamma$ acts on a compact perfect metrizable space $M$ as a
geometrically finite convergence group and the maximal parabolic subgroups are
exactly the set

$\\{\gamma P\gamma^{-1}:P\in\mathcal{P},\gamma\in\Gamma\\}.$

To avoid confusion with parabolic subgroups in the Lie group sense, we also
sometimes call a maximal parabolic subgroup of $\Gamma$ a _peripheral
subgroup_.

######  Remark 3.2.

Notice that by definition we assume that a relatively hyperbolic group is non-
elementary (i.e. $M$ is infinite) and finitely generated.

By a theorem of Bowditch [8, Thm. 9.4], given a relatively hyperbolic group
$(\Gamma,\mathcal{P})$, any two compact perfect metrizable spaces satisfying
Definition 3.1 are $\Gamma$-equivariantly homeomorphic. This unique
topological space is then denoted by $\partial(\Gamma,\mathcal{P})$ and called
the _Bowditch boundary of $(\Gamma,\mathcal{P})$_.

###  3.2. The Groves-Manning cusp space

Given a relatively hyperbolic group $(\Gamma,\mathcal{P})$, Groves and Manning
[20] constructed a Gromov hyperbolic space which $\Gamma$ acts on proper
discontinuously so that subgroups in $\mathcal{P}$ preserve “combinatorial
horoballs.” This space, now called Groves-Manning cusp spaces, are central
tools in the study of relatively hyperbolic groups. We recall their
construction here.

######  Definition 3.3.

Suppose $Y$ is a graph with the simplicial distance ${\rm d}_{Y}$. The
_combinatorial horoball_ $\mathcal{H}(Y)$ is the graph, also equipped with the
simplicial distance, that has vertex set
$Y^{(0)}\times\operatorname{\mathbb{N}}$ and two types of edges:

  * •

_vertical edges_ joining vertices $(v,n)$ and $(v,n+1)$,

  * •

_horizontal edges_ joining vertices $(v,n)$ and $(w,n)$ when ${\rm
d}_{Y}(v,w)\leq 2^{n-1}$.

######  Definition 3.4.

Suppose that $(\Gamma,\mathcal{P})$ is relatively hyperbolic. A finite
symmetric generating set $S\subset\Gamma$ is _adapted_ if $S\cap P$ is a
generating set of $P$ for every $P\in\mathcal{P}$. Given such an $S$, we let
${\mathcal{C}}(\Gamma,S)$ and ${\mathcal{C}}(P,S\cap P)$ denote the associated
Cayley graphs. Then the associated _Groves–Manning cusp space_ , denoted
${\mathcal{C}}_{GM}(\Gamma,\mathcal{P},S)$, is obtained from the Cayley graph
${\mathcal{C}}(\Gamma,S)$ by attaching, for each $P\in\mathcal{P}$ and each
coset $\gamma P\in\Gamma/P$, a copy of the combinatorial horoball
$\mathcal{H}(\gamma{\mathcal{C}}(P,S\cap P))$ by identifying
$\gamma{\mathcal{C}}(P,S\cap P)\subset{\mathcal{C}}(\Gamma,S)$ with the $n=1$
level of $\mathcal{H}(\gamma{\mathcal{C}}(P,S\cap P))$.

######  Theorem 3.5 (Groves-Manning [20, Thm. 3.25]).

If $(\Gamma,\mathcal{P})$ is relatively hyperbolic and $S$ is an adapted
finite generating set, then ${\mathcal{C}}_{GM}(\Gamma,\mathcal{P},S)$ is a
proper geodesic Gromov hyperbolic space such that

  1. (1)

$\Gamma$ acts properly discontinuously on
${\mathcal{C}}_{GM}(\Gamma,\mathcal{P},S)$ by isometries,

  2. (2)

every point in $X$ is within a uniformly bounded distance of a bi-infinite
geodesic, and

  3. (3)

there exists a $\Gamma$-equivariant homeomorphism between
$\partial_{\infty}{\mathcal{C}}_{GM}(\Gamma,\mathcal{P},S)$, the Gromov
boundary of ${\mathcal{C}}_{GM}(\Gamma,\mathcal{P},S)$, and
$\partial(\Gamma,\mathcal{P})$.

##  4\. Discrete subgroups of semisimple Lie groups

In this expository section we introduce three classes of discrete subgroups in
$\mathsf{G}$ and state some of their basic properties. In the rest of the
paper, we assume that $\theta\subset\Delta$ is symmetric.

###  4.1. Divergent groups

A discrete subgroup $\Gamma\subset\mathsf{G}$ is called
_$\operatorname{\mathsf{P}}_{\theta}$ -divergent_ if

$\lim_{n\rightarrow+\infty}\min_{\alpha\in\theta}\alpha(\kappa_{\theta}(\gamma_{n}))=+\infty$

whenever $\\{\gamma_{n}\\}$ is a sequence of distinct elements in $\Gamma$.
The _limit set_ $\Lambda_{\theta}(\Gamma)$ of such a subgroup is the set of
accumulation points of $\\{U_{\theta}(\gamma):\gamma\in\Gamma\\}$. We note
that in the literature, divergent groups are sometimes called regular groups
(e.g. [27]).

The limit set of a divergent group can be used to compactify it.

######  Lemma 4.1 (see e.g. [15, Prop. 2.3]).

If $\Gamma\subset\mathsf{G}$ is
$\operatorname{\mathsf{P}}_{\theta}$-divergent, then the set
$\Gamma\cup\Lambda_{\theta}(\Gamma)$ has a topology that makes it a
compactification of $\Gamma$. More precisely:

  1. (1)

$\Gamma\cup\Lambda_{\theta}(\Gamma)$ is a compact metrizable space.

  2. (2)

If $\Gamma$ has the discrete topology, then
$\Gamma\hookrightarrow\Gamma\cup\Lambda_{\theta}(\Gamma)$ is an embedding.

  3. (3)

If $\Lambda_{\theta}(\Gamma)$ has the subspace topology from
$\operatorname{\mathcal{F}}_{\theta}$, then
$\Lambda_{\theta}(\Gamma)\hookrightarrow\Gamma\cup\Lambda_{\theta}(\Gamma)$ is
an embedding.

  4. (4)

A sequence $\\{\gamma_{n}\\}$ in $\Gamma$ converges to $F$ in
$\Lambda_{\theta}(\Gamma)$ if and only if

$\lim_{n\rightarrow+\infty}\min_{\alpha\in\theta}\alpha(\kappa_{\theta}(\gamma_{n}))=+\infty\quad\text{and}\quad
U_{\theta}(\gamma_{n})\rightarrow F.$

  5. (5)

The natural left action of $\Gamma$ on $\Gamma\cup\Lambda_{\theta}(\Gamma)$ is
by homeomorphisms.

###  4.2. Transverse groups

A $\operatorname{\mathsf{P}}_{\theta}$-divergent subgroup
$\Gamma\subset\mathsf{G}$ is _$\operatorname{\mathsf{P}}_{\theta}$
-transverse_ if $\Lambda_{\theta}(\Gamma)$ is a transverse subset of
$\operatorname{\mathcal{F}}_{\theta}$, i.e. distinct pairs of flags in
$\Lambda_{\theta}(\Gamma)$ are transverse. We note that in the literature,
transverse groups are sometimes called regular antipodal groups (e.g. [27]).

One crucial feature of $\operatorname{\mathsf{P}}_{\theta}$-transverse groups
is that they act as a convergence group on their limit sets.

######  Proposition 4.2.

[28, Prop. 5.38] If $\Gamma$ is
$\operatorname{\mathsf{P}}_{\theta}$-transverse, then $\Gamma$ acts on
$\Lambda_{\theta}(\Gamma)$ as a convergence group. In particular, if $\Gamma$
is non-elementary, then $\Gamma$ acts on $\Lambda_{\theta}(\Gamma)$ minimally,
and $\Lambda_{\theta}(\Gamma)$ is perfect.

When $\Gamma\subset\mathsf{G}$ is
$\operatorname{\mathsf{P}}_{\theta}$-transverse, the set of conical limit
points for the action of $\Gamma$ on $\Lambda_{\theta}(\Gamma)$ is called the
$\theta$-conical limit set and is denoted $\Lambda^{\rm
con}_{\theta}(\Gamma)$.

###  4.3. Relatively Anosov subgroups

There are several equivalent definitions of relatively Anosov groups. The
definition we use comes from [26].

A $\operatorname{\mathsf{P}}_{\theta}$-transverse subgroup
$\Gamma\subset\mathsf{G}$ is _$\operatorname{\mathsf{P}}_{\theta}$ -Anosov
relative to $\mathcal{P}$_, a finite collection of subgroups of $\Gamma$, if
$(\Gamma,\mathcal{P})$ is relatively hyperbolic with Bowditch boundary
$\partial(\Gamma,\mathcal{P})$ and there is a continuous $\Gamma$-equivariant
map

$\xi\colon\partial(\Gamma,\mathcal{P})\to\operatorname{\mathcal{F}}_{\theta}$

which is a homeomorphism onto $\Lambda_{\theta}(\Gamma)$. Observe that such a
$\xi$ is unique, so we refer to it as the _limit map_ of $\Gamma$.

The next result shows that this limit maps plays nicely with the Gromov
boundary of a Groves–Manning cusp space.

######  Proposition 4.3.

Suppose $\Gamma\subset\mathsf{G}$ is
$\operatorname{\mathsf{P}}_{\theta}$-Anosov relative to $\mathcal{P}$, with
limit map
$\xi:\partial(\Gamma,\mathcal{P})\rightarrow\operatorname{\mathcal{F}}_{\theta}$.
Let $X$ be a Groves–Manning cusp space for $(\Gamma,\mathcal{P})$, and let
$b_{0}\in X$. If $\\{\gamma_{n}\\}$ is a sequence in $\Gamma$ and
$\gamma_{n}(b_{0})\rightarrow
x\in\partial_{\infty}X=\partial(\Gamma,\mathcal{P})$, then
$U_{\theta}(\gamma_{n})\rightarrow\xi(x)$.

###### Proof.

By compactness it suffices to consider the case when
$U_{\theta}(\gamma_{n})\rightarrow F^{+}$ and show that $F^{+}=\xi(x)$.
Passing to subsequence we can suppose that $\gamma_{n}^{-1}(b_{0})\rightarrow
y\in\partial(\Gamma,\mathcal{P})$ and $U_{\theta}(\gamma_{n}^{-1})\rightarrow
F^{-}$. Then, by properties of Gromov hyperbolic spaces,
$\gamma_{n}(z)\rightarrow x$ for all
$z\in\partial(\Gamma,\mathcal{P})-\\{y\\}$. Also, by Proposition 2.2,

$\lim_{n\rightarrow\infty}\gamma_{n}(F)=F^{+}$

for all $F\in\operatorname{\mathcal{F}}_{\theta}$ transverse to $F^{-}$.

Notice that $F^{-}\in\Lambda_{\theta}(\Gamma)$ and so $F^{-}=\xi(y^{\prime})$
for some $y^{\prime}\in\partial(\Gamma,\mathcal{P})$. Fix
$z\in\partial(\Gamma,\mathcal{P})-\\{y,y^{\prime}\\}$. Then

$\xi(x)=\lim_{n\rightarrow\infty}\xi(\gamma_{n}(z))=\lim_{n\rightarrow\infty}\gamma_{n}\xi(z)=F^{+}$

since $\xi(z)$ is transverse to $F^{-}=\xi(y^{\prime})$. ∎

The following theorem was established in [41] when
$\mathsf{G}=\operatorname{\mathsf{SL}}(d,\operatorname{\mathbb{R}})$. In
Appendix B we will explain why it is also true in the following setting. In
the following theorem and elsewhere in the paper, given a Lie group
$\mathsf{H}$ let $\mathsf{H}^{0}$ denote the connected component of the
identity in $\mathsf{H}$.

######  Theorem 4.4.

Assume $Z(\mathsf{G})$ is trivial and $\operatorname{\mathsf{P}}_{\theta}$
contains no simple factors of $\mathsf{G}$. Suppose $\Gamma\subset\mathsf{G}$
is a non-elementary $\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup
relative to $\mathcal{P}$.

  1. (1)

If $X$ is a Groves–Manning cusp space for $(\Gamma,\mathcal{P})$ and
$M:=\mathsf{G}/\operatorname{\mathsf{K}}$ is a Riemannian symmetric space
associated to $\mathsf{G}$, then there exist $c>1$, $C>0$ such that

$\frac{1}{c}{\rm
d}_{M}(\gamma\operatorname{\mathsf{K}},\operatorname{\mathsf{K}})-C\leq{\rm
d}_{X}(\gamma,\operatorname{\mathrm{id}})\leq c{\rm
d}_{M}(\gamma\operatorname{\mathsf{K}},\operatorname{\mathsf{K}})+C$

for all $\gamma\in\Gamma$.

  2. (2)

If $P\in\mathcal{P}$, then $P$ is a cocompact lattice in a closed Lie group
$\mathsf{H}$ with finitely many components. Moreover,

    1. (a)

$\mathsf{H}=\mathsf{L}\ltimes\mathsf{U}$ where $\mathsf{L}$ is compact and
$\mathsf{U}$ is the unipotent radical of $\mathsf{H}$.

    2. (b)

$\mathsf{H}^{0}=\mathsf{L}^{0}\times\mathsf{U}$ and $\mathsf{L}^{0}$ is
Abelian.

Example 1.11 provides an example where $\operatorname{\mathsf{P}}_{\theta}$
contains a simple factor of $\mathsf{G}$ and the conclusions of Theorem 4.4
fail.

###  4.4. Helpful reductions

We first explain why one can often reduce to the case where the center
$Z(\mathsf{G})$ of $\mathsf{G}$ is trivial and
$\operatorname{\mathsf{P}}_{\theta}$ contains no simple factors of
$\mathsf{G}$. We then explain how one can often reduce to the case where
$\mathsf{G}=\operatorname{\mathsf{SL}}(d,\operatorname{\mathbb{R}})$.

Decompose the Lie algebra ${\mathfrak{g}}$ of $G$ into a product of simple Lie
algebras, ${\mathfrak{g}}=\oplus_{j=1}^{m}{\mathfrak{g}}_{j}$. For each $1\leq
j\leq m$, let $\mathsf{G}_{j}\subset\mathsf{G}$ denote the closed connected
normal subgroup with Lie algebra ${\mathfrak{g}}_{j}$. Then

$\mathsf{G}=\mathsf{G}_{1}\cdots\mathsf{G}_{m}$

is an almost direct product and $\mathsf{G}_{1},\dots,\mathsf{G}_{m}$ are
called the _simple factors_ of $\mathsf{G}$.

######  Proposition 4.5 ([15, Prop. 2.9]).

Suppose $\theta\subset\Delta$ is symmetric and
$\mathsf{H}:=Z(\mathsf{G})\prod\\{\mathsf{G}_{j}:\mathsf{G}_{j}\subset\operatorname{\mathsf{P}}_{\theta}\\}$.
Let $p:\mathsf{G}\rightarrow\mathsf{G}^{\prime}:=\mathsf{G}/\mathsf{H}$ be the
quotient map. Then:

  1. (1)

$\mathsf{G}^{\prime}$ is a semisimple Lie group without compact factors and
with trivial center.

  2. (2)

There is a Cartan decomposition
${\mathfrak{g}}^{\prime}={\mathfrak{k}}^{\prime}+{\mathfrak{p}}^{\prime}$ of
the Lie algebra of $\mathsf{G}^{\prime}$, a Cartan subspace
${\mathfrak{a}}^{\prime}\subset{\mathfrak{p}}^{\prime}$, a system of simple
restricted roots $\Delta^{\prime}\subset({\mathfrak{a}}^{\prime})^{*}$ and a
subset $\theta^{\prime}\subset\Delta^{\prime}$ such that

$p(\operatorname{\mathsf{P}}_{\theta})=\operatorname{\mathsf{P}}^{\prime}_{\theta^{\prime}}$

(where $\operatorname{\mathsf{P}}^{\prime}_{\theta^{\prime}}$ is the parabolic
subgroup of $\mathsf{G}^{\prime}$ associated to $\theta^{\prime}$). Moreover,
$\operatorname{\mathsf{P}}_{\theta^{\prime}}^{\prime}$ contains no simple
factors of $\mathsf{G}^{\prime}$.

  3. (3)

${\rm d}p$ induces an isomorphism of the partial Cartan subspaces
${\mathfrak{a}}_{\theta}$ and ${\mathfrak{a}}_{\theta^{\prime}}^{\prime}$.
Moreover, the partial Cartan projections satisfies

${\rm
d}p(\kappa_{\theta}(g))=\kappa_{\theta^{\prime}}^{\prime}(p(g))\quad\text{for
all $g\in\mathsf{G}$}.$

  4. (4)

The map
$\xi:\operatorname{\mathcal{F}}_{\theta}\rightarrow\operatorname{\mathcal{F}}_{\theta^{\prime}}=\mathsf{G}^{\prime}/\operatorname{\mathsf{P}}_{\theta^{\prime}}^{\prime}$
defined by
$\xi(g\operatorname{\mathsf{P}}_{\theta})=p(g)\operatorname{\mathsf{P}}_{\theta^{\prime}}^{\prime}$
is a diffeomorphism which preserves transversality. Moreover, the partial
Iwasawa cocycles satisfies

${\rm
d}p(B_{\theta}(g,F))=B^{\prime}_{\theta^{\prime}}(p(g),\xi(F))\quad\text{for
all $g\in\mathsf{G}$ and $F\in\operatorname{\mathcal{F}}_{\theta}$}.$

Using the discussion in [21, Section 3] it is possible to prove the following
result which allows one to reduce many calculations to the linear case, see
[15, Prop. B.1] for details. The statement of the result uses the notation
introduced in Section 2.5.

######  Proposition 4.6.

For any symmetric $\theta\subset\Delta$ and
$\chi\in\sum_{\alpha\in\theta}\operatorname{\mathbb{N}}\cdot\omega_{\alpha}$
there exist $d\in\operatorname{\mathbb{N}}$, an irreducible linear
representation
$\Phi:\mathsf{G}\rightarrow\mathsf{SL}(d,\operatorname{\mathbb{R}})$ and a
$\Phi$-equivariant smooth embedding

$\xi:\operatorname{\mathcal{F}}_{\theta}\rightarrow\operatorname{\mathcal{F}}_{1,d-1}(\operatorname{\mathbb{R}}^{d})$

such that:

  1. (1)

$F_{1},F_{2}\in\operatorname{\mathcal{F}}_{\theta}$ are transverse if and only
if $\xi(F_{1})$ and $\xi(F_{2})$ are transverse.

  2. (2)

There exists $N\in\operatorname{\mathbb{N}}$ such that

$\log\sigma_{1}(\Phi(g))=N\chi(\kappa(g))$

for all $g\in\mathsf{G}$.

  3. (3)

$\alpha_{1}(\kappa(\Phi(g)))=\min_{\alpha\in\theta}\alpha(\kappa(g))$ for all
$g\in\mathsf{G}$.

  4. (4)

If $\min_{\alpha\in\theta}\alpha(\kappa(g))>0$, then

$\xi(U_{\theta}(g))=U_{1,d-1}(\Phi(g)).$

  5. (5)

$\Gamma\subset\mathsf{G}$ is $\operatorname{\mathsf{P}}_{\theta}$-divergent
(respectively $\operatorname{\mathsf{P}}_{\theta}$-transverse) if and only if
$\Phi(\Gamma)$ is $\operatorname{\mathsf{P}}_{1,d-1}$-divergent (respectively
$\operatorname{\mathsf{P}}_{1,d-1}$-transverse). Moreover, in this case

$\xi(\Lambda_{\theta}(\Gamma))=\Lambda_{1,d-1}(\Phi(\Gamma)).$

  6. (6)

If $\Gamma\subset\mathsf{G}$ is discrete and $\mathcal{P}$ is a finite
collection of subgroups in $\Gamma$, then $\Gamma\subset\mathsf{G}$ is
$\operatorname{\mathsf{P}}_{\theta}$-Anosov relative to $\mathcal{P}$ if and
only if $\Phi(\Gamma)$ is $\operatorname{\mathsf{P}}_{1,d-1}$-Anosov relative
to $\mathcal{P}^{\prime}:=\\{\Phi(P):P\in\mathcal{P}\\}$.

######  Remark 4.7.

Part (6) is not explicitly stated in [15, Prop. B.1], however it follows
immediately from part (5) and the definitions.

##  5\. Multiplicative estimates

It is a general principle from linear algebra that if $U_{\theta}(A^{-1})$ is
uniformly transverse to $U_{\theta}(B)$, then $\kappa_{\theta}(AB)$ is
coarsely equal to $\kappa_{\theta}(A)+\kappa_{\theta}(B)$, see, for example,
[6, Lem. A.7]. We make use of two manifestations of this principle, the first
in the context of transverse groups and the second in the context of
relatively Anosov groups. Previous instances of this principle in our work
include [14, Lem. 6.2] and [15, Prop. 6.3].

In the results of this section, $\left\|\cdot\right\|$ denotes the norm on
$\mathfrak{a}$ induced by the restriction of the Killing from to
$\mathfrak{a}$.

######  Proposition 5.1.

Suppose $\Gamma\subset\mathsf{G}$ is
$\operatorname{\mathsf{P}}_{\theta}$-transverse and ${\rm
d}_{\operatorname{\mathcal{F}}_{\theta}}$ is a distance on
$\operatorname{\mathcal{F}}_{\theta}$ which is induced by a Riemannian metric.
For any $\epsilon>0$ there exists $C=C(\epsilon)>0$ such that: if
$\gamma,\eta\in\Gamma$ and ${\rm
d}_{\operatorname{\mathcal{F}}_{\theta}}(U_{\theta}(\gamma^{-1}),U_{\theta}(\eta))>\epsilon$,
then

$\left\|\kappa_{\theta}(\gamma\eta)-\kappa_{\theta}(\gamma)-\kappa_{\theta}(\eta)\right\|\leq
C.$

###### Proof.

We use the following special case of [6, Lem. A.7].

######  Lemma 5.2.

Let $e_{1},\dots,e_{d}$ denote the standard basis of
$\operatorname{\mathbb{R}}^{d}$. Suppose
$g_{1},g_{2}\in\operatorname{\mathsf{SL}}(d,\operatorname{\mathbb{R}})$ have
singular value decomposition $g_{1}=m_{1}a_{1}\ell_{1}$ and
$g_{2}=m_{2}a_{2}\ell_{2}$. Then

$\sigma_{1}(g_{1})\sigma_{1}(g_{2})\sin(\theta)\leq\sigma_{1}(g_{1}g_{2})\leq\sigma_{1}(g_{1})\sigma_{1}(g_{2})$

where $\theta:=\angle\left(m_{2}\left\langle
e_{1}\right\rangle,\ell_{1}^{-1}\left\langle
e_{2},\dots,e_{d}\right\rangle\right)$ is the Euclidean angle between the
subspaces $m_{2}\left\langle e_{1}\right\rangle$ and
$\ell_{1}^{-1}\left\langle e_{2},\dots,e_{d}\right\rangle$.

If the proposition fails, then for every $n\geq 1$ there exist
$\gamma_{n},\eta_{n}\in\Gamma$ where

${\rm
d}_{\operatorname{\mathcal{F}}_{\theta}}(U_{\theta}(\gamma_{n}^{-1}),U_{\theta}(\eta_{n}))>\epsilon$

and

$\left\|\kappa_{\theta}(\gamma_{n}\eta_{n})-\kappa_{\theta}(\gamma_{n})-\kappa_{\theta}(\eta_{n})\right\|\geq
n.$

Lemma 2.1 implies that $\\{\gamma_{n}\\}$ and $\\{\eta_{n}\\}$ are escaping
sequences in $\Gamma$. So by passing to a subsequence we can suppose that
$U_{\theta}(\gamma_{n}^{-1})\rightarrow F_{1}\in\Lambda_{\theta}(\Gamma)$ and
$U_{\theta}(\eta_{n})\rightarrow F_{2}\in\Lambda_{\theta}(\Gamma)$. Then ${\rm
d}_{\operatorname{\mathcal{F}}_{\theta}}(F_{1},F_{2})\geq\epsilon$ and hence
$F_{1}$ and $F_{2}$ are transverse (since $\Gamma$ is
$\operatorname{\mathsf{P}}_{\theta}$-transverse).

Since $\\{\omega_{\alpha}|_{\mathfrak{a}_{\theta}}:\alpha\in\theta\\}$ is a
basis for $\mathfrak{a}_{\theta}^{*}$, after passing to a subsequence there
exists some
$\chi\in\sum_{\alpha\in\theta}\operatorname{\mathbb{N}}\cdot\omega_{\alpha}$
such that

$\lim_{n\rightarrow\infty}\left|\chi\Big{(}\kappa_{\theta}(\gamma_{n}\eta_{n})-\kappa_{\theta}(\gamma_{n})-\kappa_{\theta}(\eta_{n})\Big{)}\right|=\infty.$

Let $N\in\operatorname{\mathbb{N}}$,
$\Phi:\mathsf{G}\rightarrow\mathsf{PSL}(d,\operatorname{\mathbb{R}})$ and
$\xi:\operatorname{\mathcal{F}}_{\theta}\rightarrow\operatorname{\mathcal{F}}_{1,d-1}(\operatorname{\mathbb{R}}^{d})$
satisfy Proposition 4.6 for $\chi$. Then by Proposition 4.6 Claim (2),

$\log\sigma_{1}(\Phi(g))=N\chi(\kappa(g))=N\chi(\kappa_{\theta}(g))$

for all $g\in\mathsf{G}$.

Let $\hat{\gamma}_{n}:=\Phi(\gamma_{n})$ and $\hat{\eta}_{n}:=\Phi(\eta_{n})$.
Since $\Gamma$ is $\operatorname{\mathsf{P}}_{\theta}$-transverse,
$\alpha(\kappa(\gamma_{n}))\rightarrow+\infty$ for all $\alpha\in\theta$. So
by Proposition 4.6 Claim (4),

$\lim_{n\rightarrow\infty}U_{1,d-1}(\hat{\gamma}_{n}^{-1})=\lim_{n\rightarrow\infty}\xi(U_{\theta}(\gamma_{n}^{-1}))=\xi(F_{1}).$

Likewise, $U_{1,d-1}(\hat{\eta}_{n})\rightarrow\xi(F_{2})$. Since $F_{1}$ and
$F_{2}$ are transverse, Proposition 4.6 Claim (1) implies that $\xi(F_{1})$
and $\xi(F_{2})$ are transverse. So Lemma 5.2 implies that there exists a
constant $C>0$ such that

$\left|\log\sigma_{1}(\hat{\gamma}_{n}\hat{\eta}_{n})-\log\sigma_{1}(\hat{\gamma}_{n})-\log\sigma_{1}(\hat{\eta}_{n})\right|\leq
C$

for all $n\geq 1$. So,

$\left|\chi\Big{(}\kappa_{\theta}(\gamma_{n}\eta_{n})-\kappa_{\theta}(\gamma_{n})-\kappa_{\theta}(\eta_{n})\Big{)}\right|\leq\frac{C}{N}$

and we have a contradiction. ∎

Our result for relatively Anosov groups involves a choice of Groves-Manning
cusp space.

######  Proposition 5.3.

Suppose $\Gamma\subset\mathsf{G}$ is
$\operatorname{\mathsf{P}}_{\theta}$-Anosov relative to $\mathcal{P}$ and $X$
is a Groves–Manning cusp space for $(\Gamma,\mathcal{P})$. There exists $C>0$
such that: if $f:[0,T]\rightarrow X$ is a geodesic with
$f(0)=\operatorname{\mathrm{id}}$ and $f(T)\in\Gamma$, then

$\left\|\kappa_{\theta}(f(T))-\kappa_{\theta}(f(t))-\kappa_{\theta}(f(t)^{-1}f(T))\right\|\leq
C$

whenever $t\in[0,T]$ and $f(t)\in\Gamma$.

###### Proof.

Suppose not. Then for every $n\geq 1$ there is a geodesic
$f_{n}:[0,T_{n}]\rightarrow X$ and some $t_{n}\in[0,T_{n}]$ such that
$f_{n}(0)=\operatorname{\mathrm{id}}$, $f_{n}(t_{n})\in\Gamma$,
$f_{n}(T_{n})\in\Gamma$, and

$\left\|\kappa_{\theta}(f_{n}(T_{n}))-\kappa_{\theta}(f_{n}(t_{n}))-\kappa_{\theta}(f_{n}(t_{n})^{-1}f_{n}(T_{n}))\right\|\geq
n.$

Let $\gamma_{n}:=f(t_{n})$ and $\eta_{n}:=f_{n}(t_{n})^{-1}f_{n}(T_{n})$. By
Lemma 2.1, both $\\{\gamma_{n}\\}$ and $\\{\eta_{n}\\}$ are escaping sequences
in $\Gamma$. So by passing to a subsequence we can suppose that
$\gamma_{n}^{-1}\rightarrow x\in\partial_{\infty}X$ and $\eta_{n}\rightarrow
y\in\partial_{\infty}X$. Since $t\mapsto f_{n}(t_{n})^{-1}f_{n}(t)$ is a
geodesic in $X$ passing through $\operatorname{\mathrm{id}}$ and joining
$\gamma_{n}^{-1}$ to $\eta_{n}$, we must have $x\neq y$.

Let $\xi:\partial_{\infty}X\rightarrow\Lambda_{\theta}(\Gamma)$ be the limit
map. Proposition 4.3 implies that
$U_{\theta}(\gamma_{n}^{-1})\rightarrow\xi(x)$ and
$U_{\theta}(\eta_{n})\rightarrow\xi(y)$. Since $\xi(x)\neq\xi(y)$, Proposition
5.1 implies that there exists $C>0$ such that

$\displaystyle\left\|\kappa_{\theta}(f_{n}(T_{n}))-\kappa_{\theta}(f_{n}(t_{n}))-\kappa_{\theta}(f_{n}(t_{n})^{-1}f_{n}(T_{n}))\right\|=\left\|\kappa_{\theta}(\gamma_{n}\eta_{n})-\kappa_{\theta}(\gamma_{n})-\kappa_{\theta}(\eta_{n})\right\|\leq
C$

for all $n\geq 1$. So we have a contradiction. ∎

##  6\. Resolution of singularities

In this section we study the asymptotic behavior of proper positive functions
which are products of powers of rational functions. This is the key technical
step needed to prove that the Poincaré series of a peripheral subgroup of a
relatively Anosov subgroup diverges at its critical exponent, see Theorem 7.1.

Let $\lambda$ denote the Lebesgue measure on $\operatorname{\mathbb{R}}^{d}$,
and $\left\|\cdot\right\|$ the standard Euclidean norm on
$\operatorname{\mathbb{R}}^{d}$. The main result of this section is the
following theorem. Its proof is motivated by arguments of Benoist-Oh [3, Prop.
7.2], which implies the special case when $m=1$.

######  Theorem 6.1.

Suppose
$R_{1},\dots,R_{m}:\operatorname{\mathbb{R}}^{d}\rightarrow\operatorname{\mathbb{R}}$
are rational functions which are everywhere defined and positive, and let

$R:=R_{1}^{\ell_{1}}\cdots R_{m}^{\ell_{m}}$

where $\ell_{1},\dots,\ell_{m}\in\operatorname{\mathbb{R}}$. If $R$ is a
proper function, then:

  1. (1)

There exists $\delta=\delta(R)>0$ such that

$\int_{\operatorname{\mathbb{R}}^{d}}R^{-s}d\lambda$

converges when $s\in(\delta,+\infty)$ and diverges when $s\in[0,\delta]$.

  2. (2)

There exist $c=c(R)>0$ and $\epsilon=\epsilon(R)>0$ such that

$R(x)\geq c(1+\left\|x\right\|)^{\epsilon}$

for all $x\in\operatorname{\mathbb{R}}^{d}$.

To prove Theorem 6.1 , we compactify $\operatorname{\mathbb{R}}^{d}$ by
identifying it with the affine subspace

$\operatorname{\mathbb{A}}_{1}:=\\{[x_{0}:\cdots:x_{d}]:x_{0}\neq
0\\}\subset\operatorname{\mathbb{P}}(\operatorname{\mathbb{R}}^{d+1})$

via the coordinate chart
$\psi_{1}:\operatorname{\mathbb{A}}_{1}\to\operatorname{\mathbb{R}}^{d}$ given
by

$\psi_{1}:[1:x_{1}:\cdots:x_{d}]\mapsto\left(x_{1},\dots,x_{d}\right).$

It now suffices to prove the following lemma, which is the analog of Theorem
6.1 on “neighborhoods of infinity.”

######  Lemma 6.2.

For each
$p\in\operatorname{\mathbb{P}}(\operatorname{\mathbb{R}}^{d+1})-\operatorname{\mathbb{A}}_{1}$,
there exist $\delta_{p},c_{p},\epsilon_{p}>0$ and an open neighborhood
$\mathcal{O}_{p}\subset\operatorname{\mathbb{P}}(\operatorname{\mathbb{R}}^{d+1})$
of $p$ such that:

  1. (1)

The integral

$\int_{\psi_{1}(\mathcal{O}_{p}\cap\operatorname{\mathbb{A}}_{1})}R^{-s}d\lambda$

converges when $s\in(\delta_{p},+\infty)$ and diverges when
$s\in[0,\delta_{p}]$.

  2. (2)

$R(x)\geq c_{p}(1+\left\|x\right\|)^{\epsilon_{p}}$ for all
$x\in\psi_{1}(\mathcal{O}_{p}\cap\operatorname{\mathbb{A}}_{1})\subset\operatorname{\mathbb{R}}^{d}$.

Assuming Lemma 6.2, we prove Theorem 6.1.

###### Proof of Theorem 6.1.

Since
$\operatorname{\mathbb{P}}(\operatorname{\mathbb{R}}^{d+1})-\operatorname{\mathbb{A}}_{1}$
is compact, there exist finitely many points $p_{1},\dots,p_{n}$ in
$\operatorname{\mathbb{P}}(\operatorname{\mathbb{R}}^{d+1})-\operatorname{\mathbb{A}}_{1}$
such that

$\operatorname{\mathbb{P}}(\operatorname{\mathbb{R}}^{d+1})-\operatorname{\mathbb{A}}_{1}\subset\bigcup_{j=1}^{n}\mathcal{O}_{p_{j}}.$

Proof of (1). Let $\delta:=\max\\{\delta_{p_{j}}:1\leq j\leq n\\}$. Since $R$
is positive,

$\displaystyle\max_{1\leq j\leq
n}\int_{\psi_{1}(\mathcal{O}_{p_{j}}\cap\operatorname{\mathbb{A}}_{1})}R^{-s}d\lambda$
$\displaystyle\leq\int_{\operatorname{\mathbb{R}}^{d}}R^{-s}d\lambda\leq\int_{\psi_{1}(\operatorname{\mathbb{A}}_{1}-\bigcup_{j=1}^{n}\mathcal{O}_{p_{j}})}R^{-s}d\lambda+\sum_{j=1}^{n}\int_{\psi_{1}(\mathcal{O}_{p_{j}}\cap\operatorname{\mathbb{A}}_{1})}R^{-s}d\lambda.$

Since $\operatorname{\mathbb{A}}_{1}-\bigcup_{j=1}^{n}\mathcal{O}_{p_{j}}$ is
compact, the integral
$\int_{\psi_{1}(\operatorname{\mathbb{A}}_{1}-\bigcup_{j=1}^{n}\mathcal{O}_{p_{j}})}R^{-s}d\lambda$
is finite, so Claim (1) of Lemma 6.2 implies that (1) holds.

Proof of (2). Let $\epsilon:=\min\\{\epsilon_{p_{j}}:1\leq j\leq n\\}$. Since
$R$ is positive and continuous on the compact set
$K:=\psi_{1}(\operatorname{\mathbb{A}}_{1}-\bigcup_{j=1}^{n}\mathcal{O}_{p_{j}})\subset\operatorname{\mathbb{R}}^{d}$,
there exists $c_{0}>0$ such that

$R(x)\geq c_{0}(1+\left\|x\right\|)^{\epsilon}$

for all $x\in K$. Thus, if we set
$c:=\min\left(\\{c_{0}\\}\cup\\{c_{p_{j}}:1\leq j\leq n\\}\right)$, then Claim
(2) of Lemma 6.2 implies that

$R(x)\geq c(1+\left\|x\right\|)^{\epsilon}$

for all $x\in\operatorname{\mathbb{R}}^{d}$. ∎

To prove Lemma 6.2, first note that we can assume that $p=[0:1:0:\dots:0]$ by
changing coordinates. Then $p$ lies in the affine subspace

$\operatorname{\mathbb{A}}_{2}:=\left\\{[x_{0}:\cdots:x_{d}]:x_{1}\neq
0\right\\}\subset\operatorname{\mathbb{P}}(\operatorname{\mathbb{R}}^{d+1}).$

Let $\psi_{2}:\operatorname{\mathbb{A}}_{2}\to\operatorname{\mathbb{R}}^{d}$
be the coordinate chart given by

$\psi_{2}:[y_{1}:1:y_{2}:\dots:y_{d}]\mapsto(y_{1},\dots,y_{d}).$

Observe that $\psi_{2}(p)=0$ and

$\displaystyle\psi_{2}(\operatorname{\mathbb{A}}_{2}-\operatorname{\mathbb{A}}_{1})=\mathcal{Z}:=\left\\{(y_{1},\dots,y_{d})\in\operatorname{\mathbb{R}}^{d}:y_{1}=0\right\\}.$
(3)

Thus, the restriction

$\psi_{1}\circ\psi_{2}^{-1}|_{\operatorname{\mathbb{R}}^{d}-\mathcal{Z}}:\operatorname{\mathbb{R}}^{d}-\mathcal{Z}\to\operatorname{\mathbb{R}}^{d}$

is a well-defined embedding.

Since
$\psi_{1}\circ\psi_{2}^{-1}|_{\operatorname{\mathbb{R}}^{d}-\mathcal{Z}}$ is
given by

$\displaystyle\psi_{1}\circ\psi_{2}^{-1}(y_{1},\dots,y_{d})=\left(\frac{1}{y_{1}},\frac{y_{2}}{y_{1}},\dots,\frac{y_{d}}{y_{1}}\right),$
(4)

for all $j\in\\{1,\dots,m\\}$,

$T_{j}:=R_{j}\circ\psi_{1}\circ\psi_{2}^{-1}:\operatorname{\mathbb{R}}^{d}-\mathcal{Z}\to\operatorname{\mathbb{R}},$

is a rational function that is well-defined and positive on
$\operatorname{\mathbb{R}}^{d}-\mathcal{Z}$, and the assumption that $R$ is
proper implies that $T:=T_{1}^{\ell_{1}}\dots T_{m}^{\ell_{m}}$ satisfies
$\lim_{y\to p}T(y)=\infty$ for all $p\in\mathcal{Z}$. Also, (4) implies that
the Jacobian $D(\psi_{1}\circ\psi_{2}^{-1})$ of $\psi_{1}\circ\psi_{2}^{-1}$
satisfies

$\left|\det
D(\psi_{1}\circ\psi_{2}^{-1})\right|=\frac{1}{\left|y_{1}\right|^{d+1}},$

and

$\left\|\psi_{1}\circ\psi_{2}^{-1}(y_{1},\ldots,y_{d})\right\|=\sqrt{\frac{1+y_{2}^{2}+\dots+y_{d}^{2}}{y_{1}^{2}}}.$

Thus, to prove Lemma 6.2, it now suffices to prove the following lemma.

######  Lemma 6.3.

Suppose that
$T_{1},\dots,T_{m}:\operatorname{\mathbb{R}}^{d}-\mathcal{Z}\to\operatorname{\mathbb{R}}$
are positive, rational functions, and let

$T:=T_{1}^{\ell_{1}}\dots T_{m}^{\ell_{m}}$

for some $\ell_{1},\dots,\ell_{m}\in\operatorname{\mathbb{R}}$. If $\lim_{y\to
p}T(y)=\infty$ for all $p\in\mathcal{Z}$, then there exist
$\delta,c,\epsilon>0$ and an open neighborhood
$\mathcal{O}\subset\operatorname{\mathbb{R}}^{d}$ of $0$ such that:

  1. (1)

The integral

$\int_{\mathcal{O}-\mathcal{Z}}\frac{T^{-s}}{\left|y_{1}\right|^{d+1}}{\rm
d}\lambda$

converges when $s\in(\delta,+\infty)$ and diverges when $s\in[0,\delta]$.

  2. (2)

For all $y=(y_{1},\dots,y_{d})\in\mathcal{O}-\mathcal{Z}$, we have

$T(y)\geq
c\left(1+\sqrt{\frac{1+y_{2}^{2}+\dots+y_{d}^{2}}{y_{1}^{2}}}\right)^{\epsilon}.$

For the remainder of this section, we will focus on the proof of Lemma 6.3. An
important tool used in the proof is the following version of Hironaka’s
theorem [23] on the resolution of singularities (as stated in [1, pg. 147]).

######  Theorem 6.4 (Resolution theorem).

Let $F$ be a real analytic function defined in a neighborhood of
$0\in\operatorname{\mathbb{R}}^{d}$, and let $\mathcal{Z}$ be the set of
zeroes of $F$. If $F$ is not identically zero, then there exists a
neighborhood $U$ of $0$ in $\operatorname{\mathbb{R}}^{d}$, a real analytic
manifold $M$ and a proper real analytic map $\Phi:M\rightarrow U$ such that

  1. (1)

$\Phi$ restricts to a real-analytic diffeomorphism

$M-\Phi^{-1}(\mathcal{Z})\rightarrow U-\mathcal{Z}.$

  2. (2)

For every $q\in M$ there exists an open neighborhood $V_{q}\subset M$ of $q$
and real analytic local coordinates $z_{q,1},\dots,z_{q,d}$ on $V_{q}$
centered at $q$ where

$F\circ\Phi(z_{q,1},\dots,z_{q,d})=z_{q,1}^{k_{q,1}}\cdots
z_{q,d}^{k_{q,d}}\cdot\hat{F}_{q}(z_{q,1},\dots,z_{q,d})$

for some $k_{q,1},\dots,k_{q,d}\in\operatorname{\mathbb{Z}}_{\geq 0}$ and some
nowhere-vanishing real analytic function
$\hat{F}_{q}:V_{q}\to\operatorname{\mathbb{R}}$.

The main idea of the proof of Claim (1) of Lemma 6.3 is to try to factor the
integrand $\frac{T^{-s}}{\left|y_{1}\right|^{d+1}}$ into the product of a
bounded, nowhere vanishing function on $\mathcal{O}$, and power functions on
$\mathcal{O}-\mathcal{Z}$ that are “responsible" for how
$\frac{T^{-s}}{\left|y_{1}\right|^{d+1}}$ goes to infinity or zero near
$\mathcal{Z}$. Doing so allows us to compare the required integral with the
integral of a product of power functions, whose convergence or lack thereof is
well-understood. Unfortunately, such a global factorization of
$\frac{T^{-s}}{\left|y_{1}\right|^{d+1}}$ on $\mathcal{O}$ is not possible in
general. However, Theorem 6.4 ensures that we do have such a factorization
locally, but at the cost of pre-composing the integrand with a given real
analytic function $\Phi$. We will show that the complications introduced to
this strategy by using $\Phi$ are surmountable, and that it can indeed be used
to prove Lemma 6.3.

We first write, for each $j\in\\{1,\dots,n\\}$, the rational function $T_{j}$
as

$T_{j}=\frac{f_{j}}{g_{j}}$

where $f_{j},g_{j}:\operatorname{\mathbb{R}}^{d}\to\operatorname{\mathbb{R}}$
are polynomials whose zeroes lie in the set $\mathcal{Z}$ defined by (3). Then

$\displaystyle F:=f_{1}\cdots f_{m}\cdot g_{1}\cdots g_{m}\cdot
y_{1}:\operatorname{\mathbb{R}}^{d}\to\operatorname{\mathbb{R}},$ (5)

is a polynomial whose set of zeroes is $\mathcal{Z}$. Applying Theorem 6.4 to
this $F$, we get an open neighborhood $U$ of $0\in\mathbb{R}^{d}$, a real
analytic manifold $M$, a proper real analytic map $\Phi:M\to U$ such that for
all $q\in M$, there exists an open neighborhood $V_{q}$ of $q$ with local
coordinates $\\{z_{q,r}\\}_{r=1}^{d}$ centered at $q$ such that
$F\circ\Phi|_{V_{q}}=z_{q,1}^{k_{q,1}}\cdots
z_{q,d}^{k_{q,d}}\cdot\hat{F}_{q}$ and $\hat{F}_{q}:V_{q}\to\mathbb{R}$ is a
nowhere-vanishing real analytic function. Notice that, via the local
coordinates $(z_{q,1},\dots,z_{q,d})$ on $V_{q}$, the Lebesgue measure
$\lambda$ on $\operatorname{\mathbb{R}}^{d}$ induces a measure $\lambda_{q}$
on $V_{q}$.

Fix a neighborhood $\mathcal{O}\subset\operatorname{\mathbb{R}}^{d}$ of $0$
whose closure is compact and lies in $U$. Also, for each $q\in M$, fix an open
neighborhood $V_{q}^{\prime}$ of $q$ whose closure lies in $V_{q}$. Since
$\Phi$ is proper, we can find finitely many points $q_{1},\dots,q_{n}\in M$
such that

$\displaystyle\Phi^{-1}(\mathcal{O})\subset\bigcup_{i=1}^{n}V_{q_{i}}^{\prime}.$

For all $i\in\\{1,\dots,n\\}$ and $r\in\\{1,\dots,d\\}$, set

$V_{i}:=V_{q_{i}}^{\prime}\cap\Phi^{-1}(\mathcal{O})\subset
V_{q_{i}},\quad\hat{F}_{i}:=\hat{F}_{q_{i}}:V_{q_{i}}\rightarrow\operatorname{\mathbb{R}},\quad
z_{i,r}:=z_{q_{i},r},\quad
k_{i,r}:=k_{q_{i},r}\quad\text{and}\quad\lambda_{i}:=\lambda_{q_{i}}.$

Notice that by construction,

$\overline{V_{i}}\subset V_{q_{i}}$ (6)

is compact, and

$\displaystyle\mathcal{O}=\bigcup_{i=1}^{n}\Phi(V_{i}).$ (7)

######  Remark 6.5.

In the above set up, one might be tempted to say that by shrinking
$\mathcal{O}$, one can find some $q\in M$ such that
$\mathcal{O}\subset\Phi(V_{q}^{\prime})$. However, this might not be possible:
Each $\Phi(V_{q})\subset U$ might not contain any open neighborhoods of
$\Phi(q)$ even if $V_{q}\subset M$ is an open neighborhood of $q$.

Using the sets $V_{i}$ and the map $\Phi$, we have the following local
criterion for when the integral

$\int_{\mathcal{O}-\mathcal{Z}}\frac{T^{-s}}{\left|y_{1}\right|^{d+1}}{\rm
d}\lambda$

converges.

######  Lemma 6.6.

For any $s\geq 0$, the integral

$\int_{\mathcal{O}-\mathcal{Z}}\frac{T^{-s}}{\left|y_{1}\right|^{d+1}}{\rm
d}\lambda$

converges if and only if the integral

$\int_{V_{i}-\Phi^{-1}(\mathcal{Z})}\frac{(T\circ\Phi)^{-s}}{\left|y_{1}\circ\Phi\right|^{d+1}}\left|\det
D(\Phi)\right|{\rm d}\lambda_{i}$

converges for all $i\in\\{1,\dots,n\\}$. Here, $D(\Phi)$ is the Jacobian of
$\Phi$ restricted to $V_{q_{i}}$, with respect to the local coordinates
$(z_{i,1},\dots,z_{i,d})$ of $V_{q_{i}}$.

###### Proof.

By (7),

$\displaystyle\max_{i=1,\dots,n}\int_{\Phi(V_{i})-\mathcal{Z}}\frac{T^{-s}}{\left|y_{1}\right|^{d+1}}{\rm
d}\lambda\leq\int_{\mathcal{O}-\mathcal{Z}}\frac{T^{-s}}{\left|y_{1}\right|^{d+1}}{\rm
d}\lambda\leq\sum_{i=1}^{n}\int_{\Phi(V_{i})-\mathcal{Z}}\frac{T^{-s}}{\left|y_{1}\right|^{d+1}}{\rm
d}\lambda.$

Since $\Phi$ restricts to a diffeomorphism
$M-\Phi^{-1}(\mathcal{Z})\rightarrow U-\mathcal{Z}$, we have

$\int_{V_{i}-\Phi^{-1}(\mathcal{Z})}\frac{(T\circ\Phi)^{-s}}{\left|y_{1}\circ\Phi\right|^{d+1}}\left|\det
D(\Phi)\right|{\rm
d}\lambda_{i}=\int_{\Phi(V_{i})-\mathcal{Z}}\frac{T^{-s}}{\left|y_{1}\right|^{d+1}}{\rm
d}\lambda$

for each $i\in\\{1,\dots,n\\}$. The lemma follows. ∎

In light of Lemma 6.6, we now need to understand for which values of $s\geq 0$
the integral

$\int_{V_{i}-\Phi^{-1}(\mathcal{Z})}\frac{(T\circ\Phi)^{-s}}{\left|y_{1}\circ\Phi\right|^{d+1}}\left|\det
D(\Phi)\right|{\rm d}\lambda_{i}$

converges. We do so using the local expressions of the real analytic functions
$f_{j}\circ\Phi|_{V_{q_{i}}}$, $g_{j}\circ\Phi|_{V_{q_{i}}}$,
$y_{1}\circ\Phi|_{V_{q_{i}}}$ and $\det D(\Phi)|_{V_{q_{i}}}$. By factoring
their Taylor series in the local coordinates $(z_{i,1},\dots,z_{i,d})$ on
$V_{q_{i}}$, we can write

  * •

$f_{j}\circ\Phi|_{V_{q_{i}}}=z_{i,1}^{a_{i,j,1}}\cdots
z_{i,d}^{a_{i,j,d}}\cdot\hat{f}_{i,j}$ where
$a_{i,j,1},\dots,a_{i,j,d}\in\operatorname{\mathbb{Z}}_{\geq 0}$ and
$\hat{f}_{i,j}:V_{q_{i}}\to\operatorname{\mathbb{R}}$ is not identically zero
on

$\mathcal{Z}_{i,r}:=\\{(z_{i,1},\dots,z_{i,d})\in V_{q_{i}}:z_{i,r}=0\\}$

for any $r\in\\{1,\dots,d\\}$.

  * •

$g_{j}\circ\Phi|_{V_{q_{i}}}=z_{i,1}^{b_{i,j,1}}\cdots
z_{i,d}^{b_{i,j,d}}\cdot\hat{g}_{i,j}$ where
$b_{i,j,1},\dots,b_{i,j,d}\in\operatorname{\mathbb{Z}}_{\geq 0}$ and
$\hat{g}_{i,j}:V_{q_{i}}\to\operatorname{\mathbb{R}}$ is not identically zero
on $\mathcal{Z}_{i,r}$ for any $r\in\\{1,\dots,d\\}$.

  * •

$y_{1}\circ\Phi|_{V_{q_{i}}}=z_{i,1}^{c_{i,1}}\cdots
z_{i,d}^{c_{i,d}}\cdot\hat{h}_{i}$ where
$c_{i,1},\dots,c_{i,d}\in\operatorname{\mathbb{Z}}_{\geq 0}$ and
$\hat{h}_{i}:V_{q_{i}}\to\operatorname{\mathbb{R}}$ is not identically zero on
$\mathcal{Z}_{i,r}$ for any $r\in\\{1,\dots,d\\}$.

  * •

$\det D(\Phi)|_{V_{q_{i}}}=z_{i,1}^{\gamma_{i,1}}\cdots
z_{i,d}^{\gamma_{i,d}}\cdot\hat{J}_{i}$, where
$\gamma_{i,1},\dots,\gamma_{i,d}\in\operatorname{\mathbb{Z}}_{\geq 0}$ and
$\hat{J}_{i}:V_{q_{i}}\to\operatorname{\mathbb{R}}$ is not identically zero on
$\mathcal{Z}_{i,r}$ for any $r\in\\{1,\dots,d\\}$.

Using Theorem 6.4, we deduce the following lemma about the functions
$\hat{f}_{i,j}$, $\hat{g}_{i,j}$, $\hat{h}_{i}$ and

$\hat{W}_{i,s}:=\left(\frac{\hat{g}_{i,1}^{\ell_{1}}\cdots\hat{g}_{i,m}^{\ell_{m}}}{\hat{f}_{i,1}^{\ell_{1}}\cdots\hat{f}_{i,m}^{\ell_{m}}}\right)^{s}\frac{\hat{J}_{i}}{\hat{h}_{i}^{d+1}}$

for all $s\geq 0$.

######  Lemma 6.7.

Fix $i\in\\{1,\dots,n\\}$.

  1. (1)

$\hat{F}_{i}=\hat{f}_{i,1}\cdots\hat{f}_{i,m}\cdot\hat{g}_{i,1}\cdots\hat{g}_{i,m}\cdot\hat{h}_{i}$.
In particular, $\hat{f}_{i,1}$, $\dots$, $\hat{f}_{i,m}$, $\hat{g}_{i,1}$,
$\dots$, $\hat{g}_{i,m}$ and $\hat{h}_{i}$ are all nowhere-vanishing on
$V_{q_{i}}$.

  2. (2)

For any $s\geq 0$, $\hat{W}_{i,s}$ is a real analytic function which is
bounded on $V_{i}$ and not identically zero on $\mathcal{Z}_{i,r}$ for any
$r\in\\{1,\dots,d\\}$.

###### Proof.

Proof of (1). Observe that

$\displaystyle\frac{\hat{f}_{i,1}\cdots\hat{f}_{i,m}\cdot\hat{g}_{i,1}\cdots\hat{g}_{i,m}\cdot\hat{h}_{i}}{\hat{F}_{i}}=\prod_{r=1}^{d}z_{r}^{k_{i,r}-c_{i,r}-\sum_{j=1}^{m}(a_{i,j,r}+b_{i,j,r})}.$

By Theorem 6.4, $\hat{F}_{i}$ is nowhere vanishing, so the left hand side is
finite at every point in $V_{q_{i}}$. Also, by the definition of
$\hat{f}_{i,j}$, $\hat{g}_{i,j}$ and $\hat{h}_{i}$, the left hand side is not
identically zero on $\mathcal{Z}_{i,r}$ for all $r\in\\{1,\dots,d\\}$. Hence
we must have

$k_{i,r}-c_{i,r}-\sum_{j=1}^{m}(a_{i,j,r}+b_{i,j,r})=0$

for all $r\in\\{1,\dots,d\\}$. So
$\hat{F}_{i}=\hat{f}_{i,1}\cdots\hat{f}_{i,m}\cdot\hat{g}_{i,1}\cdots\hat{g}_{i,m}\cdot\hat{h}_{i}$.

Proof of (2). By (1) and the definition of $\hat{J}_{i}$, the function
$\hat{W}_{i,s}$ is a real analytic function on $V_{q_{i}}$ which is not
identically zero on $\mathcal{Z}_{i,r}$ for any $r\in\\{1,\dots,d\\}$. Since
$\overline{V_{i}}\subset V_{q_{i}}$ is a compact subset (see Equation (6)),
the function $\hat{W}_{i,s}$ is bounded on $V_{i}$. ∎

For all $i\in\\{1,\dots,n\\}$ and $r\in\\{1,\dots,d\\}$, set

$\beta_{i,r}:=\sum_{j=1}^{m}\ell_{j}(b_{i,j,r}-a_{i,j,r}).$

Since $T$ is positive on $\operatorname{\mathbb{R}}^{d}-\mathcal{Z}$, we can
take the absolute value of each term to conclude that

$\displaystyle\frac{(T\circ\Phi)^{-s}}{\left|y_{1}\circ\Phi\right|^{d+1}}\left|\det
D(\Phi)\right|$
$\displaystyle=\left|\hat{W}_{i,s}\right|\prod_{r=1}^{d}|z_{i,r}|^{s\beta_{i,r}-(d+1)c_{i,r}+\gamma_{i,r}}.$
(8)

The next lemma characterizes when integrals of functions with the above form
converge.

######  Lemma 6.8.

Suppose $i\in\\{1,\dots,n\\}$ and
$W:V_{i}\rightarrow\operatorname{\mathbb{R}}$ is a bounded real analytic
function which is not identically zero on the hyperplane $\mathcal{Z}_{i,r}$
for any $r\in\\{1,\dots,d\\}$. Then for
$\eta_{1},\dots,\eta_{d}\in\operatorname{\mathbb{R}}$ the integral

$\int_{V_{i}-\Phi^{-1}(\mathcal{Z})}\left|z_{i,1}\right|^{\eta_{1}}\cdots\left|z_{i,d}\right|^{\eta_{d}}\left|W\right|{\rm
d}\lambda_{i}$

converges if and only if $\eta_{r}>-1$ for all $r\in\\{1,\dots,d\\}$. In
particular,

$\int_{\mathcal{O}-\mathcal{Z}}\frac{T^{-s}}{\left|y_{1}\right|^{d+1}}{\rm
d}\lambda$

converges if and only if $s\beta_{i,r}-(d+1)c_{i,r}+\gamma_{i,r}>-1$ for all
$i\in\\{1,\dots,n\\}$ and all $r\in\\{1,\dots,d\\}$.

###### Proof.

Since $W$ is bounded, it is clear that if $\eta_{r}>-1$ for all
$r\in\\{1,\dots,d\\}$, then the integral converges.

For the other direction, it suffices to assume that $\eta_{1}\leq-1$ and then
show that the integral diverges. Since $W$ is not identically zero on
$\mathcal{Z}_{i,1}$, we can find an open set
$V^{\prime}\subset\mathcal{Z}_{i,1}$ such that $V^{\prime}\subset V_{i}$ and
$W$ is nowhere vanishing on $V^{\prime}$. By further shrinking $V^{\prime}$ we
can assume that there exists $\epsilon>0$ such that
$\overline{(0,\epsilon)\times V^{\prime}}\subset V_{i}$ and

$\overline{V^{\prime}}\cap\bigcup_{r=2}^{d}\mathcal{Z}_{i,r}=\emptyset.$

Then

$\alpha:=\min_{z\in\overline{(0,\epsilon)\times
V^{\prime}}}\left|z_{i,2}\right|^{\eta_{2}}\cdots\left|z_{i,d}\right|^{\eta_{d}}\left|W\right|$

is positive.

Since $\mathcal{Z}$ is the set of zeroes of $F$, the local formula for
$F\circ\Phi$ given by Theorem 6.4 implies that $\Phi^{-1}(\mathcal{Z})\cap
V_{i}\subset\bigcup_{r=1}^{d}\mathcal{Z}_{i,r}$. Hence,

$\displaystyle\int_{V_{i}-\Phi^{-1}(\mathcal{Z})}$
$\displaystyle\left|z_{i,1}\right|^{\eta_{1}}\cdots\left|z_{i,d}\right|^{\eta_{d}}\left|W\right|{\rm
d}\lambda_{i}\geq\int_{(0,\epsilon)\times
V^{\prime}}\left|z_{i,1}\right|^{\eta_{1}}\cdots\left|z_{i,d}\right|^{\eta_{d}}\left|W\right|{\rm
d}\lambda_{i}$ $\displaystyle\geq\alpha{\rm
Vol}(V^{\prime})\int_{0}^{\epsilon}t^{\eta_{1}}dt=+\infty$

where ${\rm Vol}$ is the measure on $\mathcal{Z}_{i,1}$ induced by the
coordinates $(z_{q,2},\dots,z_{q,d})$ on $\mathcal{Z}_{i,1}$ and the Lebesgue
measure on $\operatorname{\mathbb{R}}^{d-1}$.

To prove the second statement of the lemma, first note that by Lemma 6.7 Claim
(2), $\hat{W}_{i,s}$ is a bounded real analytic function on $V_{i}$ and not
identically zero on the hyperplane $\mathcal{Z}_{i,r}$ for any
$r\in\\{1,\dots,d\\}$. So, we may apply the first statement of the lemma to
Equation (8) and deduce that

$\int_{V_{i}-\Phi^{-1}(\mathcal{Z})}\frac{(T\circ\Phi)^{-s}}{\left|y_{1}\circ\Phi\right|^{d+1}}\left|\det
D(\varphi_{i})\right|{\rm d}\lambda_{i}$

converges if and only if

$s\beta_{i,r}-(d+1)c_{i,r}+\gamma_{i,r}>-1$

for all $r\in\\{1,\dots,d\\}$. The second statement of the lemma now follows
from Lemma 6.6. ∎

By Lemma 6.8, we now need to know the values of $s$ so that
$s\beta_{i,r}-(d+1)c_{i,r}+\gamma_{i,r}>-1$ for all $i\in\\{1,\dots,n\\}$ and
all $r\in\\{1,\dots,d\\}$. To that end, it is useful to have the following
relations between the exponents $\beta_{i,r}$, $c_{i,r}$ and $\gamma_{i,r}$.

######  Lemma 6.9.

  1. (1)

For all $i\in\\{1,\dots,n\\}$ and all $r\in\\{1,\dots,d\\}$, $\beta_{i,r}\geq
0$.

  2. (2)

For all $i\in\\{1,\dots,n\\}$,
$\\{r:c_{i,r}>0\\}\subset\\{r:\beta_{i,r}>0\\}$.

  3. (3)

There exist $i\in\\{1,\dots,n\\}$ and $r\in\\{1,\dots,d\\}$ such that

$c_{i,r}-\gamma_{i,r}\geq 1.$

In particular,

$\delta:=\max\left\\{\frac{(d+1)c_{i,r}-\gamma_{i,r}-1}{\beta_{i,r}}:i\in\\{1,\dots,n\\},\,r\in\\{1,\dots,d\\}\text{
such that }\beta_{i,r}>0\right\\}$

is a positive real number.

###### Proof.

Proof of (1). Since $T$ is positive on $U-\mathcal{Z}$ and $\lim_{y\to
p}T(y)=\infty$ for all $p\in\mathcal{Z}$, the function

$\frac{1}{T\circ\Phi}:M-\Phi^{-1}(\mathcal{Z})\to\operatorname{\mathbb{R}}$

extends to a continuous function on all of $M$. In particular, for all
$i\in\\{1,\dots,n\\}$,

$\frac{1}{T\circ\Phi}\bigg{|}_{V_{i}}=z_{i,1}^{\beta_{i,1}}\cdots
z_{i,d}^{\beta_{i,d}}\frac{\hat{g}_{i,1}^{\ell_{1}}\cdots\hat{g}_{i,m}^{\ell_{m}}}{\hat{f}_{i,1}^{\ell_{1}}\cdots\hat{f}_{i,m}^{\ell_{m}}}$
(9)

is finite. By Claim (1) of Lemma 6.7 and the fact that
$\overline{V_{i}}\subset V_{q_{i}}$ is compact, we see that
$\frac{\hat{g}_{i,1}^{\ell_{1}}\cdots\hat{g}_{i,m}^{\ell_{m}}}{\hat{f}_{i,1}^{\ell_{1}}\cdots\hat{f}_{i,m}^{\ell_{m}}}$
is bounded and nowhere vanishing on $V_{i}$, so (1) follows.

Proof of (2). Since $\lim_{y\to p}T(y)=\infty$ for all $p\in\mathcal{Z}$, it
follows that $\frac{1}{T}$ vanishes on $\mathcal{Z}$. Equivalently,
$\frac{1}{T\circ\Phi}$ vanishes wherever $y_{1}\circ\Phi$ vanishes. By Lemma
6.7 Claim (1) and the fact that $\overline{V_{i}}\subset V_{q_{i}}$ is
compact, $\hat{h}_{i}$ and
$\frac{\hat{g}_{i,1}^{\ell_{1}}\cdots\hat{g}_{i,m}^{\ell_{m}}}{\hat{f}_{i,1}^{\ell_{1}}\cdots\hat{f}_{i,m}^{\ell_{m}}}$
are bounded and nowhere vanishing on $V_{i}$, so (2) follows from Equation
(9).

Proof of (3). Observe that the integral

$\int_{\mathcal{O}-\mathcal{Z}}\frac{1}{\left|y_{1}\right|}{\rm d}\lambda$

diverges. By (7),

$\int_{\mathcal{O}-\mathcal{Z}}\frac{1}{\left|y_{1}\right|}{\rm
d}\lambda\leq\sum_{i=1}^{n}\int_{V_{i}-\Phi^{-1}(\mathcal{Z})}\frac{\left|\det
D(\Phi)\right|}{|y_{1}\circ\Phi|}{\rm
d}\lambda_{i}=\sum_{i=1}^{n}\int_{V_{i}-\Phi^{-1}(\mathcal{Z})}\left(\prod_{r=1}^{d}\left|z_{i,r}\right|^{-c_{i,r}+\gamma_{i,r}}\right)\left|\frac{\hat{J}_{i}}{\hat{h}_{i}}\right|{\rm
d}\lambda_{i},$

so there is some $i\in\\{1,\dots,n\\}$ such that the integral

$\int_{V_{i}-\Phi^{-1}(\mathcal{Z})}\left(\prod_{r=1}^{d}\left|z_{i,r}\right|^{-c_{i,r}+\gamma_{i,r}}\right)\left|\frac{\hat{J}_{i}}{\hat{h}_{i}}\right|{\rm
d}\lambda_{i}$

diverges. By Lemma 6.7 Claim (1), $\frac{\hat{J}_{i}}{\hat{h}_{i}}$ is real
analytic on $V_{q_{i}}$. Also, by the definition of $\hat{J}_{i}$,
$\frac{\hat{J}_{i}}{\hat{h}_{i}}$ is not identically zero on
$\mathcal{Z}_{i,r}$ for all $r\in\\{1,\dots,d\\}$. Since
$\overline{V_{i}}\subset V_{q_{i}}$ is compact, the function
$\frac{\hat{J}_{i}}{\hat{h}_{i}}$ is bounded on $V_{i}$. So by Lemma 6.8 there
is some $r\in\\{1,\dots,d\\}$ such that $-c_{i,r}+\gamma_{i,r}\leq-1$. This
proves (3).

We will now deduce the final claim of the lemma. By (3), there is some
$i\in\\{1,\dots,n\\}$ and $r\in\\{1,\dots,d\\}$ such that
$c_{i,r}-\gamma_{i,r}\geq 1$. Since $\gamma_{i,r}\geq 0$, it follows that
$c_{i,r}>0$, so (2) implies that $\beta_{i,r}>0$. Then

$\frac{(d+1)c_{i,r}-\gamma_{i,r}-1}{\beta_{i,r}}>0,$

which implies that $\delta>0$. ∎

Combining Lemmas 6.8 and 6.9, we may now prove Lemma 6.3.

###### Proof of Lemma 6.3.

Proof of (1). Let $\delta>0$ be the quantity specified in the statement of
Lemma 6.9.

Suppose that $s\in(\delta,+\infty)$. Pick any $i\in\\{1,\dots,n\\}$ and
$r\in\\{1,\dots,d\\}$. If $\beta_{i,r}=0$, then Lemma 6.9 Claim (2) implies
that $c_{i,r}=0$, in which case

$s\beta_{i,r}-(d+1)c_{i,r}+\gamma_{i,r}=\gamma_{i,r}\geq 0>-1.$

If $\beta_{i,r}\neq 0$, then
$s>\delta\geq\frac{(d+1)c_{i,r}-\gamma_{i,r}-1}{\beta_{i,r}}$. By Lemma 6.9
Claim (1), $\beta_{i,r}>0$, so

$s\beta_{i,r}-(d+1)c_{i,r}+\gamma_{i,r}>-1.$

It now follows from Lemma 6.8 that

$\int_{\mathcal{O}-\mathcal{Z}}\frac{T^{-s}}{\left|y_{1}\right|^{d+1}}{\rm
d}\lambda$

converges.

Next, suppose that $s\in[0,\delta]$. Then by definition, there is some
$i\in\\{1,\dots,n\\}$ and $r\in\\{1,\dots,d\\}$ such that

$s\beta_{i,r}-(d+1)c_{i,r}+\gamma_{i,r}\leq-1.$

Then by Lemma 6.8,

$\int_{\mathcal{O}-\mathcal{Z}}\frac{T^{-s}}{\left|y_{1}\right|^{d+1}}{\rm
d}\lambda$

diverges. This completes the proof of (1).

Proof of (2). Recall that

$\frac{1}{T\circ\Phi}\bigg{|}_{V_{i}}=z_{i,1}^{\beta_{i,1}}\cdots
z_{i,d}^{\beta_{i,d}}R$

where
$R=\frac{\hat{g}_{i,1}^{\ell_{1}}\cdots\hat{g}_{i,m}^{\ell_{m}}}{\hat{f}_{i,1}^{\ell_{1}}\cdots\hat{f}_{i,m}^{\ell_{m}}}$
is a nowhere vanishing analytic function (by Lemma 6.7) and that

$y_{1}\circ\Phi|_{V_{q_{i}}}=z_{i,1}^{c_{i,1}}\cdots
z_{i,d}^{c_{i,d}}\cdot\hat{h}_{i}$

where $\hat{h}_{i}$ is a nowhere vanishing analytic function (again by Lemma
6.7). Since $\overline{V_{i}}\subset V_{q_{i}}$ is compact, $\beta_{i,r}\geq
0$ for all $r$ and $\beta_{i,r}=0$ whenever $c_{i,r}=0$ (by Lemma 6.9), there
exist $C_{i},\epsilon_{i}>0$ such that

$\displaystyle\frac{1}{T\circ\Phi(z)}\leq
C_{i}\left|y_{1}\circ\Phi(z)\right|^{\epsilon_{i}}$ (10)

for all $z\in V_{i}$. Set

$\epsilon:=\max\\{\epsilon_{1},\dots,\epsilon_{n}\\}.$

Since the closure of $\mathcal{O}$ is compact, there exists $c_{0}>0$ such
that

$1\geq
c_{0}\left(\left|y_{1}\right|+\sqrt{1+y_{2}^{2}+\dots+y_{d}^{2}}\right)$

for all $y\in\mathcal{O}$. Hence

$\frac{1}{\left|y_{1}\right|}\geq
c_{0}\left(1+\sqrt{\frac{1+y_{2}^{2}+\dots+y_{d}^{2}}{y_{1}^{2}}}\right)$ (11)

for all $y\in\mathcal{O}-\mathcal{Z}$. Set

$c:=\min\left\\{\frac{c_{0}^{\epsilon_{1}}}{C_{1}},\dots,\frac{c_{0}^{\epsilon_{n}}}{C_{n}}\right\\}.$

Fix $y\in\mathcal{O}-\mathcal{Z}$. Then there exist $i\in\\{1,\dots,m\\}$ and
$z\in V_{i}$ such that $y=\Phi(z)$. Then

$T(y)\geq\frac{1}{C_{i}\left|y_{1}\right|^{\epsilon_{i}}}\geq
c\left(1+\sqrt{\frac{1+y_{2}^{2}+\dots+y_{d}^{2}}{y_{1}^{2}}}\right)^{\epsilon},$

where the first inequality holds by Equation (10) and the second inequality
holds by Equation (11).

∎

##  7\. Entropy gap for peripheral subgroups

In this section we prove that the Poincaré series associated to any peripheral
subgroup diverges at its critical exponent.

######  Theorem 7.1 (Theorem 1.10).

Suppose $\Gamma\subset\mathsf{G}$ is a
$\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative to
$\mathcal{P}$, $\phi\in\mathfrak{a}_{\theta}^{*}$ and
$\delta^{\phi}(\Gamma)<+\infty$. If $P\in\mathcal{P}$, then $Q_{P}^{\phi}$
diverges at its critical exponent.

Delaying the proof of Theorem 7.1 for a moment, we observe that it implies
that the critical exponent of the peripheral subgroup is strictly smaller than
the critical exponent of the entire group.

######  Corollary 7.2.

Suppose $\Gamma\subset\mathsf{G}$ is a
$\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative to
$\mathcal{P}$, $\phi\in\mathfrak{a}_{\theta}^{*}$ and
$\delta^{\phi}(\Gamma)<+\infty$. If $P\in\mathcal{P}$, then
$\delta^{\phi}(P)<\delta^{\phi}(\Gamma)$.

###### Proof.

Notice that $\Lambda_{\theta}(P)$ consists of a single point, namely the fixed
point of $P$ in $\Lambda_{\theta}(\Gamma)$. Hence Theorems 1.5 and 7.1 imply
that $\delta^{\phi}(P)<\delta^{\phi}(\Gamma)$. ∎

The rest of the section is devoted to the proof of Theorem 7.1, so fix
$\Gamma$, $\mathcal{P}$ and $\phi$ as in the statement of the theorem.

Let $p:\mathsf{G}\rightarrow\mathsf{G}^{\prime}$ and
$\theta^{\prime}\subset\Delta^{\prime}$ be as in Proposition 4.5. By part (4)
of that proposition, $\Gamma^{\prime}:=p(\Gamma)$ is a
$\operatorname{\mathsf{P}}^{\prime}_{\theta^{\prime}}$-Anosov subgroup
relative to $\mathcal{P}^{\prime}:=\\{p(P):P\in\mathcal{P}\\}$. Also let
$\phi^{\prime}\in({\mathfrak{a}}_{\theta^{\prime}}^{\prime})^{*}$ be the
unique functional where $\phi^{\prime}\circ{\rm d}p=\phi$. Then
$Q_{p(P)}^{\phi^{\prime}}=Q_{P}^{\phi}$ for all $P\in\mathcal{P}$. So by
replacing $\mathsf{G}$ with $\mathsf{G}^{\prime}$, we may assume that
$\mathsf{G}$ has trivial center, and that $\operatorname{\mathsf{P}}_{\theta}$
contains no simple factors of $\mathsf{G}$.

Fix $P\in\mathcal{P}$. Then by Theorem 4.4 there exists a closed subgroup
$\mathsf{H}\subset\mathsf{G}$ with finitely many components such that:

  1. (1)

$P$ is a cocompact lattice in $\mathsf{H}$.

  2. (2)

$\mathsf{H}=\mathsf{L}\ltimes\mathsf{U}$ where $\mathsf{L}$ is compact and
$\mathsf{U}$ is the unipotent radical of $\mathsf{H}$.

  3. (3)

$\mathsf{H}^{0}=\mathsf{L}^{0}\times\mathsf{U}$ and $\mathsf{L}^{0}$ is
Abelian.

Let $\mathfrak{u}$ denote the Lie algebra of $\mathsf{U}$.

Since $\omega_{\alpha}(\kappa_{\theta}(g))=\omega_{\alpha}(\kappa(g))$ for all
$\alpha\in\theta$ and $g\in\mathsf{G}$, by Proposition 2.3, for any
$\alpha\in\theta$ there exist $M_{\alpha}\in\operatorname{\mathbb{N}}$,
$C_{\alpha}>1$ and a positive everywhere defined rational function
$R_{\alpha}:\mathfrak{u}\rightarrow\operatorname{\mathbb{R}}$ where

$\frac{1}{C_{\alpha}}R_{\alpha}(Y)^{1/M_{\alpha}}\leq
e^{\omega_{\alpha}(\kappa_{\theta}(\exp(Y)))}\leq
C_{\alpha}R_{\alpha}(Y)^{1/M_{\alpha}}$

for all $Y\in\mathfrak{u}$. Write
$\phi=\sum_{\alpha\in\theta}c_{\alpha}\omega_{\alpha}$. Then define
$R:=\prod_{\alpha\in\theta}R_{\alpha}^{\left|c_{\alpha}\right|/M_{\alpha}}$
and $C_{\phi}:=\prod_{\alpha\in\theta}C_{\alpha}^{\left|c_{\alpha}\right|}$.
Note that

$C_{\phi}^{-s}R^{-s}(Y)\leq e^{-s\phi(\kappa_{\theta}(\exp(Y)))}\leq
C_{\phi}^{s}R^{-s}(Y)$

for all $s\in\operatorname{\mathbb{R}}$.

######  Lemma 7.3.

$R:\mathfrak{u}\rightarrow\operatorname{\mathbb{R}}$ is proper.

###### Proof.

Suppose that $\\{Y_{n}\\}$ is an escaping sequence in $\mathfrak{u}$. Since
$\exp:\mathfrak{u}\rightarrow\mathsf{U}$ is a diffeomorphism,
$\\{\exp(Y_{n})\\}$ is an escaping sequence in $\mathsf{U}$ (see Proposition
2.3). Since $P$ is a cocompact lattice in $\mathsf{H}$, there exists an
escaping sequence $\\{\gamma_{n}\\}$ in $P$ such that
$\\{\gamma_{n}^{-1}\exp(Y_{n})\\}$ is relatively compact in $\mathsf{H}$. So
there exists $C_{0}>0$ such that

$\left\|\kappa\left(\gamma_{n}^{-1}\exp(Y_{n})\right)\right\|\leq C_{0}$

for all $n\geq 1$. Then by Lemma 2.1

$R(Y_{n})\geq\frac{1}{C_{\phi}}e^{\phi(\kappa_{\theta}(\exp(Y_{n})))}\geq\frac{1}{C_{\phi}}e^{\phi(\kappa_{\theta}(\gamma_{n}))}e^{-\left\|\phi\right\|C_{0}},$

where $\left\|\phi\right\|$ is the operator norm of the linear map
$\phi:\mathfrak{a}\to\operatorname{\mathbb{R}}$. Since
$\delta^{\phi}(\Gamma)<+\infty$ and $\\{\gamma_{n}\\}$ is an escaping
sequence, we must have $\phi(\kappa_{\theta}(\gamma_{n}))\rightarrow+\infty$.
Hence $R(Y_{n})\rightarrow+\infty$. So $R$ is proper. ∎

Fix a measure $\lambda$ on $\mathfrak{u}$ which is obtained by pulling back
the Lebesgue measure on $\operatorname{\mathbb{R}}^{\dim\mathfrak{u}}$ via
some linear isomorphism (notice that any two measures obtained this way are
scalar multiplies of each other). Then the push-forward $\mu:=\exp_{*}\lambda$
is a Haar measure on $\mathsf{U}$, see Proposition 2.3. By Theorem 6.1 there
exists $\delta>0$ such that

$Q_{R}(s):=\int_{\mathfrak{u}}R^{-s}d\lambda$

converges when $s\in(\delta,+\infty)$ and diverges when $s\in[0,\delta]$.
Hence to complete the proof of Theorem 7.1 it suffices to show the following.

######  Lemma 7.4.

There exists a continuous function $A:\operatorname{\mathbb{R}}_{\geq
0}\to\operatorname{\mathbb{R}}_{>0}$ such that

$\frac{1}{A(s)}Q_{P}^{\phi}(s)\leq Q_{R}(s)\leq A(s)Q_{P}^{\phi}(s)$

for all $s\geq 0$.

###### Proof.

We prove the lemma via a series of estimates. First let

$P_{0}:=\mathsf{H}^{0}\cap P=\left(\mathsf{L}^{0}\times\mathsf{U}\right)\cap
P.$

Since $\mathsf{H}$ has finitely many connected components, $P_{0}$ has finite
index in $P$. Let $\gamma_{1},\dots,\gamma_{n}\in P$ such that
$P/P_{0}=\\{\gamma_{1}P_{0},\dots,\gamma_{n}P_{0}\\}$, and let

$D:=\left\|\phi\right\|\max_{i=1,\dots,n}\left\|\kappa(\gamma_{i})\right\|,$

where $\left\|\phi\right\|$ is the operator norm of the linear map
$\phi:\mathfrak{a}_{\theta}\to\operatorname{\mathbb{R}}$. Then by Lemma 2.1,

$\displaystyle\frac{e^{-Ds}}{n}Q_{P}^{\phi}(s)\leq Q_{P_{0}}^{\phi}(s)\leq
Q_{P}^{\phi}(s)$ (12)

for all $s\geq 0$.

Next, let $\pi:\mathsf{L}^{0}\times\mathsf{U}\rightarrow\mathsf{U}$ denote the
projection and let $P_{1}:=\pi(P_{0})$. Since $P_{0}$ is discrete and
$\mathsf{L}^{0}$ is compact, the kernel of $\pi|_{P_{0}}$ is finite and
$P_{1}$ is discrete. Then by Lemma 2.1,

$\displaystyle\frac{e^{-Cs}}{B}Q_{P_{0}}^{\phi}(s)\leq Q_{P_{1}}^{\phi}(s)\leq
Q_{P_{0}}^{\phi}(s)$ (13)

for all $s\geq 0$, where $B:=\\#\ker(\pi|_{P_{0}})$ and

$C:=\left\|\phi\right\|\max_{g\in\mathsf{L}^{0}}\left\|\kappa(g)\right\|.$

By (12) and (13), it now suffices to show that there exists a continuous
function $A:\operatorname{\mathbb{R}}_{\geq
0}\to\operatorname{\mathbb{R}}_{>0}$ such that

$\displaystyle\frac{1}{A(s)}Q_{P_{1}}^{\phi}(s)\leq Q_{R}(s)\leq
A(s)Q_{P_{1}}^{\phi}(s)$ (14)

for all $s\geq 0$.

Since $P$ is a cocompact lattice in $\mathsf{H}$, we see that $P_{1}$ is a
cocompact lattice in $\mathsf{U}$. Fix compact neighborhoods
$K_{1},K_{2}\subset\mathsf{U}$ of the identity such that the left
$P_{1}$-translates of $K_{1}$ are pairwise disjoint and the left
$P_{1}$-translates of $K_{2}$ cover $\mathsf{U}$. Since $K_{1}$ and $K_{2}$
have non-empty interior, both $\mu(K_{1})$ and $\mu(K_{2})$ are positive. For
both $i=1,2$, define

$E_{i}:=\left\|\phi\right\|\max_{h\in K_{i}}\left\|\kappa(h)\right\|.$

Then by Lemma 2.1,

$e^{-sE_{i}}e^{-s\phi(\kappa_{\theta}(g))}\leq
e^{-s\phi(\kappa_{\theta}(\gamma))}\leq
e^{sE_{i}}e^{-s\phi(\kappa_{\theta}(g))}$

for all $\gamma\in P_{1}$, $g\in\gamma K_{i}$ and $s\geq 0$. So

$\frac{e^{-sE_{2}}}{\mu(K_{2})}\int_{\gamma
K_{2}}e^{-s\phi(\kappa_{\theta}(g))}d\mu(g)\leq
e^{-s\phi(\kappa_{\theta}(\gamma))}\leq\frac{e^{sE_{1}}}{\mu(K_{1})}\int_{\gamma
K_{1}}e^{-s\phi(\kappa_{\theta}(g))}d\mu(g)$

for all $\gamma\in P_{1}$ and $s>0$. Then

$\displaystyle Q_{R}(s)$
$\displaystyle=\int_{\mathfrak{u}}R^{-s}(Y)d\lambda(Y)\leq
C_{\phi}^{s}\int_{\mathfrak{u}}e^{-s\phi(\kappa_{\theta}(\exp(Y)))}d\lambda(Y)=C_{\phi}^{s}\int_{\mathsf{U}}e^{-s\phi(\kappa_{\theta}(g))}d\mu(g)$
$\displaystyle\leq C_{\phi}^{s}\sum_{\gamma\in P_{1}}\int_{\gamma
K_{2}}e^{-s\phi(\kappa_{\theta}(g))}d\mu(g)\leq
C_{\phi}^{s}\mu(K_{2})e^{sE_{2}}Q_{P_{1}}^{\phi}(s).$

Likewise,

$\displaystyle Q_{R}(s)$ $\displaystyle\geq
C_{\phi}^{-s}\mu(K_{1})e^{-sE_{1}}Q_{P_{1}}^{\phi}(s),$

so (14) holds. ∎

######  Remark 7.5.

Notice that the proof implies that $\delta^{\phi}(P)$ depends only on $\phi$
and $\mathsf{U}$. One may concisely describe $\mathsf{U}$ as the unipotent
radical of the Zariski closure of $P$.

##  8\. Divergence of the Poincaré series at its critical exponent

In this section we use an argument of Dal’bo-Otal-Peigné [17] and Corollary
7.2 to prove that the Poincaré series diverges at its critical exponent. A
similar argument was used by Blayac-Zhu [5] in their study of relatively
Anosov subgroups of the projective general linear group which preserve a
properly convex domain and Patterson-Sullivan measures defined using the
Busemann functions associated to the Hilbert distance on the properly convex
domain.

######  Theorem 8.1 (Theorem 1.1).

If $\Gamma\subset\mathsf{G}$ is a $\operatorname{\mathsf{P}}_{\theta}$-Anosov
subgroup relative to $\mathcal{P}$, $\phi\in\mathfrak{a}_{\theta}^{*}$ and
$\delta^{\phi}(\Gamma)<+\infty$, then $Q_{\Gamma}^{\phi}$ diverges at its
critical exponent.

The key result needed to prove Theorem 8.1 is the following lemma.

######  Lemma 8.2.

If $\Gamma\subset\mathsf{G}$ is a $\operatorname{\mathsf{P}}_{\theta}$-Anosov
subgroup relative to $\mathcal{P}$, $\phi\in\mathfrak{a}_{\theta}^{*}$ and
$\delta^{\phi}(\Gamma)<+\infty$, then there exists a $\phi$-Patterson-Sullivan
measure $\mu$ for $\Gamma$ of dimension $\delta^{\phi}(\Gamma)$ such that
$\mu$ has no atoms.

Assuming Lemma 8.2, we prove Theorem 8.1.

###### Proof of Theorem 8.1 assuming Lemma 8.2.

By Lemma 8.2 there exists an atom-less $\phi$-Patterson-Sullivan measure $\mu$
of dimension $\delta^{\phi}(\Gamma)$. Suppose for a contradiction that
$Q_{\Gamma}^{\phi}(\delta)<+\infty$. Then Theorem 1.3 implies that
$\mu(\Lambda_{\theta}^{\rm con}(\Gamma))=0$. Since
$\Lambda_{\theta}(\Gamma)-\Lambda_{\theta}^{\rm con}(\Gamma)$ is countable and
$\mu$ has no atoms, $\mu(\Lambda_{\theta}(\Gamma))=0$. Since $\mu$ is a
probability measure on $\Lambda_{\theta}(\Gamma)$ this is impossible. ∎

We now give the proof of Lemma 8.2.

###### Proof of Lemma 8.2.

Endow $\Gamma\cup\Lambda_{\theta}(\Gamma)$ with the topology from Lemma 4.1
and for $x\in\Gamma\cup\Lambda_{\theta}(\Gamma)$ let $\mathcal{D}_{x}$ denote
the Dirac measure supported on $x$. By [34, Lem. 3.1] there exists a
continuous non-decreasing function
$h:\operatorname{\mathbb{R}}^{+}\to\operatorname{\mathbb{R}}^{+}$ such that:

  1. (1)

The series

$\hat{Q}(s):=\sum_{\gamma\in\Gamma}h\left(e^{\phi(\kappa_{\theta}(\gamma))}\right)e^{-s\phi(\kappa_{\theta}(\gamma))}$

converges for $s>\delta$ and diverges for $s\leq\delta$ where
$\delta=\delta^{\phi}(\Gamma)$.

  2. (2)

For any $\epsilon>0$ there exists $\nu_{0}>0$ such that: if $s>1$ and
$\nu>\nu_{0}$, then $h(\nu s)\leq s^{\epsilon}h(\nu)$.

(In the case when $Q_{\Gamma}^{\phi}$ diverges at its critical exponent, we
can choose $h\equiv 1$.)

For $s>\delta$ consider the probability measure

$\mu_{s}:=\frac{1}{\hat{Q}(s)}\sum_{\gamma\in\Gamma}h\left(e^{\phi(\kappa_{\theta}(\gamma))}\right)e^{-s\phi(\kappa_{\theta}(\gamma))}\mathcal{D}_{\gamma}$

on $\Gamma\cup\Lambda_{\theta}(\Gamma)$. By compactness, there exists
$\\{s_{n}\\}\subset(\delta,\infty)$ so that
$\lim_{n\rightarrow\infty}s_{n}=\delta$ and

$\mu:=\lim_{n\rightarrow\infty}\mu_{s_{n}}$

exists in the weak-$*$. topology. In the proof of [15, Prop. 3.1], we showed
that $\mu$ is a $\phi$-Patterson-Sullivan measure for $\Gamma$ of dimension
$\delta$.

We will now prove that $\mu$ has no atoms. By [15, Prop. 8.1], if
$F\in\Lambda_{\theta}^{\rm con}(\Gamma)$, then $\mu(\\{F\\})=0$. So it
suffices to fix a parabolic point $F_{0}\in\Lambda_{\theta}(\Gamma)$ and show
that $\mu(\\{F_{0}\\})=0$. By the definition of the weak-$*$ topology, it
suffices to find a sequence of open sets $\\{V_{m}\\}$ in
$\Gamma\cup\Lambda_{\theta}(\Gamma)$ each of which contains $F_{0}$, and where

$\displaystyle\liminf_{m\rightarrow\infty}\,\limsup_{s\searrow\delta}\,\mu_{s}(V_{m})=0.$
(15)

Let $\xi:\partial(\Gamma,\mathcal{P})\to\operatorname{\mathcal{F}}_{\theta}$
be the limit map of $\Gamma$. Let $P$ be the stabilizer in $\Gamma$ of $F_{0}$
(it is a conjugate of a subgroup in $\mathcal{P}$), fix an enumeration
$P=\\{g_{1},g_{2},\dots\\}$, and let $Q_{m}:=\\{g_{1},\dots,g_{m}\\}\subset P$
for each positive integer $m$.

Fix a Groves-Manning cusp space $X$ for $(\Gamma,\mathcal{P})$, and let
$\Gamma^{\prime}$ be the set of elements $\gamma\in\Gamma$ for which there is
a geodesic ray $\sigma:[0,\infty)\rightarrow X$ such that $\sigma(0)=\gamma$,
$\sigma(T)=\operatorname{\mathrm{id}}$ for some $T\geq 0$ and
$\lim_{t\rightarrow\infty}\sigma(t)=\xi^{-1}(F_{0})\in\partial_{\infty}X=\partial(\Gamma,\mathcal{P})$.
The next lemma implies that for all positive integers $m$,

$V_{m}:=(\Gamma\cup\Lambda_{\theta}(\Gamma))-Q_{m}\cdot\overline{\Gamma^{\prime}}$

is an open neighborhood of $F_{0}$, where $\overline{\Gamma^{\prime}}$ denotes
the closure of $\Gamma^{\prime}$ in $\Gamma\cup\Lambda_{\theta}(\Gamma)$.

######  Lemma 8.3.

The closed subset
$\overline{\Gamma^{\prime}}\subset\Gamma\cup\Lambda_{\theta}(\Gamma)$ does not
contain $F_{0}$. In particular, for all positive integers $m$,
$V_{m}\subset\Gamma\cup\Lambda_{\theta}(\Gamma)$ is an open neighborhood of
$F_{0}$.

###### Proof.

Suppose for a contradiction that there exists a sequence $\\{\gamma_{n}\\}$ in
$\Gamma^{\prime}$ converging to $F_{0}$. Then Lemma 4.1 implies that
$U_{\theta}(\gamma_{n})\rightarrow F_{0}$. Hence, if we view
$\\{\gamma_{n}\\}$ as a sequence in $X$, then by Proposition 4.3 and taking a
subsequence if necessary, we must have
$\gamma_{n}=\gamma_{n}(\operatorname{\mathrm{id}})\rightarrow\xi^{-1}(F_{0})$.

For each $n$ fix a geodesic ray $\sigma_{n}:[0,\infty)\rightarrow X$ such that
$\sigma_{n}(0)=\gamma_{n}$, $\sigma_{n}(T_{n})=\operatorname{\mathrm{id}}$ for
some $T_{n}\geq 0$ and
$\lim_{t\rightarrow\infty}\sigma_{n}(t)=\xi^{-1}(F_{0})$. Passing to a
subsequence we can suppose that the geodesics $\\{\sigma_{n}(\cdot+T_{n})\\}$
converges locally uniformly to a geodesic
$\sigma:\operatorname{\mathbb{R}}\rightarrow X$. Then, since
$\gamma_{n}\rightarrow\xi^{-1}(F_{0})$, we have

$\lim_{t\rightarrow-\infty}\sigma(t)=\lim_{n\rightarrow\infty}\sigma_{n}(0)=\lim_{n\rightarrow\infty}\gamma_{n}=\xi^{-1}(F_{0})=\lim_{t\rightarrow+\infty}\sigma(t),$

which is impossible.

Since $Q_{m}$ is finite for all $m$, the second claim follows immediately from
the first. ∎

Since $P$ is conjugate to a subgroup in $\mathcal{P}$, by Corollary 7.2,

$\epsilon:=\frac{\delta^{\phi}(\Gamma)-\delta^{\phi}(P)}{2}>0.$

To prove that Equation (15) holds, we prove the following lemma. This will be
used to give an upper bound on $\mu_{s}(V_{m})$ involving the tail of the
Poincaré series $Q_{P}^{\phi}(\delta^{\phi}(P)+\epsilon)$.

######  Lemma 8.4.

  1. (1)

$P\cdot\Gamma^{\prime}=\Gamma$.

  2. (2)

There exists $C_{1}>0$ such that

$\left|\phi(\kappa_{\theta}(g\gamma))-\phi(\kappa_{\theta}(g))-\phi(\kappa_{\theta}(\gamma))\right|\leq
C_{1}$

for all $\gamma\in\Gamma^{\prime}$ and $g\in P$.

  3. (3)

There exists $C_{2}>0$ such that

$h\left(e^{\phi(\kappa_{\theta}(g_{k}))+\phi(\kappa_{\theta}(\gamma))+C_{1}}\right)\leq
C_{2}e^{\epsilon\phi(\kappa_{\theta}(g_{k}))}h\left(e^{\phi(\kappa_{\theta}(\gamma))}\right)$

for all $\gamma\in\Gamma^{\prime}$ and $k\geq 1$.

###### Proof.

Proof of (1). Fix $\gamma\in\Gamma$. Then fix a geodesic ray
$\sigma:[0,\infty)\rightarrow X$ with $\sigma(0)=\gamma$ and
$\lim_{t\rightarrow\infty}\sigma(t)=\xi^{-1}(F_{0})$. Let
$\mathcal{H}_{P}\subset X$ denote the combinatorial horoball associated to
$P$. Notice that the geodesic ray $\sigma_{0}:[0,\infty)\rightarrow X$ which
satisfies $\sigma_{0}(n)=(\operatorname{\mathrm{id}},n)\in\mathcal{H}_{P}$ for
all $n\in\operatorname{\mathbb{N}}$ also limits to $\xi^{-1}(F_{0})$. Hence

$\sup_{t\geq 0}\,{\rm d}_{X}(\sigma(t),\sigma_{0}(t))<+\infty.$

Since

$\lim_{t\rightarrow\infty}{\rm
d}_{X}(\sigma_{0}(t),X-\mathcal{H}_{P})=+\infty,$

there exists $T\geq 0$ such that $\sigma(T)\in P$ and
$\sigma(t)\in\mathcal{H}_{P}$ for all $t\geq T$. Then
$\sigma(T)^{-1}\gamma\in\Gamma^{\prime}$. Since $\gamma\in\Gamma$ was
arbitrary, (1) holds.

Proof of (2). Suppose not. Then for each $n\geq 1$ there exist
$\gamma_{n}\in\Gamma^{\prime}$ and $g_{n}\in P$ such that

$\left|\phi(\kappa_{\theta}(g_{n}\gamma_{n}))-\phi(\kappa_{\theta}(g_{n}))-\phi(\kappa_{\theta}(\gamma_{n}))\right|\geq
n.$

Lemma 2.1 implies that $\\{\gamma_{n}\\}$ and $\\{g_{n}\\}$ are both escaping
sequences in $\Gamma$. Since $\\{g_{n}\\}\subset P$, in the topology on
$\Gamma\cup\Lambda_{\theta}(\Gamma)$ we have $g_{n}^{-1}\rightarrow F_{0}$.
Passing to a subsequence we can also assume that

$F:=\lim_{n\rightarrow\infty}\gamma_{n}\in\Lambda_{\theta}(\Gamma)$

exists. By Lemma 8.3, $F_{0}\neq F$. So by Lemma 4.1 and Proposition 5.1, we
have

$\limsup_{n\rightarrow\infty}\left|\phi(\kappa_{\theta}(g_{n}\gamma_{n}))-\Big{(}\phi(\kappa_{\theta}(g_{n}))+\phi(\kappa_{\theta}(\gamma_{n}))\Big{)}\right|<+\infty$

and hence a contradiction.

Proof of (3). Let $\nu_{0}>0$ be a constant such that if $s>1$ and
$\nu>\nu_{0}$, then $h(\nu s)\leq s^{\epsilon}h(\nu)$. Let $C_{1}>0$ be the
constant from (2), and fix $C_{1}^{\prime}\geq C_{1}$ such that

$\phi(\kappa_{\theta}(g_{k}))+C_{1}^{\prime}>0\quad\text{and}\quad\phi(\kappa_{\theta}(\gamma))+C_{1}^{\prime}>\log\nu_{0}$

for all $\gamma\in\Gamma^{\prime}$ and $k\geq 1$. Also let

$\Gamma^{\prime\prime}:=\\{\gamma\in\Gamma^{\prime}:\phi(\kappa_{\theta}(\gamma))\leq\log\nu_{0}\\}.$

If $\gamma\notin\Gamma^{\prime\prime}$, then

$h\left(e^{\phi(\kappa_{\theta}(g_{k}))+\phi(\kappa_{\theta}(\gamma))+C_{1}}\right)\leq
h\left(e^{\phi(\kappa_{\theta}(g_{k}))+\phi(\kappa_{\theta}(\gamma))+C_{1}^{\prime}}\right)\leq
e^{\epsilon
C_{1}^{\prime}}e^{\epsilon\phi(\kappa_{\theta}(g_{k}))}h\left(e^{\phi(\kappa_{\theta}(\gamma))}\right).$

On the other hand, if $\gamma\in\Gamma^{\prime\prime}$, then

$h\left(e^{\phi(\kappa_{\theta}(g_{k}))+\phi(\kappa_{\theta}(\gamma))+C_{1}}\right)\leq
h\left(e^{\phi(\kappa_{\theta}(g_{k}))+\phi(\kappa_{\theta}(\gamma))+2C_{1}^{\prime}}\right)\leq
e^{\epsilon
C_{1}^{\prime}}e^{\epsilon\phi(\kappa_{\theta}(g_{k}))}h\left(e^{\phi(\kappa_{\theta}(\gamma))+C_{1}^{\prime}}\right).$

So (3) holds with

$C_{2}:=e^{\epsilon
C_{1}^{\prime}}\max\left\\{\frac{h\left(e^{\phi(\kappa_{\theta}(\gamma))+C_{1}^{\prime}}\right)}{h\left(e^{\phi(\kappa_{\theta}(\gamma))}\right)}:\gamma\in\Gamma^{\prime\prime}\right\\},$

which is finite since $\Gamma^{\prime\prime}$ is finite. ∎

If $s>\delta$, then by Lemma 8.4,

$\displaystyle\mu_{s}(V_{m})$
$\displaystyle\leq\frac{1}{\hat{Q}(s)}\sum_{k>m}\sum_{\gamma\in\Gamma^{\prime}}h\left(e^{\phi(\kappa_{\theta}(g_{k}\gamma))}\right)e^{-s\phi(\kappa_{\theta}(g_{k}\gamma))}$
$\displaystyle\leq\frac{e^{sC_{1}}}{\hat{Q}(s)}\sum_{k>m}\sum_{\gamma\in\Gamma^{\prime}}h\left(e^{\phi(\kappa_{\theta}(g_{k}))+\phi(\kappa_{\theta}(\gamma))+C_{1}}\right)e^{-s\phi(\kappa_{\theta}(g_{k}))-s\phi(\kappa_{\theta}(\gamma))}$
$\displaystyle\leq\frac{C_{2}e^{sC_{1}}}{\hat{Q}(s)}\sum_{k>m}\sum_{\gamma\in\Gamma^{\prime}}e^{-(s-\epsilon)\phi(\kappa_{\theta}(g_{k}))}h\left(e^{\phi(\kappa_{\theta}(\gamma))}\right)e^{-s\phi(\kappa_{\theta}(\gamma))}$
$\displaystyle\leq
C_{2}e^{sC_{1}}\sum_{k>m}e^{-(\delta^{\phi}(P)+\epsilon)\phi(\kappa_{\theta}(g_{k}))}.$

Since $Q_{P}^{\phi}(\delta^{\phi}(P)+\epsilon)<+\infty$, it follows that

$\liminf_{m\rightarrow\infty}\,\limsup_{s\searrow\delta}\,\mu_{s}(V_{m})\leq
C_{2}e^{\delta
C_{1}}\liminf_{m\rightarrow\infty}\sum_{k>m}e^{-(\delta^{\phi}(P)+\epsilon)\phi(\kappa_{\theta}(g_{k}))}=0,$

so Equation (15) holds. ∎

##  9\. Relatively quasiconvex subgroups

Suppose that $\Gamma$ is hyperbolic relative to $\mathcal{P}$. Given a
subgroup $\Gamma_{0}\subset\Gamma$, the _limit set of $\Gamma_{0}$_, denoted
$\Lambda(\Gamma_{0})$, is the set of all points
$x\in\partial(\Gamma,\mathcal{P})$ where there is a sequence $\\{h_{n}\\}$ in
$\Gamma_{0}$ and $y\in\partial(\Gamma,\mathcal{P})$ such that
$h_{n}(z)\rightarrow x$ for all $z\in\partial(\Gamma,\mathcal{P})-\\{y\\}$.
Notice that the action of $\Gamma_{0}$ on $\Lambda(\Gamma_{0})$ is a
convergence group action. Then $\Gamma_{0}$ is _relatively quasiconvex_ if
$\Gamma_{0}$ acts geometrically finitely on $\Lambda(\Gamma_{0})$, see [24,
Defn. 6.2]. In this section we prove that infinite index, relatively
quasiconvex subgroups of a relatively Anosov group have strictly lower
critical exponent.

######  Corollary 9.1 (Corollary 1.6).

Suppose $\Gamma\subset\mathsf{G}$ is a
$\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative to
$\mathcal{P}$, $\phi\in{\mathfrak{a}}_{\theta}^{*}$ and
$\delta^{\phi}(\Gamma)<+\infty$. If $\Gamma_{0}$ is infinite index relatively
quasiconvex subgroup of $(\Gamma,\mathcal{P})$, then

$\delta^{\phi}(\Gamma)>\delta^{\phi}(\Gamma_{0}).$

######  Remark 9.2.

Since we will be using results from Hruska [24] extensively in this section,
we remark that the definition of relatively hyperbolic group used in [24,
Defn. 3.1] is strictly weaker than Definition 3.1. To obtain Definition 3.1
from Hruska’s definition, one needs to further require that $\Gamma$ be non-
elementary and finitely generated, and that every $P\in\mathcal{P}$ is
finitely generated. By [24, Cor. 9.2], these additional conditions follow if
$\Gamma$ is non-elementary and every $P\in\mathcal{P}$ is _slender_ , i.e. any
subgroup of $P$ is finitely generated.

As mentioned in the Introduction, the two results needed to deduce Corollary
9.1 from Theorems 1.1 and 1.5 are stated as Lemma 9.3 and Proposition 9.4
below. They both follow from results of Hruska [24].

######  Lemma 9.3.

Suppose $\Gamma$ is hyperbolic relative to $\mathcal{P}$ and
$\Gamma_{0}\subset\Gamma$ is relatively quasiconvex. If $\Gamma_{0}$ has
infinite index in $\Gamma$, then $\Lambda(\Gamma_{0})$ is a closed proper
subset of $\partial(\Gamma,\mathcal{P})$.

###### Proof.

We will prove the contrapositive: if
$\Lambda(\Gamma_{0})=\partial(\Gamma,\mathcal{P})$, then $\Gamma_{0}$ has
finite index in $\Gamma$.

Let $X$ be a Groves–Manning cusp space for $(\Gamma,\mathcal{P})$. Given a
subset $A\subset X$, we will let $\mathcal{N}_{r}(A)$ denote the closed
$r$-neighborhood of $A$ in $X$. Since $X$ is Gromov hyperbolic, there exists
$c>0$ such that if $\sigma_{1},\sigma_{2}:\operatorname{\mathbb{R}}\rightarrow
X$ are geodesics with

$\lim_{t\rightarrow\pm\infty}\sigma_{1}(t)=\lim_{t\rightarrow\pm\infty}\sigma_{2}(t),$

then

$\sigma_{1}\subset\mathcal{N}_{c}(\sigma_{2})\quad\text{and}\quad\sigma_{2}\subset\mathcal{N}_{c}(\sigma_{1}).$

Fix a geodesic $\sigma_{0}:\operatorname{\mathbb{R}}\rightarrow X$. Then fix
$r>0$ such that

$\operatorname{\mathrm{id}}\in\mathcal{N}_{r}(\sigma_{0}).$

An equivalent definition [24, Defn. 6.6] of relatively quasiconvex subgroups
implies that there exists $R>0$ such that: if $s:[0,T]\rightarrow X$ is a
geodesic segment with endpoints in $\Gamma_{0}$, then

$s\cap\mathcal{N}_{c+r+1}(\Gamma)\subset\mathcal{N}_{R}(\Gamma_{0}),$

see [24, Prop. 7.5 and 7.6].

Fix $g\in\Gamma$ and let
$\sigma:=g\circ\sigma_{0}:\operatorname{\mathbb{R}}\rightarrow X$. Since
$\Lambda(\Gamma_{0})=\partial(\Gamma,\mathcal{P})$, there exist sequences
$\\{h_{n}^{-}\\}$, $\\{h_{n}^{+}\\}$ in $\Gamma_{0}$ such that

$\lim_{n\rightarrow\infty}h_{n}^{\pm}=\lim_{t\rightarrow\pm\infty}\sigma(t).$

Let $\sigma_{n}$ be a geodesic in $X$ joining $h_{n}^{-}$ to $h_{n}^{+}$.
After possibly reparametrizing and passing to a subsequence we may suppose
that $\sigma_{n}$ converges locally uniformly to a geodesic
$\sigma_{\infty}:\operatorname{\mathbb{R}}\rightarrow X$. Then

$\sigma_{\infty}\cap\mathcal{N}_{c+r}(\Gamma)\subset\mathcal{N}_{R}(\Gamma_{0}).$

Further,

$\lim_{t\rightarrow\pm\infty}\sigma_{\infty}(t)=\lim_{n\rightarrow\infty}h_{n}^{\pm}=\lim_{t\rightarrow\pm\infty}\sigma(t),$

and so $g\in\mathcal{N}_{r}(\sigma)\subset\mathcal{N}_{c+r}(\sigma_{\infty})$.
Hence,

$g\in\mathcal{N}_{R+c+r}(\Gamma_{0}).$

Since $g\in\Gamma$ was arbitrary,

$\Gamma\subset\mathcal{N}_{R+c+r}(\Gamma_{0})$

and thus $\Gamma_{0}\subset\Gamma$ has finite index. ∎

######  Proposition 9.4.

Suppose $\Gamma\subset\mathsf{G}$ is a
$\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative to
$\mathcal{P}$. Assume $\Gamma_{0}\subset\Gamma$ is non-elementary and
relatively quasiconvex. Let $\mathcal{P}_{0}$ denote a set of representatives
of the conjugacy classes in $\Gamma_{0}$ of the intersection of the peripheral
subgroups of $\Gamma$ with $\Gamma_{0}$. Then:

  1. (1)

$(\Gamma_{0},\mathcal{P}_{0})$ is relatively hyperbolic (in the sense of
Definition 3.1).

  2. (2)

$\Gamma_{0}$ is a $\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup
relative to $\mathcal{P}_{0}$.

###### Proof.

Proof of (1). Hruska [24, Thm. 9.1] proved that relatively quasiconvex
subgroups of relatively hyperbolic groups in his weaker sense (see Remark 9.2)
are also relatively hyperbolic. Thus, it suffices to show that every
$P\in\mathcal{P}_{0}$ is slender.

######  Lemma 9.5.

Suppose $\Gamma\subset\mathsf{G}$ is a
$\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative to
$\mathcal{P}$. If $P\in\mathcal{P}$ and $Q\subset P$ is a subgroup, then $Q$
is finitely generated.

###### Proof.

Using Proposition 4.5 we may assume that $Z(\mathsf{G})$ is trivial and
$\operatorname{\mathsf{P}}_{\theta}$ contains no simple factors of
$\mathsf{G}$ (notice that if $p:\mathsf{G}\rightarrow\mathsf{G}^{\prime}$ is
as in the proposition, then $p|_{\Gamma}$ has finite kernel and hence $Q$ is
finitely generated if and only if $p(Q)$ is finitely generated). By Theorem
4.4 there exists a closed subgroup $\mathsf{H}\subset\mathsf{G}$ with finitely
many components such that:

  1. (1)

$P$ is a cocompact lattice in $\mathsf{H}$.

  2. (2)

$\mathsf{H}=\mathsf{L}\ltimes\mathsf{U}$ where $\mathsf{L}$ is compact and
$\mathsf{U}$ is the unipotent radical of $\mathsf{H}$.

Then Auslander’s theorem (see for instance [26, Thm. 11.1]) implies that $Q$
is finitely generated.

∎

Proof of (2). By definition, $\Gamma_{0}$ is a
$\operatorname{\mathsf{P}}_{\theta}$-transverse subgroup of $\mathsf{G}$.
Also, by (1), $(\Gamma_{0},\mathcal{P}_{0})$ is relatively hyperbolic, and so
we may identify

$\partial(\Gamma_{0},\mathcal{P}_{0})=\Lambda(\Gamma_{0})\subset\partial(\Gamma,\mathcal{P}).$

Since $\Gamma$ is a $\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup
relative to $\mathcal{P}$, there is a $\Gamma$-equivariant homeomorphism
$\xi:\partial(\Gamma,\mathcal{P})\rightarrow\Lambda_{\theta}(\Gamma)$. Thus,
to show that $\Gamma_{0}$ is a $\operatorname{\mathsf{P}}_{\theta}$-Anosov
subgroup relative to $\mathcal{P}_{0}$, it suffices to show that
$\xi(\Lambda(\Gamma_{0}))=\Lambda_{\theta}(\Gamma_{0})$.

Fix any $F^{+}\in\Lambda_{\theta}(\Gamma_{0})$. Then by Proposition 2.2 there
is $\\{h_{n}\\}$ in $\Gamma_{0}$ and $F^{-}\in\Lambda_{\theta}(\Gamma)$ such
that $h_{n}(F)\rightarrow F^{+}$ for all
$F\in\operatorname{\mathcal{F}}_{\theta}$ transverse to $F^{-}$. By
definition, $F^{\pm}=\xi(x^{\pm})$ for some
$x^{\pm}\in\partial(\Gamma,\mathcal{P})$. Also, since $\Gamma_{0}$ acts on
$\Lambda(\Gamma_{0})$ as a convergence group, by passing to a subsequence, we
can suppose that there exist $y^{\pm}\in\Lambda(\Gamma_{0})$ such that
$h_{n}(z)\rightarrow y^{+}$ for all $z\in\Lambda(\Gamma_{0})-\\{y^{-}\\}$.
Since $\Gamma_{0}$ is non-elementary, we may fix
$z\in\Lambda(\Gamma_{0})-\\{x^{-},y^{-}\\}$. Then

$F^{+}=\lim_{n\rightarrow\infty}h_{n}(\xi(z))=\lim_{n\rightarrow\infty}\xi(h_{n}(z))=\xi(y^{+}).$

Since $F^{+}$ was arbitrary, it follows that
$\Lambda_{\theta}(\Gamma_{0})\subset\xi(\Lambda(\Gamma_{0}))$. A very similar
argument shows that
$\xi(\Lambda(\Gamma_{0}))\subset\Lambda_{\theta}(\Gamma_{0})$. ∎

We may now give the proof of Corollary 9.1.

###### Proof of Corollary 9.1.

First, notice that if $\Gamma_{0}\subset\Gamma$ is an elementary subgroup,
then it is either finite (in which case $\Lambda(\Gamma_{0})$ is empty),
conjugate to a subgroup of a peripheral subgroup of $\Gamma$ (in which case
$\Lambda(\Gamma_{0})$ is a single point) or virtually a cyclic group generated
by a hyperbolic element (in which case $\Lambda(\Gamma_{0})$ consists of the
the attracting and repelling fixed points of the hyperbolic element). If
$\Gamma_{0}$ is non-elementary, then $\Lambda(\Gamma_{0})$ is perfect. See the
discussion in [24, Sec. 3.1] for more details.

_Case 1:_ Suppose $\Gamma_{0}$ is non-elementary. Let $\mathcal{P}_{0}$ denote
a set of representatives of the conjugacy classes in $\Gamma_{0}$ of the
intersection of the peripheral subgroups of $\Gamma$ with $\Gamma_{0}$. Then
Proposition 9.4 implies that $\Gamma_{0}$ is a
$\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative to
$\mathcal{P}_{0}$. By Lemma 9.3, $\Lambda_{\theta}(\Gamma_{0})$ is a proper
subset of $\Lambda_{\theta}(\Gamma)$. By Theorem 1.1, $Q^{\phi}_{\Gamma_{0}}$
converges at its critical exponent. So, by Theorem 1.5,

$\delta^{\phi}(\Gamma_{0})<\delta^{\phi}(\Gamma).$

_Case 2:_ Assume $\\#\Lambda(\Gamma_{0})=2$. Then there is an infinite order
hyperbolic element $\gamma\in\Gamma_{0}$ such that $\langle\gamma\rangle$ has
finite index in $\Gamma_{0}$. Since $\Gamma$ acts as a convergence group on
$\Lambda_{\theta}(\Gamma)$, we can label the fixed points
$F^{+},F^{-}\in\Lambda_{\theta}(\Gamma)$ of $\gamma$ so that
$\gamma^{n}(F)\rightarrow F^{+}$ for all
$F\in\operatorname{\mathcal{F}}_{\theta}$ transverse to $F^{-}$. By
Proposition 2.2, $U_{\theta}(\gamma^{-n})\rightarrow F^{-}$ and
$U_{\theta}(\gamma^{n})\rightarrow F^{+}$. Then by Proposition 5.1 there
exists $C>0$ such that

$\phi(\kappa_{\theta}(\gamma^{n+m}))\geq\phi(\kappa_{\theta}(\gamma^{n}))+\phi(\kappa_{\theta}(\gamma^{m}))-C$

for all $n,m\geq 1$. This estimate implies that
$\delta^{\phi}(\langle\gamma\rangle)=0$, and hence that
$\delta^{\phi}(\Gamma_{0})=0$.

Since $\Gamma$ is non-elementary, it contains a free subgroup of rank two and
hence

$\delta^{\phi}(\Gamma_{0})=0<\delta^{\phi}(\Gamma).$

_Case 3:_ Assume $\\#\Lambda(\Gamma_{0})=1$. Then, after conjugating, there is
a peripheral subgroup $P\in\mathcal{P}$ with $\Gamma_{0}\subset P$. Then by
Corollary 7.2 we have

$\delta^{\phi}(\Gamma_{0})\leq\delta^{\phi}(P)<\delta^{\phi}(\Gamma).$

_Case 4:_ Assume $\\#\Lambda(\Gamma_{0})=0$. Then $\Gamma_{0}$ is finite and
so $\delta^{\phi}(\Gamma_{0})=0$. So, as in Case 2,

$\delta^{\phi}(\Gamma_{0})=0<\delta^{\phi}(\Gamma).\qed$

##  10\. Characterizing linear functions with finite critical exponent

In this section, we give a complete analysis of which linear functionals in
$\mathfrak{a}_{\theta}^{*}$ have associated Poincaré series with finite
critical exponents. This generalizes the results of Sambarino [36] for Anosov
groups.

Given a subgroup $\Gamma\subset\mathsf{G}$, the _$\theta$ -Benoist limit cone
of $\Gamma$_, denoted $B_{\theta}(\Gamma)\subset\mathfrak{a}_{\theta}$, is the
set of vectors $X\in\mathfrak{a}_{\theta}$ for which there exists a sequence
$\\{\gamma_{n}\\}$ of distinct elements of $\Gamma$ and a sequence
$\\{r_{n}\\}$ in $\mathbb{R}_{+}$ so that $r_{n}\kappa_{\theta}(\gamma_{n})$
converges to $X$.

As in Proposition 4.5, let $p:\mathsf{G}\rightarrow\mathsf{G}^{\prime}$ denote
the projection map of $\mathsf{G}$ onto
$\mathsf{G}^{\prime}:=\mathsf{G}/\mathsf{H}$, where $\mathsf{H}$ is the
product of $Z(\mathsf{G})$ and the simple factors of $\mathsf{G}$ contained in
$\operatorname{\mathsf{P}}_{\theta}$. Notice that the Benoist limit cones of a
group $\Gamma\subset\mathsf{G}$ and its projection
$\Gamma^{\prime}:=p(\Gamma)\subset\mathsf{G}^{\prime}$ satisfy

${\rm d}p(B_{\theta}(\Gamma))=B_{\theta^{\prime}}(\Gamma^{\prime}).$

We prove the following expanded version of Theorem 1.2.

######  Theorem 10.1.

Suppose $\Gamma\subset\mathsf{G}$ is a
$\operatorname{\mathsf{P}}_{\theta}$-Anosov subgroup relative to $\mathcal{P}$
and $\phi\in\mathfrak{a}_{\theta}^{*}$. The following are equivalent:

  1. (1)

$\lim_{n\rightarrow\infty}\phi(\kappa_{\theta}(\gamma_{n}))=+\infty$ for every
sequence of distinct elements $\\{\gamma_{n}\\}$ in $\Gamma$.

  2. (2)

$\delta^{\phi}(\Gamma)<+\infty$.

  3. (3)

If $x_{0}\in
M^{\prime}:=\mathsf{G}^{\prime}/\operatorname{\mathsf{K}}^{\prime}$, where
$\mathsf{K}^{\prime}:=p(\mathsf{K})$, then there exist constants $c\geq
1,C\geq 0$ such that

$\frac{1}{c}{\rm
d}_{M^{\prime}}(\gamma(x_{0}),x_{0})-C\leq\phi(\kappa_{\theta}(\gamma))\leq
c{\rm d}_{M^{\prime}}(\gamma(x_{0}),x_{0})+C$

for all $\gamma\in\Gamma$ (where ${\rm d}_{M^{\prime}}$ is the distance
defined on $M^{\prime}$ in Section 2).

  4. (4)

$\phi(Y)>0$ for all $Y\in B_{\theta}(\Gamma)-\\{0\\}$.

  5. (5)

If $X$ is a Groves–Manning cusp space for $(\Gamma,\mathcal{P})$, then there
exist constants $c,C>0$ such that

$\phi(\kappa_{\theta}(\gamma))\geq c{\rm
d}_{X}(\gamma,\operatorname{\mathrm{id}})-C$

for all $\gamma\in\Gamma$.

Indeed, statements (1) and (2) of Theorem 1.2 are the same as statements (1)
and (2) of Theorem 10.1. Furthermore, when
$\operatorname{\mathsf{P}}_{\theta}$ contains no simple factors of
$\mathsf{G}$, then $\mathsf{G}^{\prime}=\mathsf{G}/Z(\mathsf{G})$. Thus, in
this case $M=M^{\prime}$ and so statement (3) of Theorem 1.2 is equivalent to
statement (3) of Theorem 10.1.

We will now prove Theorem 10.1. By part (3) of Proposition 4.5, we may assume
that $\mathsf{G}$ has trivial center and $\operatorname{\mathsf{P}}_{\theta}$
contains no simple factors of $\mathsf{G}$, in which case
$M^{\prime}=M:=\mathsf{G}/\operatorname{\mathsf{K}}$. Then note that any one
of (2), (3), (4) or (5) immediately imply (1), and (3) also immediately
implies (4). Since $\operatorname{\mathsf{P}}_{\theta}$ contains no simple
factors of $\mathsf{G}$, the equivalence of (3) and (5) follows from Theorem
4.4. So it suffices to show that (3) implies (2) and (1) implies (5).

Proof of (3) $\Rightarrow$ (2): Without loss of generality we may assume that
$x_{0}:=\operatorname{\mathsf{K}}\in M=\mathsf{G}/\operatorname{\mathsf{K}}$.
By assumption, there exist constants $c,C>0$ such that

$\phi(\kappa_{\theta}(\gamma))\geq c{\rm d}_{M}(\gamma(x_{0}),x_{0})-C$

for all $\gamma\in\Gamma$. Then
$\delta^{\phi}(\Gamma)\leq\frac{1}{c}\delta_{M}(\Gamma)$, where

$\delta_{M}(\Gamma):=\lim_{T\to\infty}\frac{\log\\#\left\\{\gamma\in\Gamma:{\rm
d}_{M}(\gamma(x_{0}),x_{0})<T\right\\}}{T}.$

Recall that the volume growth entropy of $M$ is

$h(M):=\limsup_{T\rightarrow\infty}\frac{\log{\rm Vol}_{M}(B_{T}(x_{0}))}{T}$

where ${\rm Vol}_{M}$ is the Riemannian volume on $M$ and $B_{r}(x_{0})\subset
M$ is the open ball of radius $r>0$. Since $M$ has bounded sectional
curvature, volume comparison theorems imply that $h(M)<+\infty$.

Fix $r_{0}>0$ and for $T>0$ let $\Gamma_{T}:=\left\\{\gamma\in\Gamma:{\rm
d}_{M}(\gamma(x_{0}),x_{0})<T\right\\}$. Then

$\\#\Gamma_{T}=\frac{1}{{\rm
Vol}_{M}(B_{r_{0}}(x_{0}))}\sum_{\gamma\in\Gamma_{T}}{\rm
Vol}_{M}(B_{r_{0}}(\gamma x_{0}))\leq\frac{\\#\Gamma_{2r_{0}}}{{\rm
Vol}_{M}(B_{r_{0}}(x_{0}))}{\rm Vol}_{M}(B_{T+r_{0}}(x_{0})).$

Thus $\delta_{M}(\Gamma)\leq h(M)<+\infty$. ∎

The proof that (1) implies (5) is more technical, so we provide a brief
outline. We first use Proposition 2.3 to provide a lower bound for
$\phi\circ\kappa_{\theta}$ on peripheral subgroups, see Lemma 10.2. We then
divide a geodesic joining $\operatorname{\mathrm{id}}$ to $\gamma\in\Gamma$ in
the Groves-Manning cusp space $X$ into segments
$\overline{\gamma_{i}\gamma_{i+1}}$ with endpoints in $\Gamma$ which either
(a) have a pre-chosen size guaranteeing that
$\phi(\kappa_{\theta}(\rho(\gamma_{i+1}\gamma_{i}^{-1})))$ is large enough, or
(b) are at least as long as the pre-chosen size and lie entirely in a cusped
portion of $X$. We then apply Proposition 5.3 to show that the image of the
Cartan projections are roughly additive along the segment.

Proof of (1) $\Rightarrow$ (5): Suppose that

$\lim_{n\rightarrow\infty}\phi(\kappa_{\theta}(\gamma_{n}))=+\infty$

for every sequence of distinct elements $\\{\gamma_{n}\\}$ in $\Gamma$.

Fix a Groves–Manning cusp space $X$ for $(\Gamma,\mathcal{P})$. We first
control the growth of peripheral elements.

######  Lemma 10.2.

There exist $c_{1},C_{1}>0$ such that: if $P\in\mathcal{P}$ and $\upsilon\in
P$, then

$\phi(\kappa_{\theta}(\upsilon))\geq c_{1}{\rm
d}_{X}(\upsilon,\operatorname{\mathrm{id}})-C_{1}.$

###### Proof.

Recall that ${\rm d}_{M}$ was defined so that ${\rm
d}_{M}(g\operatorname{\mathsf{K}},\operatorname{\mathsf{K}})=\left\|\kappa(g)\right\|$
for all $g\in\mathsf{G}$. Then by Theorem 4.4 it suffices to find
$c_{1},C_{1}>0$ such that: if $P\in\mathcal{P}$ and $\upsilon\in P$, then

$\phi(\kappa_{\theta}(\upsilon))\geq
c_{1}\left\|\kappa(\upsilon)\right\|-C_{1}.$

Then, since $\mathcal{P}$ is finite, it is enough to fix $P\in\mathcal{P}$ and
find constants $c_{P},C_{P}>0$ such that: if $\upsilon\in P$, then

$\phi(\kappa_{\theta}(\upsilon))\geq
c_{P}\left\|\kappa(\upsilon)\right\|-C_{P}.$

By Theorem 4.4 there exists a closed subgroup $\mathsf{H}\subset\mathsf{G}$
with finitely many components such that:

  1. (1)

$P$ is a cocompact lattice in $\mathsf{H}$.

  2. (2)

$\mathsf{H}=\mathsf{L}\ltimes\mathsf{U}$ where $\mathsf{L}$ is compact and
$\mathsf{U}$ is the unipotent radical of $\mathsf{H}$.

Let $\mathfrak{u}$ denote the Lie algebra of $\mathsf{U}$. By Proposition 2.3,
$\mathsf{U}=\exp(\mathfrak{u})$. By the same proposition, for any
$\alpha\in\theta$ there exist $M_{\alpha}\in\operatorname{\mathbb{N}}$,
$C_{\alpha}>1$ and a positive everywhere defined rational function
$R_{\alpha}:\mathfrak{u}\rightarrow\operatorname{\mathbb{R}}$ where

$\frac{1}{C_{\alpha}}R_{\alpha}(Y)^{1/M_{\alpha}}\leq
e^{\omega_{\alpha}(\kappa(\exp(Y)))}\leq
C_{\alpha}R_{\alpha}(Y)^{1/M_{\alpha}}$

for all $Y\in\mathfrak{u}$.

Write $\phi=\sum_{\alpha\in\theta}c_{\alpha}\omega_{\alpha}$, and define
$R:=\prod_{\alpha\in\theta}R_{\alpha}^{\left|c_{\alpha}\right|/M_{\alpha}}$
and $C_{\phi}:=\prod_{\alpha\in\theta}C_{\alpha}^{\left|c_{\alpha}\right|}$.
Then

$\frac{1}{C_{\phi}}R(Y)\leq e^{\phi(\kappa_{\theta}(\exp(Y)))}\leq
C_{\phi}R(Y)$

for all $s\in\operatorname{\mathbb{R}}$. We proved in Lemma 7.3 that $R$ is
positive and proper, so by Theorem 6.1 there exist $c_{2},\epsilon>0$ such
that

$R(Y)\geq c_{2}(1+\left\|Y\right\|)^{\epsilon}$

for all $Y\in\mathfrak{u}$. By Proposition 2.3, there exist $A>0$ such that

$\left\|\kappa(\exp(Y))\right\|\leq A+A\log(1+\left\|Y\right\|)$

for all $Y\in\mathfrak{u}$. Finally, let

$M:=\max\\{\left\|\kappa(\ell)\right\|:\ell\in\mathsf{L}\\}.$

If $\upsilon\in P$, then $\upsilon=\ell\exp(Y)$ for some $\ell\in\mathsf{L}$
and $Y\in\mathfrak{u}$. So by Lemma 2.1,

$\displaystyle\phi(\kappa_{\theta}(\upsilon))$
$\displaystyle\geq\phi(\kappa_{\theta}(\exp(Y)))-M\left\|\phi\right\|$
$\displaystyle\geq\log R(Y)-M\left\|\phi\right\|-\log C_{\phi}$
$\displaystyle\geq\epsilon\log(1+\left\|Y\right\|)-M\left\|\phi\right\|-\log
C_{\phi}+\log c_{2}$
$\displaystyle\geq\frac{\epsilon}{A}\left\|\kappa(\exp(Y))\right\|-\frac{\epsilon}{A}-M\left\|\phi\right\|-\log
C_{\phi}+\log c_{2}$
$\displaystyle\geq\frac{\epsilon}{A}\left\|\kappa(\upsilon)\right\|-\frac{\epsilon}{A}M-\frac{\epsilon}{A}-M\left\|\phi\right\|-\log
C_{\phi}+\log c_{2}.\qed$

By Proposition 5.3 there exists $C_{0}>0$ such that: if $f:[0,T]\rightarrow X$
is a geodesic with $f(0)=\operatorname{\mathrm{id}}$, and
$f(t_{1}),f(t_{2})\in\Gamma$ for some $0\leq t_{1}\leq t_{2}\leq T$, then

$\left|\phi\big{(}\kappa_{\theta}(f(t_{2}))\big{)}-\phi\big{(}\kappa_{\theta}(f(t_{1}))\big{)}-\phi\big{(}\kappa_{\theta}(f(t_{1})^{-1}f(t_{2}))\big{)}\right|\leq
C_{0}.$ (16)

By hypothesis,
$\lim_{n\rightarrow\infty}\phi(\kappa_{\theta}(\gamma_{n}))=+\infty$ for every
sequence of distinct elements $\\{\gamma_{n}\\}$ in $\Gamma$, so there exists
$T_{0}>0$ such that: if $\gamma\in\Gamma$ and ${\rm
d}_{X}(\gamma,\operatorname{\mathrm{id}})\geq T_{0}$, then

$\phi(\kappa_{\theta}(\gamma))>1+C_{0}.$ (17)

Then let

$C_{2}:=\max\\{\left|\phi(\kappa_{\theta}(\gamma))\right|:{\rm
d}_{X}(\gamma,\operatorname{\mathrm{id}})<T_{0}\\},$

let

$B:=\frac{2}{c_{1}T_{0}}\left(3C_{0}+C_{1}+2C_{2}+c_{1}T_{0}\right)>2,$

and let

$c:=\min\left\\{\frac{c_{1}}{2},\frac{1}{(B+2)T_{0}}\right\\}\quad\text{and}\quad
C:=C_{2}+cT_{0}.$

We will show that (2) holds with $c$ and $C$ as described above.

Fix $\gamma\in\Gamma$ and let $f:[0,T]\rightarrow X$ be a geodesic with
$f(0)=\operatorname{\mathrm{id}}$ and $f(T)=\gamma$. If $T<T_{0}$, then