1 Introduction↩︎

One of the basic themes in geometric group theory and dynamics is that the large-scale geometry of a discrete group action is reflected in the way its orbits accumulate at infinity. Patterson-Sullivan theory makes this principle quantitative: starting from the exponential growth of an orbit, it produces natural measures on the boundary, and these measures reveal the asymptotic distribution of the orbit through shadow lemmas, counting estimates, and equidistribution phenomena. In real hyperbolic geometry, this theory has been a fundamental tool in the study of Kleinian groups and negatively curved manifolds. This paper develops a higher-rank analogue for relatively Morse subgroups of semisimple Lie groups, where the visual boundary is replaced by an appropriate flag manifold. The main goal is to prove a global shadow lemma for the corresponding Patterson-Sullivan measures. We begin with the classical real hyperbolic case, both as motivation and as a guide to the higher-rank relative setting considered later.

Let \(\Gamma < \operatorname{Isom}^+(\mathbb{H}^n)\) be a non-elementary discrete subgroup, let \(o \in \mathbb{H}^n\), and let \(\Lambda \subset \partial \mathbb{H}^n\) be the limit set of \(\Gamma\). Patterson [1] and Sullivan [2] constructed a Borel probability measure \(\nu\) supported on \(\Lambda\) whose transformation rule is governed by the critical exponent \(\delta_\Gamma > 0\): \[\frac{d\gamma_*\nu}{d\nu}(\xi) = e^{\delta_\Gamma \beta_\xi(o,\gamma o)} \quad \text{for all } \gamma \in \Gamma \text{ and } \nu\text{-a.e. } \xi \in \Lambda .\] Here \(\beta_\xi\) denotes the Busemann function. The measure \(\nu\) is now referred to as the Patterson-Sullivan measure of \(\Gamma\).

A fundamental feature of this measure is Sullivan’s shadow lemma. For \(x,y\in \mathbb{H}^n\) and \(R>0\), let \(O_R(x,y)\subset \partial \mathbb{H}^n\) denote the shadow of the ball \(B(y,R)\) seen from \(x\), namely the set of endpoints \(\xi\) such that the geodesic ray \([x,\xi] \subset \mathbb{H}^n\) intersects the \(R\)-neighborhood of \(y\).

Thus the Patterson-Sullivan measure of a shadow is comparable to the exponential of minus the orbit distance: \(\nu(O_R(o,\gamma o)) \asymp e^{-\delta_\Gamma d(o,\gamma o)},\) with implied constants independent of \(\gamma\). This estimate is the basic prototype for the global shadow lemma proved in this paper.

Geometrically finite groups↩︎

For geometrically finite groups, Sullivan’s shadow lemma admits a global version due to Stratmann-Velani [3]. Let \(\Gamma<\operatorname{Isom}^+(\mathbb{H}^n)\) be geometrically finite with limit set \(\Lambda \subset \partial \mathbb{H}^n\), and let \(\operatorname{hull}\Lambda\subset \mathbb{H}^n\) be its convex hull. Choose a \(\Gamma\)-invariant family \(\mathcal{B}\) of pairwise disjoint cusp horoballs in \(\operatorname{hull}\Lambda\) whose complement has compact quotient. Let \(\mathcal{P}\) be a set of representatives of the corresponding maximal parabolic subgroups, and write \(\delta_{\mathsf P}\) for the critical exponent of \(\mathsf P\in\mathcal{P}\). It is known that \(\delta_{\mathsf P}<\delta_{\Gamma}\) ([2], [4]).

In \(\operatorname{hull}\Lambda\) but away from the cusp horoballs, the quantity \(d(\Gamma o,x)\) is uniformly bounded, and the estimate reduces to the usual shadow lemma \(\nu(O_R(o,x))\asymp e^{-\delta_\Gamma d(o,x)}\).

Relatively Anosov and relatively Morse groups↩︎

Relatively Anosov subgroups provide a higher-rank analogue of geometrically finite Kleinian groups. Let \(G\) be a connected semisimple real algebraic group with Cartan decomposition \(G=KA^+K\) where \(K\) is a maximal compact subgroup. Let \(X:=G/K\) denote the associated Riemannian symmetric space. Let \(\mathfrak a^+:=\log A^+\), let \(\Pi\) be the corresponding set of simple roots, and fix a non-empty subset \(\theta\subset\Pi\). We denote by \(P_\theta\) the associated standard parabolic subgroup and by \[\mathcal{F}_\theta:=G/P_\theta\] the corresponding flag variety. We also set \[\mathfrak a_\theta := \bigcap_{\alpha\in\Pi-\theta}\ker\alpha\] and regard \(\mathfrak a_\theta^*\) as a subspace of \(\mathfrak a^*\) via the canonical projection \(p_\theta:\mathfrak a\to\mathfrak a_\theta\). After replacing \(\theta\) by \(\theta\cup\operatorname{i}(\theta)\), if necessary, we assume that \[\theta=\operatorname{i}(\theta),\] where \(\operatorname{i}\) is the opposition involution on \(\mathfrak a\).

Let \(\Gamma<G\) be a discrete subgroup which is hyperbolic relative to a finite collection \(\mathcal{P}\) of finitely generated infinite subgroups. We write \(\partial(\Gamma,\mathcal{P})\) for its Bowditch boundary. A Gromov model for \((\Gamma,\mathcal{P})\) is a proper geodesic Gromov hyperbolic space \(Y\) on which \(\Gamma\) acts properly discontinuously, together with a \(\Gamma\)-invariant collection \(\mathcal{B}\) of disjoint horoballs whose stabilizers are conjugates of subgroups in \(\mathcal{P}\), and on whose complement \(\Gamma\) acts cocompactly. We identify \(\partial Y\) with \(\partial(\Gamma,\mathcal{P})\), and we assume that it contains at least three points, i.e., \((\Gamma, \mathcal{P})\) is non-elementary.

We say that \(\Gamma\) is \(\theta\)-Anosov relative to \(\mathcal{P}\) if it is \(\theta\)-regular and admits a transverse \(\Gamma\)-equivariant boundary map \[\zeta:\partial Y\to\mathcal{F}_\theta .\] See section 4 for its precise definition. The image of this map is the \(\theta\)-limit set, denoted by \(\Lambda_\theta\).

For \(\psi\in\mathfrak a_\theta^*\), a \((\Gamma,\psi)\)-Patterson-Sullivan measure is a Borel probability measure \(\nu\) on \(\Lambda_\theta\) satisfying \[\frac{d\gamma_*\nu}{d\nu}(\xi) = e^{\psi(\beta^\theta_\xi(e,\gamma))} \quad \text{for all } \gamma\in\Gamma \text{ and } \nu\text{-a.e. } \xi\in\Lambda_\theta ,\] where \(\beta^\theta\) is the \(\mathfrak a_\theta\)-valued Busemann map; see 3 . This notion of higher-rank Patterson-Sullivan measure was introduced by Quint [5].

Let \(\mu : G \to \mathfrak a^+\) be the Cartan projection. Let \(\mathcal{L}_\Gamma\subset\mathfrak a^+\) denote the asymptotic cone of the Cartan projection \(\mu(\Gamma)\) of \(\Gamma\), called the limit cone of \(\Gamma\). The shadow lemma for Patterson-Sullivan measures was proved for shadows in the higher-rank symmetric space \(X=G/K\) in [6] and [7]. For relatively \(\theta\)-Anosov subgroups, the compatibility between shadows in the Gromov model \(Y\) and shadows in \(X\) was established in [8].¹ Combining these results gives the following orbit-shadow estimate.

The relatively Morse condition strengthens the relative Anosov condition by requiring the relative geometry of \(Y\) to be realized inside the symmetric space. We say that \(\Gamma\) is \(\theta\)-Morse relative to \(\mathcal{P}\) if there exists a \(\Gamma\)-equivariant quasi-isometric embedding \[f:Y\to X\] such that, writing \(\mathcal{L}_f\subset\mathfrak a^+\) for the asymptotic cone of the Cartan projections \[\{ \mu(f(x)^{-1}f(y)) : x,y\in Y \},\] we have \[\mathcal{L}_f\cap\ker\alpha=\{0\} \quad\text{for every } \alpha\in\theta .\] Thus being relative Anosov gives a boundary map into the flag variety, whereas the relatively Morse condition gives a coarse geometric model in the symmetric space whose Cartan projections stay uniformly away from the walls indexed by \(\theta\). By the higher-rank Morse lemma of Kapovich-Leeb-Porti [9], the map \(f\) extends continuously to a transverse \(\Gamma\)-equivariant homeomorphism \[f:\partial Y\to\Lambda_\theta.\]

Fix a basepoint \(o_Y\in Y\). For \(\psi\in\mathfrak a_\theta^*\) positive on \(\mathcal{L}_f-\{0\}\), define \[\mathsf{d}_\psi(x,y) : = \psi(\mu(f(x)^{-1}f(y))) \quad \text{for } x, y \in Y,\] and, for subsets \(E,F\subset Y\), set \[\mathsf{d}_\psi(E,F) := \inf_{x\in E,\,y\in F}\mathsf{d}_\psi(x,y).\]

Theorem 3 estimates shadows centered at orbit points. The main result of this paper is a global version for relatively Morse groups, where the center of the shadow may be an arbitrary point along a geodesic ray in the Gromov model.

For a subgroup \(H<\Gamma\), denote by \(\delta_\psi(H)\) the critical exponent of the Poincaré series \[s\mapsto \sum_{\gamma\in H} e^{-s\psi(\mu(\gamma))}.\] In the setting below, the strict inequality \(\delta_{\psi}(\mathsf{P}) < \delta_{\psi}(\Gamma)\) was proved by Canary-Zhang-Zimmer [10].

The following theorem is the higher-rank relatively Morse analogue of the global shadow lemma of Stratmann–Velani [3].

We also show that when \(\mathsf P\) is virtually cyclic or when \(G\) has rank one, then \(a_{\bar\psi}(\mathsf P)\) is zero (Theorem 13). It would be interesting to know whether \(a_{\bar\psi}(\mathsf P)\) is always zero in the general setting of this theorem.

If \(x\) lies in the thick part \(Y-\bigcup_{B\in\mathcal{B}}B\), then Theorem 4 recovers the usual orbit-shadow estimate. The content of the theorem is therefore the precise correction term that appears when \(x\) penetrates a cusp.

Local properties of Patterson-Sullivan measures↩︎

The global shadow lemma also gives local information on Patterson-Sullivan measures. Assume, in addition, that \(\psi=\psi\circ\operatorname{i}\). For \(\xi,\eta\in\Lambda_\theta\), set \[d_\psi(\xi,\eta):=e^{-\psi(\mathcal{G}^\theta(\xi,\eta))}\] where \(\mathcal{G}^\theta\) is the \(\mathfrak a_\theta\)-valued Gromov product. This is a higher-rank visual quasi-metric on \(\Lambda_\theta\); denote by \(B_\psi(\xi,r)\) the corresponding balls. As applications, we obtain the following uniform local estimates for Patterson-Sullivan measures.

This corollary follows immediately from Theorems 25 and 26; it records the resulting two-sided comparison at the scale \(L_\kappa\), while Theorem 25 gives the lower bound for every fixed scale \(L>1\).

Finally, we compare \(\nu\) with the Hausdorff measure defined by the quasi-metric \(d_\psi\) in Theorem 27. The resulting criterion is governed by the parabolic critical exponents \(\delta_\psi(\mathsf P)\), which measure the growth of peripheral subgroups, compared to the ambient exponent \(\delta_\psi(\Gamma)\).

2 Preliminaries↩︎

Let \(G\) be a connected semisimple real algebraic group. Let \(P\) be a minimal parabolic subgroup with a fixed Langlands decomposition \(P=MAN\) where \(A\) is a maximal real split torus of \(G\), \(M\) is the maximal compact subgroup of \(P\) commuting with \(A\) and \(N\) is the unipotent radical of \(P\). Let \(\mathfrak g\) and \(\mathfrak a\) denote, respectively, the Lie algebras of \(G\) and \(A\). Fix a positive Weyl chamber \(\mathfrak a^+\subset \mathfrak a\) so that \(\log N\) consists of positive root subspaces and set \(A^+=\exp \mathfrak a^+\). We fix a maximal compact subgroup \(K< G\) such that the Cartan decomposition \(G=K A^+ K\) holds. We denote by \(\mu : G \to \mathfrak a^+\) the Cartan projection defined by the condition \(g\in K\exp \mu(g) K\) for \(g \in G\). Let \(X = G/K\) be the associated Riemannian symmetric space and \(o=[K]\in X\). Fix a \(K\)-invariant norm \(\| \cdot \|\) on \(\mathfrak g\). This induces the left \(G\)-invariant Riemannian metric \(d\) on \(X\).

Let \(\Pi\) denote the set of simple roots determined by \(\mathfrak a^+\). We fix a non-empty subset \(\theta\subset \Pi\). Let \(P_\theta\) denote the corresponding standard parabolic subgroup with the convention that \(P_\Pi=P\), and set \(\mathcal{F}_\theta:=G/P_\theta\). We also set \[\mathfrak{a}_\theta :=\bigcap_{\alpha \in \Pi-\theta} \ker \alpha \quad \text{and} \quad \mathfrak a_\theta^+ :=\mathfrak a_\theta\cap \mathfrak a^+.\] Let \[\label{att} p_\theta:\mathfrak{a}\to\mathfrak{a}_\theta\tag{1}\] denote the projection invariant under all Weyl elements fixing \(\mathfrak a_\theta\) pointwise. We write \(\mu_{\theta} := p_{\theta} \circ \mu :G\to \mathfrak a_\theta^+.\) We identify \(\mathfrak a_\theta^*=\operatorname{Hom}(\mathfrak a_\theta, \mathbb{R})\) with the subspace of \(\mathfrak a^*\) consisting of linear forms invariant under \(p_\theta\).

Abusing notation, for \(p, q \in X\), we set \[\mu(p) := \mu(g) \quad \text{and} \quad \mu(p^{-1}q) := \mu(g^{-1} h)\] for \(g, h \in G\) such that \(go = p\) and \(ho = q\). This definition is independent of the choice of \(g\) and \(h\), and we similarly define \(\mu_{\theta}(p)\) and \(\mu_{\theta}(p^{-1}q)\).

Let \(w_0\in K\) represent the longest Weyl element. The opposition involution is \[\operatorname{i}:=-\operatorname{Ad}_{w_0}:\mathfrak a\to\mathfrak a,\] and it induces an involution of \(\Pi\), again denoted by \(\operatorname{i}\).

Limit set↩︎

The subgroup \(K\) acts transitively on \(\mathcal{F}_\theta\), and hence \(\mathcal{F}_\theta\simeq K/ M_\theta\) where \(M_\theta:=P_\theta\cap K\). Set \(\xi_\theta:=[M_\theta]\in \mathcal{F}_\theta\).

The \(\theta\)-limit set of a discrete subgroup \(\Gamma\) can be defined as follows: \[\label{def46limitset} \Lambda_{\theta}=\Lambda_{\theta}(\Gamma):=\{\lim {\gamma}_i\in \mathcal{F}_\theta: {\gamma}_i\in \Gamma\}\tag{2}\] where \(\lim \gamma_i\) is defined as in Definition 1. If \(\Gamma\) is Zariski dense, this is the unique \(\Gamma\)-minimal subset of \(\mathcal{F}_{\theta}\) [19].

Busemann map↩︎

The \(\mathfrak a\)-valued Busemann map \(\beta: \mathcal{F}_\Pi \times G \times G \to\mathfrak a\) is defined as follows: for \(\xi\in \mathcal{F}\) and \(g, h\in G\), \[\beta_\xi ( g, h):=\sigma (g^{-1}, \xi)-\sigma(h^{-1}, \xi)\] where \(\sigma(g^{-1},\xi)\in \mathfrak a\) is the unique element such that \(g^{-1}k \in K \exp (\sigma(g^{-1}, \xi)) N\) for any \(k\in K\) with \(\xi=kP\). For \((\xi,g,h)\in \mathcal{F}_\theta\times G\times G\), we define \[\label{Bu} \beta_{\xi}^\theta (g, h): = p_\theta ( \beta_{\xi_0} (g, h))\tag{3}\] for any \(\xi_0\in \mathcal{F}_\Pi\) projecting to \(\xi\). This is well-defined independent of the choice of \(\xi_0\) [5]. Moreover, since product map \(K\times A \times N\to G\) is a diffeomorphism, Busemann maps are continuous.

Gromov product↩︎

Two points \(\xi \in \mathcal{F}_{\theta}\) and \(\eta \in \mathcal{F}_{\operatorname{i}(\theta)}\) are said to be transverse or be in general position if \[\label{gp} \xi=gP_{\theta}\quad \text{ and } \quad \eta = gw_0 P_{\operatorname{i}(\theta)} \quad \text{ for some g\in G}.\tag{4}\] We set \[\label{fgp} \mathcal{F}_\theta^{(2)}=\{(\xi,\eta)\in \mathcal{F}_\theta\times \mathcal{F}_{\operatorname{i}(\theta)}: \text{\xi, \eta are in general position} \}\tag{5}\] which is the unique open \(G\)-orbit in \(\mathcal{F}_\theta\times \mathcal{F}_{\operatorname{i}(\theta)}\) under the diagonal \(G\)-action.

For \((\xi, \eta) \in \mathcal{F}_{\theta}^{(2)}\), we define the \(\mathfrak a_{\theta}\)-valued Gromov product as \[\label{eqn46Gromovproduct} \mathcal{G}^{\theta} (\xi, \eta) = \frac{1}{2} \left( \beta_{\xi}^{\theta}(e, g) + \operatorname{i}\beta_{\eta}^{\operatorname{i}(\theta)}(e, g) \right)\tag{6}\] where \(g \in G\) satisfies \((gP_\theta, gw_0P_{\operatorname{i}(\theta)}) = (\xi, \eta)\). This is independent of the choice of \(g\) [7].

Patterson-Sullivan measures↩︎

For \(\psi\in \mathfrak a_\theta^*\), a \((\Gamma, \psi)\)-conformal measure is a Borel probability measure on \(\mathcal{F}_\theta\) such that \[\label{eqn46psmeas} \frac{d \gamma_*\nu}{d\nu}(\xi)=e^{\psi(\beta_\xi^\theta(e,\gamma))} \quad \text{for all \gamma \in \Gamma and\xi \in \mathcal{F}_\theta}\tag{7}\] where \({\gamma}_* \nu(D) = \nu(\gamma^{-1}D)\) for any Borel subset \(D\subset \mathcal{F}_\theta\) and \(\beta_\xi^\theta\) denotes the \(\mathfrak a_\theta\)-valued Busemann map defined in 3 . A \((\Gamma, \psi)\)-conformal measure supported on \(\Lambda_\theta\) is called a \((\Gamma, \psi)\)-Patterson-Sullivan measure.

3 Relatively hyperbolic groups and Gromov models↩︎

In this section, we recall relatively hyperbolic groups, Gromov hyperbolic spaces, and Gromov models.

Relatively hyperbolic groups↩︎

Let \(\Gamma\) be a countable group acting by homeomorphisms on a compact metrizable space \(\mathcal{X}\). The action is called a convergence group action if, for every sequence of distinct elements \(\gamma_n\in\Gamma\), there exist a subsequence \(\gamma_{n_k}\) and points \(a,b\in\mathcal{X}\) such that \(\gamma_{n_k}(x)\) converges to \(a\) for all \(x\in\mathcal{X}-\{b\}\), uniformly on compact subsets.

An infinite-order element \(\gamma\in\Gamma\) is called loxodromic if it fixes exactly two points of \(\mathcal{X}\), and parabolic if it fixes exactly one point. An infinite subgroup \(\mathsf P<\Gamma\) is called parabolic if it fixes a point of \(\mathcal{X}\) and every infinite-order element of \(\mathsf P\) is parabolic.

A point \(\xi\in\mathcal{X}\) is called a conical limit point if there exist a sequence of distinct elements \(\gamma_n\in\Gamma\) and distinct points \(a,b\in\mathcal{X}\) such that \[\gamma_n^{-1}\xi\to a \quad\text{and}\quad \gamma_n^{-1}\eta\to b \quad\text{for all } \eta\in\mathcal{X}-\{\xi\}.\] A point \(\xi\in\mathcal{X}\) is called a parabolic limit point if it is fixed by a parabolic subgroup of \(\Gamma\). Such a point is called bounded parabolic if \[\operatorname{Stab}_{\Gamma}(\xi)\backslash(\mathcal{X}-\{\xi\})\] is compact. The action of \(\Gamma\) on \(\mathcal{X}\) is called a geometrically finite convergence group action if every point of \(\mathcal{X}\) is either conical or bounded parabolic. A typical example is the action of a geometrically finite Kleinian group on its limit set.

Let \(\Gamma\) be a finitely generated group and let \(\mathcal{P}\) be a finite collection of finitely generated infinite subgroups of \(\Gamma\). We say that \(\Gamma\) is hyperbolic relative to \(\mathcal{P}\), or that \((\Gamma,\mathcal{P})\) is relatively hyperbolic, if \(\Gamma\) admits a geometrically finite convergence group action on a compact perfect metrizable space \(\mathcal{X}\) whose maximal parabolic subgroups are precisely \[\mathcal{P}^\Gamma := \{\gamma\mathsf P\gamma^{-1}:\mathsf P\in\mathcal{P},\;\gamma\in\Gamma\}.\]

Bowditch [20] showed that, if \(\Gamma\) is hyperbolic relative to \(\mathcal{P}\), then the space \(\mathcal{X}\) satisfying the above condition is unique up to \(\Gamma\)-equivariant homeomorphism. This space is called the Bowditch boundary and is denoted by \(\partial(\Gamma,\mathcal{P})\). Since \(\partial(\Gamma,\mathcal{P})\) is assumed to be perfect, we have \(\#\partial(\Gamma,\mathcal{P})\ge 3\); equivalently, \((\Gamma,\mathcal{P})\) is non-elementary.

Gromov hyperbolic spaces↩︎

A proper geodesic metric space \((Y,d)\) is called Gromov hyperbolic if there exists \(\delta>0\) such that every geodesic triangle in \(Y\) is \(\delta\)-thin; that is, each side is contained in the \(\delta\)-neighborhood of the union of the other two sides. The Gromov boundary \(\partial Y\) is the set of equivalence classes of geodesic rays, where two rays are equivalent if they have finite Hausdorff distance. We write \[\overline{Y} := Y\cup\partial Y\] for the corresponding compactification.

For \(C_1,C_2\ge 1\) and an interval \(I\subset\mathbb{R}\), a map \(\sigma:I\to Y\) is called a \((C_1,C_2)\)-quasi-geodesic if \[C_1^{-1}|t-s|-C_2 \le d(\sigma(t),\sigma(s)) \le C_1|t-s|+C_2 \quad\text{for all } t,s\in I.\] We also call the image \(\sigma(I)\) a \((C_1,C_2)\)-quasi-geodesic. We shall use the following standard stability property.

We use the following notation.

It is well known that, for any \(x\in\pi_{z_1,z_2}(y)\), the concatenation \([y,x]\cup[x,z_1]\) is a \((1,O(\delta))\)-quasi-geodesic, where \(O(\delta)\) denotes a constant depending only on \(\delta\).

Given \(o_Y\in Y\) and two distinct points \(\xi,\eta\in Y\cup\partial Y\), define their Gromov product with respect to \(o_Y\) by \[\langle \xi,\eta\rangle_{o_Y} := \sup \liminf_{i,j\to\infty} \frac{1}{2}\bigl(d(o_Y,x_i)+d(o_Y,y_j)-d(x_i,y_j)\bigr),\] where the supremum is taken over all sequences \(x_i,y_j\in Y\) such that \(x_i\to\xi\) and \(y_j\to\eta\). This quantity measures the distance from \(o_Y\) to \(\pi_{\xi,\eta}(o_Y)\) up to a uniform additive error depending only on \(\delta\). We shall use the standard inequality \[\label{eqn46Gromovineq} \langle \xi,\eta\rangle_{o_Y} \ge \min\left( \langle \xi,\zeta\rangle_{o_Y}, \langle \zeta,\eta\rangle_{o_Y} \right) - O(\delta)\tag{8}\] for all \(\xi,\eta,\zeta\in Y\cup\partial Y\).

The following standard comparison between shadows and Gromov products will be used repeatedly.

Gromov models↩︎

For the rest of this section, let \((\Gamma,\mathcal{P})\) be a relatively hyperbolic group. Following [21], we recall the basic properties of Gromov models. Such a model is a proper geodesic Gromov hyperbolic space on which \(\Gamma\) acts in a way analogous to the action of a geometrically finite Kleinian group on the real hyperbolic space.

Let \((Y,d)\) be a proper geodesic Gromov hyperbolic space. For any \(\eta\in\partial Y\), a horofunction based at \(\eta\) is obtained as follows: if \(y_n \in Y\) is a sequence converging to \(\eta\), then, after passing to a subsequence, there exist \(t_n\to\infty\) and a function \(h:Y\to\mathbb{R}\) such that \[h(x)=\lim_{n\to\infty} d(x,y_n)-t_n ,\] uniformly on compact subsets. There exists a constant \(C>0\), depending only on \(Y\), such that every such function satisfies \[\label{hii} \left| \bigl(h(\ell_{\eta}(t_2))-h(\ell_{\eta}(t_1))\bigr) - (t_1-t_2) \right| \le C \quad\text{for all } t_1,t_2\ge 0,\tag{9}\] for every unit-speed geodesic ray \(\ell_{\eta}:[0,\infty)\to Y\) asymptotic to \(\eta\). By a horofunction at \(\eta\), we mean any function satisfying 9 for every such ray.

A subset \(H\subset Y\) is called a horoball at \(\eta\) if there exists a horofunction \(h\) at \(\eta\) such that \[\{h\le 0\}\subset H\subset \{h\le 10C\}.\] Then \(\overline{H}\cap\partial Y=\{\eta\}\), and horoballs are uniformly quasi-convex.

By the uniqueness of the Bowditch boundary, the Gromov boundary of a Gromov model \(Y\) for \((\Gamma,\mathcal{P})\) is \(\Gamma\)-equivariantly homeomorphic to \(\partial(\Gamma,\mathcal{P})\). By [22], the action of \(\Gamma\) on \(\partial Y\) is a convergence group action.

For each \(B\in\mathcal{B}\), write \[\{p\}=\overline{B}\cap\partial Y.\] We call \(p\) the basepoint of \(B\) and also write \(B=B_p\). Given \(\mathsf P\in\mathcal{P}^\Gamma\), we denote its basepoint by \(\xi_{\mathsf P}\in\partial Y\).

These basepoints are precisely the parabolic limit points of \((\Gamma,\mathcal{P})\) in \(\partial Y\). We shall use the following standard properties of parabolic subgroups, whose proofs are included for completeness.

Finally, a point \(\xi\in\partial Y\) is conical if and only if, for any \(o_Y\in Y\), there exist \(R>0\) and an infinite sequence \(\gamma_n\in\Gamma\) such that \[\xi\in O_R(o_Y,\gamma_n o_Y) \quad\text{for all } n.\] In this case, we say that \(\gamma_n o_Y\) converges conically to \(\xi\). It is easy to see that this notion of conicality is equivalent to the conicality defined in terms of convergence action.

4 Relatively Morse groups↩︎

Let \(\Gamma<G\) be a discrete subgroup which is hyperbolic relative to a finite collection \(\mathcal{P}\) of finitely generated infinite subgroups of \(\Gamma\). We fix a non-empty subset \(\theta\subset\Pi\).

For a Morse embedding \(f:Y\to X\), we call \(\mathcal{L}_f\) the Morse limit cone of \(f\). We also set \[\mathcal{L}_{\theta,f}:=p_\theta(\mathcal{L}_f)\] and call it the Morse \(\theta\)-limit cone. Since \(\mathcal{L}_f=\operatorname{i}(\mathcal{L}_f)\), being \(\theta\)-Morse relative to \(\mathcal{P}\) is equivalent to being \(\theta\cup\operatorname{i}(\theta)\)-Morse relative to \(\mathcal{P}\). Thus, without loss of generality, we assume \[\theta=\operatorname{i}(\theta)\] throughout the rest of this section.

A subgroup \(\Gamma<G\) which is hyperbolic relative to \(\mathcal{P}\) is called \(\theta\)-Anosov relative to \(\mathcal{P}\) if:

• \(\Gamma\) is \(\theta\)-regular, meaning that \[\min_{\alpha\in\theta}\alpha(\mu(\gamma_n))\to\infty\] for every infinite sequence \(\gamma_n\in\Gamma\);

• there exists a transverse \(\Gamma\)-equivariant embedding \[\zeta:\partial Y\to\mathcal{F}_\theta,\] i.e., it sends distinct points of \(\partial Y\) to points in general position.

If \(\Gamma\) is \(\theta\)-Morse relative to \(\mathcal{P}\), then \(\Gamma\) is \(\theta\)-Anosov relative to \(\mathcal{P}\). Indeed, \(\theta\)-regularity follows from the fact that \(f\) is a quasi-isometric embedding and \[\mathcal{L}_f\cap\bigcup_{\alpha\in\theta}\ker\alpha=\{0\}.\] Moreover, by [9], the Morse embedding \(f:Y\to X\) extends continuously to a transverse \(\Gamma\)-equivariant embedding \[f:\partial Y\to\mathcal{F}_\theta.\] Its image is the \(\theta\)-limit set \(\Lambda_\theta\).

Geometric properties of relatively Morse groups↩︎

For the rest of this section, let \(\Gamma<G\) be \(\theta\)-Morse relative to \(\mathcal{P}\), with Morse embedding \(f:Y\to X\) from a Gromov model \((Y,d)\) and continuous extension \(f:\partial Y\to\mathcal{F}_\theta\). We fix a basepoint \(o_Y\in Y\) and we may assume that \(f(o_Y)=o\).

The following proposition is a key ingredient in the proof of the global shadow lemma. Its second assertion follows from Lemma 6 below.

We first collect several lemmas. The following was proved in [6] when \(\Gamma\) is Borel Anosov, using the Morse property of the orbit map of an Anosov group into \(X\) [9]. The same argument applies in our setting, replacing the Cayley graph of a hyperbolic group by the Gromov hyperbolic space \(Y\).

The following was proved in [23]. Although the original statement applies to the values of a linear form on the Cartan projections, the same proof gives the following vector-valued version.

We also need the following elementary observation.

We will use the following coarse midpoint estimate in the Gromov model:

As an intermediate step in the proof of Proposition 10, we prove the following. We write \(\approx\) for equality up to a uniform additive error.

Proof of Proposition 10↩︎

There exists \(R=R(Q)>0\) such that every geodesic \([\xi,p]\) with \(\xi\in Q\) intersects \(B(o_Y,R)\). Let \(\gamma\in \mathsf{P}\).

Let \(\xi\in Q\), and choose a geodesic \([\xi,p]\). Choose \(x\in[\xi,p]\cap B(o_Y,R)\), and let \(w\in[\xi,p]\) be such that \[\gamma w\in\pi_{\gamma[\xi,p]}(x).\] Since \(x\in B(o_Y,R)\), the sets \(\pi_{\gamma[\xi,p]}(x)\) and \(\pi_{\gamma[\xi,p]}(o_Y)\) have uniformly bounded Hausdorff distance. Hence, by Lemmas 1 and 6, \[\mathcal{G}^\theta(f(p),\gamma f(\xi)) \approx \frac{1}{2} \left( \mu_\theta(f(\gamma w)) + \operatorname{i}\mu_\theta(f(\gamma w)) \right).\]

By Lemma 10, there exists \(y\in[x,\gamma x]\) such that \(y\), \(w\), and \(\gamma w\) are uniformly close. Since \(y\in[x,\gamma x]\), Lemma 7 gives \[\mu_\theta(f(x)^{-1}f(\gamma x)) \approx \mu_\theta(f(x)^{-1}f(y)) + \mu_\theta(f(y)^{-1}f(\gamma x)).\] Since \(f\) is a quasi-isometric embedding, the points \(f(y)\), \(f(w)\), and \(f(\gamma w)\) are uniformly close. Since \(x\in B(o_Y,R)\), the point \(f(x)\) remains in a uniformly bounded subset of \(X\). Therefore, by Lemma 1, \[\begin{align} \mu_\theta(\gamma) &\approx \mu_\theta(f(x)^{-1}f(\gamma x)) \\ &\approx \mu_\theta(f(x)^{-1}f(\gamma w)) + \mu_\theta(f(w)^{-1}f(x)) \\ &\approx \mu_\theta(f(x)^{-1}f(\gamma w)) + \operatorname{i}\mu_\theta(f(x)^{-1}f(w)) \\ &\approx \mu_\theta(f(\gamma w)) + \operatorname{i}\mu_\theta(f(\gamma w)) \\ &\approx 2\mathcal{G}^\theta(f(p),\gamma f(\xi)). \end{align}\] This finishes the proof of the main claim.

For the second assertion, note that by Lemma 6, \[\mathcal{G}^\theta(f(p),\gamma f(\xi)) \approx \frac{1}{2}\left( \mu_\theta(f(z_\gamma)) + \operatorname{i}\mu_\theta(f(z_\gamma)) \right)\] for some \(z_\gamma\in\pi_{p,\gamma\xi}(o_Y)\). Hence \[\mu_\theta(\gamma) \approx \mu_\theta(f(z_\gamma)) + \operatorname{i}\mu_\theta(f(z_\gamma)).\] Since the right-hand side is \(\operatorname{i}\)-invariant, this implies that \(\mu_\theta(\gamma)-\operatorname{i}\mu_\theta(\gamma)\) is uniformly bounded. 0◻

A metric-like function on \(Y\)↩︎

We identify \(\mathfrak a_\theta^*\) with the subspace of \(\mathfrak a^*\) obtained by precomposing with the projection \(p_\theta:\mathfrak a\to\mathfrak a_\theta\). Let \(\psi\in\mathfrak a_\theta^*\) be such that \[\psi>0 \quad\text{on } \mathcal{L}_f-\{0\}.\] Define \[\mathsf{d}_\psi(x,y) := \psi(\mu(f(x)^{-1}f(y))) \quad\text{for } x,y\in Y.\] It was shown in [23] that \(\mathsf{d}_\psi\) behaves like a metric on \(Y\); for example, it satisfies a coarse triangle inequality. The following is a consequence of Lemma 7 and the Gromov hyperbolicity of \(Y\). Since \(\psi>0\) on \(\mathcal{L}_f-\{0\}\), there exists \(c>0\) such that \[\label{eqn46lbdpsi} \mathsf{d}_\psi(x,y)>-c \quad\text{for all } x,y\in Y.\tag{12}\]

We also note that the metric-like function \(\mathsf{d}_{\psi}\) behaves quasi-isometrically to the Gromov model \((Y, d)\) as follows:

For subsets \(E,F\subset Y\), set \[\mathsf{d}_\psi(E,F) := \inf_{x\in E,\,y\in F}\mathsf{d}_\psi(x,y).\]

\(\mathfrak a_\theta\)-valued Gromov products in shadows↩︎

We now translate the elementary comparison between shadows and Gromov products in the Gromov hyperbolic space \(Y\) into a comparison involving \(\mathfrak a_\theta\)-valued Gromov products on the flag variety. The main input from the relatively Morse property is that nearest-point projections in \(Y\) coarsely control the higher-rank Gromov product \(\mathcal{G}^\theta(f(\xi),f(\eta))\) through Cartan projections; see Lemma 6. Together with the coarse additivity of Cartan projections along geodesics in \(Y\), Lemma 7, this gives the following analogue of Lemma 3. For Anosov groups, this comparison was proved in [23]; the same argument applies in the relatively Morse setting.

For \(\psi\in\mathfrak a_\theta^*\), write \[\bar\psi:=\frac{\psi+\psi\circ\operatorname{i}}{2}.\]

5 Counting in parabolic subgroups↩︎

Throughout this section, let \(\Gamma<G\) be \(\theta\)-Anosov relative to a finite collection \(\mathcal{P}\). Let \(\mathcal{L}_\Gamma\subset\mathfrak a^+\) denote the limit cone of \(\Gamma\), that is, the asymptotic cone of \(\mu(\Gamma)\).

Fix \(\psi\in\mathfrak a_\theta^*\) that is positive on \(\mathcal{L}_\Gamma-\{0\}\). For a subgroup \(H<\Gamma\), let \(\delta_\psi(H)\) denote the abscissa of convergence of the Poincaré series \[s\mapsto \sum_{\gamma\in H} e^{-s\psi(\mu(\gamma))}.\] We prove the following counting estimate for parabolic subgroups.

We deduce Theorem 13 from the following counting estimate. We write \(\ll\) for inequality up to a uniform multiplicative constant, and similarly for \(\gg\). We write \(\asymp\) when we have both \(\ll\) and \(\gg\).

The key input is a volume estimate of Benoist-Oh [24]. We first recall the structural description of parabolic subgroups in the relatively Anosov setting. After the reduction in [10], we may assume that \(P_\theta\) contains no simple factor of \(G\). Let \(\mathsf P\in\mathcal{P}\). By [10], there exists a closed subgroup \(H<G\) with finitely many connected components such that \(\mathsf P\) is a cocompact lattice in \(H\). Moreover, if \(U\) denotes the unipotent radical of \(H\), then \[\label{eqn46structure} H=L\ltimes U \quad \text{and} \quad H^\circ=L^\circ\times U,\tag{18}\] where \(L<H\) is compact and its identity component \(L^\circ\) is abelian.

Let \(\mathfrak u=\operatorname{Lie}(U)\), and let \(\ell\) denote Lebesgue measure on \(\mathfrak u\). We shall use the following consequence of [24]:

Proof of Proposition 14↩︎

By [10], for each \(\alpha\in\theta\) there exist \(m_\alpha\in\mathbb{N}\), \(C_\alpha>1\), and an everywhere-defined positive rational function \(R_\alpha:\mathfrak u\to\mathbb{R}\) such that, for all \(Y\in\mathfrak u\), \[C_\alpha^{-1}R_\alpha(Y)^{1/m_\alpha} \le e^{\omega_\alpha(\mu(\exp Y))} \le C_\alpha R_\alpha(Y)^{1/m_\alpha},\] where \(\omega_\alpha\in\mathfrak a_\theta^*\) is the fundamental weight associated to \(\alpha\).

Write \[\psi=\sum_{\alpha\in\theta} c_\alpha\omega_\alpha\] for coefficients \(c_\alpha\in\mathbb{R}\), \(\alpha \in \theta\). Set \[R_\psi:=\prod_{\alpha\in\theta} R_\alpha^{c_\alpha/m_\alpha} \quad \text{and} \quad C_\psi:=\prod_{\alpha\in\theta} C_\alpha^{|c_\alpha|}.\] Then, for all \(Y\in\mathfrak u\), \[C_\psi^{-1}R_\psi(Y)^{-1} \le e^{-\psi(\mu(\exp Y))} \le C_\psi R_\psi(Y)^{-1}.\] By [10], the function \(R_\psi\) is proper.

Let \[\mathsf P_1 := \pi(\mathsf P\cap H^\circ),\] where \(\pi:H^\circ\to U\) is the projection. Then \(\mathsf P_1\) is a cocompact lattice in \(U\). Since \(\mathsf P\cap H^\circ\) has finite index in \(\mathsf P\) and \(\ker\pi\) is compact, we have \[\#\{g\in\mathsf P_1:\psi(\mu(g))\le T\} \asymp \#\{\gamma\in\mathsf P:\psi(\mu(\gamma))\le T\}\] for all sufficiently large \(T\). It therefore suffices to count elements of \(\mathsf P_1\).

Let \(\lambda_U\) denote a Haar measure on \(U\). Let \(Q_1\subset U\) be a bounded open set such that the translates \(gQ_1\), \(g\in\mathsf P_1\), are pairwise disjoint. Then \[\begin{align} \#\{g\in\mathsf P_1:\psi(\mu(g))\le T\} &\ll \lambda_U\left( \bigcup_{\substack{g\in\mathsf P_1\\ \psi(\mu(g))\le T}} gQ_1 \right) \\ &\ll \lambda_U\left( \{u\in U:\psi(\mu(u))\le T+c'\} \right) \\ &\ll \ell\left( \{Y\in\mathfrak u:R_\psi(Y)\le c e^T\} \right) \end{align}\] for some constants \(c',c>1\).

Similarly, choose a compact set \(Q_2\subset U\) such that \(\mathsf P_1Q_2=U\). Then, after increasing \(c',c\) with \(c>1\), if necessary, \[\begin{align} \#\{g\in\mathsf P_1:\psi(\mu(g))\le T\} &\gg \lambda_U\left( \bigcup_{\substack{g\in\mathsf P_1\\ \psi(\mu(g))\le T}} gQ_2 \right) \\ &\gg \lambda_U\left( \{u\in U:\psi(\mu(u))\le T-c'\} \right) \\ &\gg \ell\left( \{Y\in\mathfrak u:R_\psi(Y)\le c^{-1}e^T\} \right). \end{align}\] The proposition now follows from Theorem 15, together with the fact that the exponential growth rate of \[\#\{g\in\mathsf P_1:\psi(\mu(g))\le T\}\] is \(\delta_\psi(\mathsf P)\). If \(\mathsf P\) is virtually cyclic, then the corresponding unipotent group is one-dimensional, and the volume asymptotic has no logarithmic factor. Hence \(a_\psi(\mathsf P)=0\). See Proposition 17 for the claim about the case \(\operatorname{rank}G =~1\). 0◻

Before proving Theorem 13, we record the following entropy gap for parabolic subgroups, due to Canary-Zhang-Zimmer.

Proof of Theorem 13↩︎

By Proposition 14, there exist \(c>0\) and \(T_0>0\) such that, for all \(T>T_0\), \[e^{\delta_\psi(\mathsf P)T-c}(1+T)^{a_\psi(\mathsf P)} \le \#\{\gamma\in\mathsf P:\psi(\mu(\gamma))<T\} \le e^{\delta_\psi(\mathsf P)T+c}(1+T)^{a_\psi(\mathsf P)}.\] For simplicity, write \[\delta=\delta_\psi(\mathsf P) \quad \text{and} \quad a=a_\psi(\mathsf P),\] and define \[N(T):=\#\{\gamma\in\mathsf P:\psi(\mu(\gamma))<T\}.\]

Choose \(k\in\mathbb{N}\) so large that \[e^{\delta k-c}-e^c>0,\] which is possible since \(\delta >0\) by Theorem 16. Set \(S:=T+kn\). Then \[\begin{align} N(S+k)-N(S) &\le e^{\delta(S+k)+c}(1+S+k)^a - e^{\delta S-c}(1+S)^a \\ &= e^{\delta S}(1+S)^a \left[ e^{\delta k+c} \left(1+\frac{k}{1+S}\right)^a - e^{-c} \right] \\ &\le C e^{\delta S}(1+S)^a \end{align}\] for a constant \(C>1\) independent of \(T\) and \(n\). Similarly, \[\begin{align} N(S+k)-N(S) &\ge e^{\delta(S+k)-c}(1+S+k)^a - e^{\delta S+c}(1+S)^a \\ &= e^{\delta S}(1+S)^a \left[ e^{\delta k-c} \left(1+\frac{k}{1+S}\right)^a - e^c \right] \\ &\ge C^{-1} e^{\delta S}(1+S)^a, \end{align}\] after increasing \(C\), if necessary. Since \[N(S+k)-N(S) = \#\{\gamma\in\mathsf P:S\le \psi(\mu(\gamma))<S+k\},\] and \(S=T+kn\), this proves the theorem. 0◻

Rank-one parabolic subgroups↩︎

We record a rank-one refinement of Proposition 14, which shows the absence of the polynomial term in Theorem 13 for rank-one \(G\). In this subsection, assume that \(G\) has real rank one. Let \(\Pi=\{\alpha\}\), so that \(\theta=\{\alpha\}\) and \(\mathfrak a_\theta=\mathfrak a\simeq \mathbb{R}\). Let \(\mathfrak{n} := \operatorname{Lie}(N)\) and write \[\mathfrak n=\mathfrak g_\alpha\oplus\mathfrak g_{2\alpha},\] where \(\mathfrak{g}_{\alpha}\) and \(\mathfrak{g}_{2\alpha}\) are the corresponding root spaces, with the convention that \(\mathfrak g_{2\alpha}=0\) if \(2\alpha\) is not a root.

Let \(\mathsf P\in\mathcal{P}\). Let \(H=L\ltimes U\) be the subgroup associated to \(\mathsf P\) as in 18 . Up to conjugation, we may assume that \(U<N\), and use the same notation \(\mathfrak u:=\operatorname{Lie}(U)\) as before. Define \[V_{\mathsf P}:=\operatorname{pr}_{\mathfrak g_\alpha}(\mathfrak u), \quad Z_{\mathsf P}:=\mathfrak u\cap\mathfrak g_{2\alpha},\quad \text{and} \quad Q(\mathsf P):=\dim V_{\mathsf P}+2\dim Z_{\mathsf P}.\]

Since \(\mathfrak a\) is one-dimensional, \(\mathcal{L}_{\Gamma} = \mathfrak a^+\) and hence any \(\psi\in \mathfrak a^*\) positive on \(\mathcal{L}_\Gamma-\{0\}\) is a multiplication by a positive real number. The norm \(\|\psi\|\) is given by \(\psi(H_0)\) where \(H_0\in \mathfrak a^+\) is the unique unit vector.

6 Shadow estimates at parabolic limit points↩︎

Let \(\Gamma<G\) be \(\theta\)-Morse relative to \(\mathcal{P}\), with Morse embedding \(f:Y\to X\) of a Gromov model \((Y,d)\) for \((\Gamma,\mathcal{P})\). Let \(\psi\in\mathfrak a_\theta^*\) be positive on \(\mathcal{L}_f-\{0\}\). We normalize \(\psi\) so that \(\delta_\psi(\Gamma)=1\). Then the existence of Patterson-Sullivan measure is a consequence of the work of Canary-Zhang-Zimmer.

Let \(\nu\) be a \((\Gamma,\psi)\)-Patterson-Sullivan measure on \(\Lambda_\theta\) given in Theorem 18. In this section we prove the global shadow lemma for shadows along geodesic rays towards parabolic limit points and provide estimates on their complements. Since all shadows we consider are taken in \(Y\cup\partial Y\), we identify \(\nu\) with its pullback to \(\partial Y\) under the \(\Gamma\)-equivariant homeomorphism \(f:\partial Y\to~\Lambda_\theta\).

Suppose \(\theta = \operatorname{i}(\theta)\) and set \[\bar\psi:=\frac{\psi+\psi\circ\operatorname{i}}{2}.\] Since \(\psi > 0\) on \(\mathcal{L}_f - \{0\}\), so is \(\bar \psi\), and hence there exists \(C_{\bar\psi}>0\) such that \[\label{eqn46C95barpsi} \bar\psi(v)>1-C_{\bar\psi} \quad \text{for all } v\in \mu(f(Y)^{-1}f(Y)).\tag{19}\]

Shadows at parabolic limit points↩︎

We first estimate shadows along geodesic rays ending at parabolic limit points. Recall that \(o_Y\in Y\) is chosen so that \(f(o_Y)=o\). For \(\xi\in\partial Y\) and \(t\ge 0\), let \(\xi_t\in[o_Y,\xi]\) denote the point with \(d(o_Y,\xi_t)=t\), after choosing a geodesic ray \([o_Y,\xi]\).

For an arbitrary parabolic limit point, we obtain the following translated form.

For \(g\in G\), define the translated measure \(\nu_g\) by \[d\nu_g(\eta) = e^{\psi(\beta_\eta^\theta(e,g))}\,d\nu(\eta).\] Then, for \(g,h\in G\), \[d\nu_g(\eta) = e^{\psi(\beta_\eta^\theta(h,g))}\,d\nu_h(\eta).\] For \(x\in Y\), we write \[\nu_x:=\nu_g\] where \(g\in G\) satisfies \(f(x)=go\). This is independent of the choice of \(g\).

The next lemma relates \(\nu\) to its translate at the center of a shadow.

Lemma 13 is an immediate consequence of the following comparison between Busemann maps and Cartan projections inside shadows. The corresponding statement for shadows in the symmetric space \(X=G/K\) was proved in [6]; the present version follows from comparing shadows in \(Y\) with shadows in \(X\) under the Morse embedding \(f\).

Since \(\Gamma\) acts cocompactly on \(Y-\bigcup\mathcal{B}\), Lemma 13 and the ordinary shadow lemma imply the following thick-part estimate.

The main estimate needed for Theorems 19 and 20 is the following.

Proof of Theorem 19↩︎

Let \(C>0\) be the constant from Proposition 21. Let \(k\) and \(T_0\) be as in Theorem 13, applied to \(\bar\psi\).

First suppose that \[2\mathsf{d}_{\bar\psi}(o_Y,\xi_t)-C>T_0.\] Set \[T:=2\mathsf{d}_{\bar\psi}(o_Y,\xi_t)-C.\] By Proposition 10, \(\psi(\mu(g))\approx\bar\psi(\mu(g))\) for \(g\in\mathsf P\). Since \(\delta_{\psi\circ\operatorname{i}}(\Gamma)=\delta_\psi(\Gamma)=1\), it follows that \(\delta_{\bar\psi}(\Gamma)\le 1\). Together with the entropy gap (Theorem 16), this implies \(\delta_{\bar\psi}(\mathsf P)<1\). Hence Theorem 13 gives \[\begin{align} & \sum_{\substack{g\in\mathsf P\\ \bar\psi(\mu(g))\ge T}} e^{-\psi(\mu(g))} \\ & \qquad \ll \sum_{n=0}^{\infty} e^{-(T+kn)} \#\left\{ g\in\mathsf P: T+kn\le \bar\psi(\mu(g))<T+k(n+1) \right\} \\ &\qquad \ll \sum_{n=0}^{\infty} e^{(\delta_{\bar\psi}(\mathsf P)-1)(T+kn)} (1+T+kn)^{a_{\bar\psi}(\mathsf P)} \\ &\qquad \asymp e^{(\delta_{\bar\psi}(\mathsf P)-1)T} (1+T)^{a_{\bar\psi}(\mathsf P)} \\ &\qquad \asymp e^{2(\delta_{\bar\psi}(\mathsf P)-1)\mathsf{d}_{\bar\psi}(o_Y,\xi_t)} \bigl(C_{\bar\psi}+\mathsf{d}_{\bar\psi}(o_Y,\xi_t)\bigr)^{a_{\bar\psi}(\mathsf P)}. \end{align}\] The lower bound is proved in the same way. Thus by Proposition 21, \[\begin{align} \nu_{\xi_t}(O_R(o_Y,\xi_t)) \asymp{} e^{\mathsf{d}_\psi(o_Y,\xi_t)} e^{2(\delta_{\bar\psi}(\mathsf P)-1)\mathsf{d}_{\bar\psi}(o_Y,\xi_t)} \cdot \bigl(C_{\bar\psi}+\mathsf{d}_{\bar\psi}(o_Y,\xi_t)\bigr)^{a_{\bar\psi}(\mathsf P)} . \end{align}\] Applying Lemma 13 proves the theorem in this case.

It remains to consider the case when \(2\mathsf{d}_{\bar\psi}(o_Y,\xi_t)-C\le T_0\). Since \(\bar\psi\) is positive on \(\mathcal{L}_f-\{0\}\), this implies that \(d(o_Y,\xi_t)\) is uniformly bounded. Hence, after increasing \(R\) by a uniform amount, \(O_R(o_Y,\xi_t)=\partial Y\), and the desired estimate is trivial. 0◻

Proof of Theorem 20↩︎

Let \(\xi=\gamma_0\xi_{\mathsf P}\) for some \(\gamma_0\in\Gamma\), and let \(B_\xi\in\mathcal{B}\) be the horoball based at \(\xi\). Then \(\operatorname{Stab}_\Gamma(B_\xi)=\gamma_0\mathsf P\gamma_0^{-1}\). Since \(\gamma_0\mathsf P\gamma_0^{-1}\) acts cocompactly on \(\partial B_\xi\), there exist a uniform constant \(c>0\), an element \(\gamma_1\in\mathsf P\), and \(t_0\ge 0\) such that \[d(\xi_{t_0},\gamma_0\gamma_1o_Y)<c.\] Set \(\gamma:=\gamma_0\gamma_1\).

For \(s\ge 0\), let \(\eta_s\in[\gamma o_Y,\xi]\) be the point satisfying \(d(\gamma o_Y,\eta_s)=s\). Increasing \(c\) by a uniform amount, if necessary, we have \[d(\xi_t,\eta_{t-t_0})<c \quad\text{for all } t\ge t_0.\] Thus, there exists \(c' > 0\) so that for all sufficiently large \(R\), \[\label{eqn46linecut} O_{R-c'}(\gamma o_Y,\eta_{t-t_0}) \subset O_R(o_Y,\xi_t) \subset O_{R+c'}(\gamma o_Y,\eta_{t-t_0})\tag{21}\] for all large \(t \ge t_0\).

Moreover, \[O_R(o_Y,\xi_t)\subset O_{R+c}(o_Y,\gamma o_Y).\] Using Lemma 14, we obtain \[\begin{align} \nu(O_R(o_Y,\xi_t)) &= \nu\bigl(\gamma\,\gamma^{-1}O_R(o_Y,\xi_t)\bigr) \\ &= \int_{\gamma^{-1}O_R(o_Y,\xi_t)} e^{\psi(\beta_x^\theta(e,\gamma^{-1}))}\,d\nu(x) \\ &= \int_{\gamma^{-1}O_R(o_Y,\xi_t)} e^{-\psi(\beta_{\gamma x}^\theta(e,\gamma))}\,d\nu(x) \\ &\asymp e^{-\psi(\mu(\gamma))} \nu(\gamma^{-1}O_R(o_Y,\xi_t)). \end{align}\] By 21 and Theorem 19, \[\begin{align} \nu(O_R(o_Y,\xi_t)) \asymp{} e^{-\psi(\mu(\gamma))} e^{2(\delta_{\bar\psi}(\mathsf P)-1) \mathsf{d}_{\bar\psi}(\gamma o_Y,\eta_{t-t_0})} \cdot \bigl(C_{\bar\psi} +\mathsf{d}_{\bar\psi}(\gamma o_Y,\eta_{t-t_0})\bigr)^{a_{\bar\psi}(\mathsf P)} . \end{align}\] Since 19 holds, the factor \(C_{\bar\psi}+\mathsf{d}_{\bar\psi}\) may be replaced, up to multiplicative constants, by \(C+\mathsf{d}_{\bar\psi}\) for any \(C\ge C_{\bar\psi}\). Finally, \(\xi_t\) and \(\eta_{t-t_0}\) are uniformly close, so Lemma 1 gives \[\mathsf{d}_{\bar\psi}(\gamma o_Y,\eta_{t-t_0}) \approx \mathsf{d}_{\bar\psi}(\gamma o_Y,\xi_t).\] This proves the theorem. 0◻

Complements of parabolic shadows↩︎

We also need an estimate for the complement of a shadow based at a parabolic limit point.

7 General form of the global shadow lemma↩︎

Continuing with the setting of section 6, we now prove the general form of the global shadow lemma. The estimate below should be viewed as the cusp version of the usual shadow lemma: when the point under consideration stays outside the horoballs in \(\mathcal{B}\), one recovers the ordinary orbit-shadow estimate (Theorem 3).

Recall that, for \(\xi\in\partial Y\) and \(t\ge 0\), we write \(\xi_t\in[o_Y,\xi]\) for the point satisfying \(d(o_Y,\xi_t)=t\).

8 Local properties of Patterson-Sullivan measures↩︎

In this section, we apply the global shadow lemma to local properties of Patterson-Sullivan measures. As before, let \(\Gamma<G\) be \(\theta\)-Morse relative to \(\mathcal{P}\), with Morse embedding \(f:Y\to X\) of a Gromov model \((Y,d)\) for \((\Gamma,\mathcal{P})\). Let \(\psi\in\mathfrak a_\theta^*\) be such that \[\psi>0 \quad\text{on } \mathcal{L}_f-\{0\} \quad \text{and} \quad \delta_\psi(\Gamma)=1.\] Let \(\nu\) be a \((\Gamma,\psi)\)-Patterson-Sullivan measure on \(\Lambda_\theta\). Throughout this section, we also assume that \[\theta = \operatorname{i}(\theta) \quad \text{and} \quad \psi=\psi\circ\operatorname{i}.\] Thus \(\bar\psi=\psi\).

The visual quasi-metric↩︎

We first define the higher-rank visual quasi-metric associated to \(\psi\). For distinct \(\xi,\eta\in\Lambda_\theta\), set \[\label{eqn46visualmetric} d_\psi(\xi,\eta) := e^{-\psi(\mathcal{G}^\theta(\xi,\eta))},\tag{26}\] and put \(d_\psi(\xi,\xi)=0\). Note that this is not the same object as the metric-like function \(\mathsf{d}_\psi\) on \(Y\). This function behaves like a metric: there exists \(c>0\) such that \[\label{eqn46trianglevisual} d_\psi(\xi,\eta) \le c\bigl(d_\psi(\xi,\zeta)+d_\psi(\zeta,\eta)\bigr)\tag{27}\] for all \(\xi,\eta,\zeta\in\Lambda_\theta\). For Anosov groups, this was proved in [6]; the same argument applies to a general Morse embedding.

For \(r>0\) and \(\xi\in\Lambda_\theta\), let \[B_\psi(\xi,r) := \{\eta\in\Lambda_\theta:d_\psi(\xi,\eta)<r\}.\]

We identify \(\partial Y\) with \(\Lambda_\theta\) via the \(\Gamma\)-equivariant homeomorphism \(f:\partial Y\to\Lambda_\theta\). Since \(\psi = \bar \psi\), Proposition 12 implies that, for all sufficiently large \(R>0\), there exist constants \(c_1,c_2>0\) such that \[\label{eqn46compareshadowsandballs} B_\psi\bigl(\xi,c_1 e^{-\mathsf{d}_\psi(o_Y,\xi_t)}\bigr) \subset O_R(o_Y,\xi_t) \subset B_\psi\bigl(\xi,c_2 e^{-\mathsf{d}_\psi(o_Y,\xi_t)}\bigr)\tag{28}\] for all \(\xi\in\Lambda_\theta\) and all \(t\ge 0\), where \(\xi_t\in[o_Y,\xi]\) is the point satisfying \(d(o_Y,\xi_t)=t\).

For later use, we rewrite the global shadow lemma in a compact form. If \(x\in Y\) lies in a horoball based at a point of \(\Gamma\xi_{\mathsf P}\), choose \(\gamma_x\in\Gamma\) so that \(\gamma_x o_Y\) is a closest orbit point to \(x\), and set \[h(x):=\mathsf{d}_\psi(\gamma_x o_Y,x),\quad \delta(x):=\delta_\psi(\mathsf P),\quad \text{and} \quad a(x):=a_\psi(\mathsf P).\]

If \(x\) lies outside the horoballs, we set \[h(x)=0,\quad \delta(x)=0,\quad \text{and} \quad a(x)=0.\] By the hypothesis \(\psi = \psi \circ \operatorname{i}\), the global shadow lemma, together with the ordinary shadow estimate in the thick part, gives the uniform estimate \[\label{eqn46uniformshadowlocal} \nu(O_R(o_Y,x)) \asymp e^{-\mathsf{d}_\psi(o_Y,x)} e^{(2\delta(x)-1)h(x)} \bigl(C_\psi+h_\psi(x)\bigr)^{a(x)} .\tag{29}\] The implied constants are independent of \(x\).

Local doubling↩︎

We first prove that \(\nu\) is locally doubling with respect to the visual quasi-metric \(d_\psi\).

Local reverse doubling↩︎

The next result is a local reverse doubling estimate: after shrinking the radius by a sufficiently large factor, the mass drops by any prescribed factor, uniformly in the center and the scale.

9 Hausdorff measure↩︎

In this section, we characterize when Patterson-Sullivan measures are Hausdorff measures, using the global shadow lemma we obtained. As in the previous sections, let \(\Gamma<G\) be \(\theta\)-Morse relative to \(\mathcal{P}\), with Morse embedding \[f:Y\to X\] from a Gromov model \((Y,d)\) for \((\Gamma,\mathcal{P})\). Let \(\psi\in\mathfrak a_\theta^*\) satisfy \[\psi>0 \quad \text{on } \mathcal{L}_f-\{0\} \quad \text{and} \quad \delta_\psi(\Gamma)=1.\] Let \(\nu\) be a \((\Gamma,\psi)\)-Patterson-Sullivan measure on \(\Lambda_\theta\). Throughout this section, we also assume that \[\theta = \operatorname{i}(\theta) \quad \text{and} \quad \psi=\psi\circ\operatorname{i}.\]

We equip \(\Lambda_\theta\) with the visual quasi-metric \(d_\psi\) defined in 26 . Since \(d_\psi\) satisfies the triangle inequality up to a multiplicative constant, as in 27 , the Vitali covering lemma holds for \(d_\psi\) by the standard proof; see, for instance, [6].

For \(s>0\), \(\varepsilon>0\), and \(B\subset\Lambda_\theta\), define \[\mathcal{H}_{\psi,\varepsilon}^s(B) := \inf \left\{ \sum_i (\operatorname{diam}_\psi U_i)^s : B\subset\bigcup_i U_i,\; \sup_i \operatorname{diam}_\psi U_i\le \varepsilon \right\},\] where \(\operatorname{diam}_\psi U := \sup_{\xi,\eta\in U} d_\psi(\xi,\eta)\). Then \[\mathcal{H}_\psi^s(B) := \lim_{\varepsilon\to 0} \mathcal{H}_{\psi,\varepsilon}^s(B)\] defines an outer measure and hence a Borel measure on \(\Lambda_\theta\); see [27] and [28]. We call \(\mathcal{H}_\psi^s\) the \(s\)-dimensional Hausdorff measure associated to \(d_\psi\). For \(s=1\), we write simply \[\mathcal{H}_\psi:=\mathcal{H}_\psi^1 .\]

The rest of this section is devoted to the proof of Theorem 27. Recall the hypothesis that \(\delta_\psi(\Gamma)=1\). First, the Hausdorff measure \(\mathcal{H}_\psi\) has the same conformality rule as the Patterson-Sullivan measure: for \(\gamma\in\Gamma\), \[\frac{d\gamma_*\mathcal{H}_\psi}{d\mathcal{H}_\psi}(\xi) = e^{\psi(\beta_\xi^\theta(e,\gamma))}.\] This was proved in [23] for Anosov subgroups, and the same proof applies in the present relatively Morse setting. Therefore, by the uniqueness of the \((\Gamma,\psi)\)-Patterson-Sullivan measure (Theorem 18), it suffices to prove that \[0<\mathcal{H}_\psi(\Lambda_\theta)<\infty.\]

Finite positive Hausdorff measure↩︎

We first establish local upper and lower estimates for \(\nu\) with respect to the visual quasi-metric. We identify \(\partial Y\) with \(\Lambda_\theta\) via \(f:\partial Y\to\Lambda_\theta\), and denote by \(\Lambda_{\theta}^{\rm con}\) the \(f\)-image of the conical limit set in \(\partial Y\).

Now the following finishes the proof of Theorem 27.

References↩︎

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1. The reference [8] treats a specific Gromov model, but the same proof applies to the Gromov models considered here.↩︎