Patterson–Sullivan Measures and Growth of Relatively Hyperbolic Groups

Local limit theorems in relatively hyperbolic groups II: the non-spectrally degenerate case

“…Before proving this proposition, let us introduce some notions of geometric measure theory from [MYJ20] and some constructions of [DG20] and [Yan22]. Let be a metric space.…”

“…Let be a conical limit point. Following Yang [Yan22], the partial shadow at is the set of points in the Bowditch boundary such that there is a geodesic ray in containing an -transition point in the ball .…”

“…Recall that an (η 1 , η 2 )-transition point on a geodesic α in the Cayley graph Cay(Γ, S) is a point γ such that for any coset γ 0 H of a parabolic subgroup, the part of α consisting of points at distance at most η 2 from γ is not contained in the η 1 -neighborhood of γ 0 H. Let ξ be a conical limit point. Following Yang [Yan22], the partial shadow Ω η 1 ,η 2 (γ) at γ ∈ Γ is the set of points ξ in the Bowditch boundary such that there is a geodesic ray [e, ξ) in Cay(Γ, S) containing an (η 1 , η 2 )-transition point in the ball B(γ, 2η 2 ).…”

Counting conjugacy classes in groups with contracting elements

“…Let $$h_-, h_+$$ be the fixed points of $$h$$ in the Bowditch boundary. By [55, Lemma 2.20], there exists a sequence of $$(\epsilon , R)$$‐transitional points along any bi‐infinite geodesic $$\gamma$$ between $$h_-, h_+$$, where $$\epsilon , R$$ are uniform depending only on $$(G, \mathcal {P})$$. Moreover, if pickup any such transitional point $$x$$, then $$\langle h\rangle x$$ is uniformly close to $$\gamma$$.…”

“…Sketch of the proof We refer to [55] for related notions and [23, Proposition 7.8] for a similar argument. Let $$h_-, h_+$$ be the fixed points of $$h$$ in the Bowditch boundary.…”