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A global shadow lemma and logarithm law for geometrically finite Hilbert geometries

Harrison Bray, Giulio Tiozzo

TL;DR

The paper develops a global shadow lemma for Patterson–Sullivan measures in geometrically finite Hilbert geometries, and derives a Sullivan-type logarithm law for cusp excursions. It builds a general hyperbolic-space framework with δ-tempered parabolic subgroups, establishes a Dirichlet-type horoball packing, and transfers these results to strictly convex Hilbert geometries with C^1 boundary. The main contributions include a global shadow lemma, a Dirichlet-type covering theorem, a horoball counting/Khinchin-type theorem, and an explicit logarithm law linking cusp depth to the limit-set dimension via δ_Γ and δ_max. These results provide quantitative control of geodesic excursion into cusps and connect cusp dynamics to the geometric and dynamical structure of the limit set, with applications to Hilbert metrics and their Patterson–Sullivan theory.

Abstract

For geometrically finite group actions on hyperbolic metric spaces and under certain assumptions on the growth of parabolic subgroups, we prove a global shadow lemma for Patterson-Sullivan measures, as well as a Dirichlet-type theorem and a logarithm law for excursion of geodesics into cusps. We then apply these results to geometrically finite quotients of strictly convex Hilbert geometries with C^1 boundary.

A global shadow lemma and logarithm law for geometrically finite Hilbert geometries

TL;DR

The paper develops a global shadow lemma for Patterson–Sullivan measures in geometrically finite Hilbert geometries, and derives a Sullivan-type logarithm law for cusp excursions. It builds a general hyperbolic-space framework with δ-tempered parabolic subgroups, establishes a Dirichlet-type horoball packing, and transfers these results to strictly convex Hilbert geometries with C^1 boundary. The main contributions include a global shadow lemma, a Dirichlet-type covering theorem, a horoball counting/Khinchin-type theorem, and an explicit logarithm law linking cusp depth to the limit-set dimension via δ_Γ and δ_max. These results provide quantitative control of geodesic excursion into cusps and connect cusp dynamics to the geometric and dynamical structure of the limit set, with applications to Hilbert metrics and their Patterson–Sullivan theory.

Abstract

For geometrically finite group actions on hyperbolic metric spaces and under certain assumptions on the growth of parabolic subgroups, we prove a global shadow lemma for Patterson-Sullivan measures, as well as a Dirichlet-type theorem and a logarithm law for excursion of geodesics into cusps. We then apply these results to geometrically finite quotients of strictly convex Hilbert geometries with C^1 boundary.

Paper Structure

This paper contains 24 sections, 27 theorems, 116 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Shadow lemma for hyperbolic metric spaces
  3. Dirichlet Theorem
  4. Applications
  5. Historical remarks
  6. Structure of the paper
  7. Acknowledgements
  8. Hyperbolic metric spaces
  9. Gromov product and inner triangle
  10. Busemann functions
  11. Extension to the boundary and horoballs
  12. Projections
  13. Shadows
  14. Fellow traveling
  15. Projections and Busemann functions
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let $\Omega$ be a strictly convex domain in $\mathbb RP^n$ with $C^1$ boundary and $\Gamma<\mathop{\mathrm{PSL}}\nolimits(n+1,\mathbb R)$ a discrete geometrically finite group which preserves $\Omega$. Assume the convex hull of the limit set $C_\Gamma$ is hyperbolic with respect to the Hilbert metri for any $t > 0$, where $\Pi = \{ \textup{id}\}$ if $\xi_t$ lies in the non-cuspidal part, and othe

Figures (5)

  • Figure 1: Inner triangles in Gromov hyperbolic metric spaces. The point $b$ is such that $\langle y, z \rangle_x = d(x, b) = d(x, c)$.
  • Figure 2: An approximate tree for the proof of Lemma \ref{['L:sqrt']}.
  • Figure 3: For the proof of Lemma \ref{['L:average_is_gromov']}, in the case that $q_1 \in [o, x]$. Note that $x$ and $z$ are within $O(\alpha)$ of the inner triangle $\Delta(o,\xi_1,\xi_2)$.
  • Figure 4: Left: the space in Example \ref{['ex:GM_space']}, constructed by attaching combinatorial horoballs (in red) to the Cayley graph a free group. Right: a detail of a combinatorial horoball, with a geodesic path from $g$ to $ga^l$.
  • Figure 5: The set-up of Lemma \ref{['L:geometriclemma']}.

Theorems & Definitions (50)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5: Dirichlet-type Theorem
  • Theorem 1.6: Logarithm Law
  • Corollary 1.7: Singularity with harmonic measure
  • Proposition 2.1: see e.g. MaherTiozzo, Proposition 2.2
  • Lemma 2.2
  • proof
  • ...and 40 more