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

Harrison Bray; Giulio Tiozzo

doi:10.3934/jmd.2025008

Article Contents

2025, Volume 21: 443-496. Doi: 10.3934/jmd.2025008

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

Harrison Bray^1, and
Giulio Tiozzo^2,

1.
Mathematical Sciences Department, George Mason University, 4400 University Dr., Fairfax, VA 22030, USA
2.
Department of Mathematics, University of Toronto, 40 St. George St., Toronto, ON, M5S 2E4, Canada

Received: January 22, 2023

Revised: December 20, 2024

Published online: July 9, 2025

HB: Partially supported by the Simons Foundation.
GT: Partially supported by NSERC and an Ontario Early Researcher Award.

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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.

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Mathematics Subject Classification: Primary: 37D40; Secondary: 20F67, 53B40.

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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) $

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Figure 2. An approximate tree for the proof of Lemma 2.15

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Figure 3. For the proof of Lemma 2.16, 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) $

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Figure 4. Left: the space in Example 3.7, constructed by attaching combinatorial horoballs (in red) to the Cayley graph of a free group. Right: a detail of a combinatorial horoball, with a geodesic path from $ g $ to $ ga^l $

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Figure 5. The set-up of Lemma 3.8

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Figure 6. Case 2 in the proof of Theorem 1.4

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Figure 7. For the proof of Theorem 6.9. The red horoballs correspond to the collection of horoballs $ \tilde H_p $ which have been rescaled by $ O(\alpha) $ so that if some choice of geodesic $ [o,\xi) $ cuts a horoball $ H_p $, then any other choice of geodesic cuts $ \tilde H_p $

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