The EMS Publishing House is now EMS Press and has its new home at

Please find all EMS Press journals and articles on the new platform.

Groups, Geometry, and Dynamics

Full-Text PDF (227 KB) | Metadata | Table of Contents | GGD summary
Online access to the full text of Groups, Geometry, and Dynamics is restricted to the subscribers of the journal, who are encouraged to communicate their IP-address(es) to their agent or directly to the publisher at
Volume 15, Issue 1, 2021, pp. 35–55
DOI: 10.4171/GGD/590

Published online: 2020-12-24

Limiting distribution of geodesics in a geometrically finite quotients of regular trees

Sanghoon Kwon[1] and Seonhee Lim[2]

(1) Catholic Kwandong University, Gangneung, Republic of Korea
(2) Seoul National University, Republic of Korea

Let $\mathcal T$ be a $(q+1)$-regular tree and let $\Gamma$ be a geometrically finite discrete subgroup of the group $\operatorname{Aut}(\mathcal T)$ of automorphisms of $\mathcal T$. In this article, we prove an extreme value theorem on the distribution of geodesics in a non-compact quotient graph $\Gamma\backslash\mathcal{T}$. Main examples of such graphs are quotients of a Bruhat–Tits tree by non-cocompact discrete subgroups $\Gamma$ of $\operatorname{PGL}(2,\mathbf{K})$ of a local field $\mathbf{K}$ of positive characteristic.

We investigate, for a given time $T$, the measure of the set of $\Gamma$-equivalent classes of geodesics with distance at most $N(T)$ from a sufficiently large fixed compact subset $D$ of $\Gamma\backslash\mathcal{T}$ up to time $T$. We show that there exists a function $N(T)$ such that for Bowen–Margulis measure $\mu$ on the space $\Gamma\backslash\mathcal{GT}$ of geodesics and the critical exponent $\delta$ of $\Gamma$, $$\lim_{T\to\infty}\mu(\{[l]\in\Gamma\backslash\mathcal{GT}\colon \underset{0\le t \le T}{\textrm{max}}d(D,l(t))\le N(T)+y\})=e^{-q^y/e^{2\delta y}}.$$ In fact, we obtain a precise formula for $N(T)$: there exists a constant $C$ depending on $\Gamma$ and $D$ such that $$N(T)=\log_{e^{2\delta/q}}\Big(\frac{T(e^{2\delta-q)}}{2e^{2\delta}-C(e^{2\delta}-q)}\Big).$$

Keywords: Extreme value theory, trees, geodesic flow

Kwon Sanghoon, Lim Seonhee: Limiting distribution of geodesics in a geometrically finite quotients of regular trees. Groups Geom. Dyn. 15 (2021), 35-55. doi: 10.4171/GGD/590