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Groups, Geometry, and Dynamics


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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