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

Abstract

Let $\mathcal{T}$ be a $(q+1)$-regular tree and let $\Gamma$ be a geometrically finite discrete subgroup of the group $\mathrm{Aut}(\mathcal{T})$ 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 $\mathrm{PGL}(2,\mathbf{K})$ 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 $\Gamma \backslash \mathcal{G}\mathcal{T}$ of geodesics and the critical exponent $\delta$ of $\Gamma$, ${\lim }_{T\to \infty }\mu (\{[l]\in \Gamma \backslash \mathcal{G}\mathcal{T}:\underset{0\le t\le T}{\max }d(D,l(t))\le N(T)+y\})=e^{-q^y/e^{2\delta y}}.$ In fact, we obtain a precise formula for $N(T)$: there exists a constant $C$ depending on $\Gamma$ and $D$ such that $N(T)={\log }_{e^{2\delta /q}}\left(\frac{T(e^{2\delta -q)}}{2e^{2\delta }-C(e^{2\delta }-q)}\right).$