JournalsggdVol. 15, No. 1pp. 35–55

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

  • Sanghoon Kwon

    Catholic Kwandong University, Gangneung, Republic of Korea
  • Seonhee Lim

    Seoul National University, Republic of Korea
Limiting distribution of geodesics in a geometrically finite quotients of regular trees cover
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Abstract

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

We investigate, for a given time TT, the measure of the set of Γ\Gamma-equivalent classes of geodesics with distance at most N(T)N(T) from a sufficiently large fixed compact subset DD of Γ\T\Gamma\backslash\mathcal{T} up to time TT. We show that there exists a function N(T)N(T) such that for Bowen–Margulis measure μ\mu on the space Γ\GT\Gamma\backslash\mathcal{GT} of geodesics and the critical exponent δ\delta of Γ\Gamma,

limTμ({[l]Γ\GT ⁣:max0tTd(D,l(t))N(T)+y})=eqy/e2δy.\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)N(T): there exists a constant CC depending on Γ\Gamma and DD such that

N(T)=loge2δ/q(T(e2δq)2e2δC(e2δq)).N(T)=\log_{e^{2\delta/q}}\Big(\frac{T(e^{2\delta-q)}}{2e^{2\delta}-C(e^{2\delta}-q)}\Big).

Cite this article

Sanghoon Kwon, Seonhee Lim, Limiting distribution of geodesics in a geometrically finite quotients of regular trees. Groups Geom. Dyn. 15 (2021), no. 1, pp. 35–55

DOI 10.4171/GGD/590