${\mathrm{PGL}}_2({\mathbb{Q}}_p)$-orbit closures on a $p$-adic homogeneous space of infinite volume

Abstract

Let $\mathbb{K}$ be an unramified quadratic extension of ${\mathbb{Q}}_p$ for a fixed $p>2$. Projective general linear groups $G={\mathrm{PGL}}_2(\mathbb{K})$ and $H={\mathrm{PGL}}_2({\mathbb{Q}}_p)$ act transitively on Bruhat-Tits trees $T_G$ and $T_H$, respectively. We identify $G/H$ with the set of $H$-subtrees $G.T_H$. Let $\Gamma$ be a Schottky subgroup such that $\Gamma \backslash T_G$ is infinite volume and has an additional condition named high-branchedness, and let $\Lambda$ be its limit set. We classify $\Gamma$-orbits in $G/H$. Let $C=g_{C}H\in G/H$. As a generalization of Ratner's theorem, if $\Gamma \backslash g_C.T_H$ meets the convex core of $\Gamma \backslash T_G$, then the $\Gamma$-orbit of $C$ is either dense or closed in ${\mathcal{C}}_{\Lambda }=\{gH:\partial (g.T_H)\cap \Lambda \ne \varnothing \}$.

35 pages, 5 figures