$\operatorname{PGL}_{2}(\mathbb{Q}_{p})$-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 $G=\operatorname{PGL}_{2}(\mathbb{K})$ and $H=\operatorname{PGL}_{2}(\mathbb{Q}_{p})$ 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 $ {\cal{C}}_{\Lambda}=\{g H: \partial(g.T_{H})\cap\Lambda\neq\varnothing\}$.