we proved closed/dense dichotomy of $\operatorname{pgl}_2(\mathbb{q}_p)$-orbit closures in the renormalized frame bundle of a $p$-adic homogeneous space of infinite volume. our result is a generalization of ratner’s theorem and the result of mcmullen, mohammadi, and oh in 2017 into non-archimedean local fields.

let $\mathbb{k}$ be an unramified quadratic extension of $\mathbb{q}_p$. our homogeneous space is a quotient space of $\operatorname{\mathbb{k}}$ by a certain class of schottky subgroups. using the main tools of mcmullen, mohammadi, and oh, we introduced the necessary properties of schottky subgroups and used the bruhat-tits tree $\operatorname{pgl}_2$. in this talk, we introduce the highly-branched schottky subgroups and steps for the proof of the main theorem.

this is a joint work with seonhee lim.