Ping-pong in the projective plane over a nonarchimedean field

TL;DR

This paper demonstrates the existence of specific undistorted subgroups within lattices of SL_3 over nonarchimedean fields, revealing new examples of complex subgroup structures in nonarchimedean algebraic groups.

Contribution

It constructs the first known finitely generated Zariski-dense infinite-covolume discrete subgroups that are not virtually free in this setting.

Findings

Existence of undistorted Z^2 * Z subgroups in lattices of SL_3(k)

First examples of such subgroups with infinite covolume

Contrasts with the open case of SL_3(Z)

Abstract

We show that any lattice in $\mathrm{SL}_3(k)$, where $k$ is a nonarchimedean local field, contains an undistorted subgroup isomorphic to the free product $\mathbb{Z}^2*\mathbb{Z}$. To our knowledge, the subgroups we construct give the first examples in the literature of finitely generated Zariski-dense infinite-covolume discrete subgroups of an almost simple group over a nonarchimedean local field that are not virtually free. Our result is in contrast to the case of $\mathrm{SL}_3(\mathbb{Z})$, in which the existence of a $\mathbb{Z}^2*\mathbb{Z}$ subgroup remains open.