Ping-pong in the projective plane over a nonarchimedean field

TL;DR

The paper addresses the existence of finitely generated, Zariski-dense, infinite-covolume discrete subgroups of $\mathrm{SL}_3(k)$ over nonarchimedean local fields that are not virtually free. It constructs, from a lattice $\Lambda<\mathrm{SL}_3(k)$ and a $\mathbb{Z}^2$ subgroup $\Delta'$, a finite-index $\Delta\leqslant \Delta'$ and an infinite-order element $g\in \Lambda$ such that $\langle \Delta, g^R\rangle$ is undistorted and decomposes as $\Delta * \langle g^R\rangle$ via a ping-pong argument on the Furstenberg boundary. The method relies on regular elements with attracting/repelling flags and a Cartan-projection growth framework to realize a ping-pong dynamic, yielding a Zariski-dense subgroup with infinite covolume. Consequently, the paper provides the first examples of such subgroups in almost simple $k$-groups for nonarchimedean $k$, including a $\,\mathbb{Z}^2 * \mathbb{Z}$ subgroup of $\mathrm{SL}_3(\mathbb{F}_q[t])$, and contrasts with the (open) case of $\mathrm{SL}_3(\mathbb{Z})$. These results illuminate boundary dynamics and subgroup geometry in nonarchimedean Lie-type groups.

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.

Theorems & Definitions (9)

• Theorem 1
• Remark 2
• Corollary 3
• proof : Proof of Theorem \ref{['thm:main']}
• Remark 4
• Remark 5
• Proposition 6
• proof
• Remark 7