Lattices in products of trees and a theorem of H.C. Wang

TL;DR

This paper investigates cocompact lattices with dense projections in products of groups acting on trees, establishing finiteness results under certain local transitivity conditions.

Contribution

It proves that such lattices are contained in only finitely many discrete subgroups when each factor acts on a regular tree with specific local properties.

Findings

Finiteness of discrete subgroups containing the lattice

Conditions under which lattices are contained in finitely many discrete groups

Application of local transitivity conditions to lattice structure

Abstract

We study cocompact lattices with dense projections in a product $G_1 \times G_2$ of locally compact groups and show, under the assumption that each $G_i$ is a closed subgroup of the automorphism group $Aut(T_i)$ of a regular tree satisfying certain local transitivity conditions, that such a lattice is contained in only finitely many discrete subgroups of $G_1 \times G_2$.