Indicability, residual finiteness, and simple subquotients of groups acting on trees

TL;DR

This paper presents three new results on groups acting on trees, including conditions for virtual indicability, non-residual finiteness of certain lattices, and existence of simple quotients in universal groups, advancing understanding of group actions on trees.

Contribution

It introduces novel theorems linking group actions on trees with properties like indicability, residual finiteness, and simple quotients, with applications to longstanding questions.

Findings

Locally compact groups acting on trees are virtually indicable under certain conditions.

Irreducible cocompact lattices in product groups are shown to be non-residually finite.

Universal groups of automorphisms of trees have compactly generated subgroups with simple quotients.

Abstract

We establish three independent results on groups acting on trees. The first implies that a compactly generated locally compact group which acts continuously on a locally finite tree with nilpotent local action and no global fixed point is virtually indicable; that is to say, it has a finite index subgroup which surjects onto $\mathbf{Z}$. The second ensures that irreducible cocompact lattices in a product of non-discrete locally compact groups such that one of the factors acts vertex-transitively on a tree with a nilpotent local action cannot be residually finite. This is derived from a general result, of independent interest, on irreducible lattices in product groups. The third implies that every non-discrete Burger-Mozes universal group of automorphisms of a tree with an arbitrary prescribed local action admits a compactly generated closed subgroup with a non-discrete simple quotient. As applications, we answer a question of D. Wise by proving the non-residual finiteness of a certain lattice in a product of two regular trees, and we obtain a negative answer to a question of C. Reid, concerning the structure theory of locally compact groups.