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

Pierre-Emmanuel Caprace and Phillip Wesolek

Geometry & Topology 22 (2018) 4163–4204

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 . The second ensures that irreducible cocompact lattices in a product of nondiscrete 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 nondiscrete Burger–Mozes universal group of automorphisms of a tree with an arbitrary prescribed local action admits a compactly generated closed subgroup with a nondiscrete simple quotient. As applications, we answer a question of D Wise by proving the nonresidual 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.

trees, lattices in products, locally compact groups
Mathematical Subject Classification 2010
Primary: 20E08
Secondary: 22D05
Received: 22 August 2017
Accepted: 13 April 2018
Published: 6 December 2018
Proposed: Martin Bridson
Seconded: Benson Farb, Walter Neumann
Pierre-Emmanuel Caprace
Institut de Recherche en Mathématiques et Physique
Université catholique de Louvain
Phillip Wesolek
Department of Mathematical Sciences
Binghamton University
Binghamton, NY
United States