Geometry & Topology
- Geom. Topol.
- Volume 22, Number 7 (2018), 4163-4204.
Indicability, residual finiteness, and simple subquotients of groups acting on trees
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.
Geom. Topol., Volume 22, Number 7 (2018), 4163-4204.
Received: 22 August 2017
Accepted: 13 April 2018
First available in Project Euclid: 14 December 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 20E08: Groups acting on trees [See also 20F65]
Secondary: 22D05: General properties and structure of locally compact groups
Caprace, Pierre-Emmanuel; Wesolek, Phillip. Indicability, residual finiteness, and simple subquotients of groups acting on trees. Geom. Topol. 22 (2018), no. 7, 4163--4204. doi:10.2140/gt.2018.22.4163. https://projecteuclid.org/euclid.gt/1544756697