## Geometry & Topology

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

#### 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 $ℤ$. 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.

#### Article information

Source
Geom. Topol., Volume 22, Number 7 (2018), 4163-4204.

Dates
Accepted: 13 April 2018
First available in Project Euclid: 14 December 2018

https://projecteuclid.org/euclid.gt/1544756697

Digital Object Identifier
doi:10.2140/gt.2018.22.4163

Mathematical Reviews number (MathSciNet)
MR3890774

Zentralblatt MATH identifier
06997386

#### Citation

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

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