New simple lattices in products of trees and their projections

Abstract

Let $\Gamma \le \mathrm{Aut}(T_{d_1})\times \mathrm{Aut}(T_{d_2})$ be a group acting freely and transitively on the product of two regular trees of degree $d_1$ and $d_2$. We develop an algorithm which computes the closure of the projection of $\Gamma$ on $\mathrm{Aut}(T_{d_t})$ under the hypothesis that $d_t\ge 6$ is even and that the local action of $\Gamma$ on $T_{d_t}$ contains $\mathrm{Alt}(d_t)$. We show that if $\Gamma$ is torsion-free and $d_1=d_2=6$, exactly seven closed subgroups of $\mathrm{Aut}(T_6)$ arise in this way. We also construct two new infinite families of virtually simple lattices in $\mathrm{Aut}(T_6)\times \mathrm{Aut}(T_{4n})$ and in $\mathrm{Aut}(T_{2n})\times \mathrm{Aut}(T_{2n+1})$ respectively, for all $n\ge 2$. In particular we provide an explicit presentation of a torsion-free infinite simple group on $5$ generators and $10$ relations, that splits as an amalgamated free product of two copies of $F_3$ over $F_{11}$. We include information arising from computer-assisted exhaustive searches of lattices in products of trees of small degrees. In an appendix by Pierre-Emmanuel Caprace, some of our results are used to show that abstract and relative commensurator groups of free groups are almost simple, providing partial answers to questions of Lubotzky and Lubotzky-Mozes-Zimmer.

With an appendix by Pierre-Emmanuel Caprace

References in corpus (2)

• Groups acting on products of trees, tiling systems and analytic K-theory
• Indicability, residual finiteness, and simple subquotients of groups acting on trees

Cited by in corpus (4)

• Lattices in a product of trees, hierarchically hyperbolic groups, and virtual torsion-freeness
• Graphs and complexes of lattices
• Amenable uniformly recurrent subgroups and lattice embeddings
• Approximating simple locally compact groups by their dense locally compact subgroups