2026/06/02 by Pierre-Emmanuel Caprace, Justin Vast
Mathematics · #math.GR
Let Γ be a group acting properly and cocompactly on the product of two trees T1 and T2. An anti-torus is a non-periodic flat plane in T1 × T2 that is the convex hull of two secant periodic lines. That notion was introduced by Dani Wise as a tool to show that Γ is irreducible. We establish a new criterion ensuring the existence of anti-tori, and use it to prove that if Γ is an S-arithmetic lattice in a product of simple algebraic groups of rank one, then T1× T2 contains anti-tori. As a byproduct, we obtain a sufficient condition ensuring that a group defined by a bi-reversible automaton contains non-abelian free sub-semigroups. We also introduce a new class of irreducible lattices acting regularly on the vertex set of a product of two trees, and containing anti-tori that are fractal aperiodic tilings of the plane. This establishes a connection between lattices in products of trees and substitution tilings.