Fractal anti-tori

Abstract

Let $\Gamma$ be a group acting properly and cocompactly on the product of two trees $T_1$ and $T_2$. An anti-torus is a non-periodic flat plane in $T_1 \times T_2$ that is the convex hull of two secant periodic lines. That notion was introduced by Dani Wise as a tool to show that $\Gamma$ is irreducible. We establish a new criterion ensuring the existence of anti-tori, and use it to prove that if $\Gamma$ is an $S$-arithmetic lattice in a product of simple algebraic groups of rank one, then $T_1\times T_2$ 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.