Scale-invariant groups

TL;DR

This paper investigates the concept of scale-invariance in infinite groups, disproves a conjecture linking scale-invariance to polynomial growth, and constructs examples of scale-invariant tilings in certain groups.

Contribution

It demonstrates that several well-known groups are scale-invariant, countering previous conjectures, and constructs new scale-invariant tilings for specific Cayley graphs.

Findings

Certain lamplighter groups are scale-invariant.

Baumslag-Solitar groups are scale-invariant.

Scale-invariant tilings exist for some Cayley graphs of the Heisenberg group.

Abstract

Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite group G to be scale-invariant if there is a nested sequence of finite index subgroups G_n that are all isomorphic to G and whose intersection is a finite group. He conjectured that every scale-invariant group has polynomial growth, hence is virtually nilpotent. We disprove his conjecture by showing that the following groups (mostly finite-state self-similar groups) are scale-invariant: the lamplighter groups F\wr\Z, where F is any finite Abelian group; the solvable Baumslag-Solitar groups BS(1,m); the affine groups A\ltimes\Z^d, for any A\leq GL(\Z,d). However, the conjecture remains open with some natural stronger notions of scale-invariance for groups and transitive graphs. We construct scale-invariant tilings of certain Cayley graphs of the discrete Heisenberg group, whose existence is not immediate just from the scale-invariance of the group. We also note that torsion-free non-elementary hyperbolic groups are not scale-invariant.