Scale-invariant groups

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 $\mathrm{F}\wr \mathbb{Z}$, where $\mathrm{F}$ is any finite Abelian group; the solvable Baumslag–Solitar groups $\mathrm{BS}(1,m)$; the affine groups $A$ ⋉ ${\mathbb{Z}}^d$, for any $A\le \mathrm{GL}(\mathbb{Z},d)$. 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.