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 Gn 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, where F is any finite Abelian group; the solvable Baumslag-Solitar groups BS(1,m); the affine groups Ad, for any A≤ GL(,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.