Scale - invariant groups

Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite groupG to be scale-invariant if there is a nested sequence of finite index subgroupsGn that are all isomorphic toG 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 o Z, where F is any finite Abelian group; the solvable Baumslag–Solitar groups BS.1;m/; the affine groups AËZ , for anyA 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. Mathematics Subject Classification (2010). 20E08, 20F18, 05B45, 60B99, 60K35.