ar X iv : 0 81 1 . 02 20 v 3 [ m at h . G R ] 1 3 A ug 2 00 9 SCALE - INVARIANT GROUPS

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 ≀ Z, where F is any finite Abelian group; the solvable Baumslag-Solitar groups BS(1, m); the affine groups A ⋉ Z d , for any A ≤ 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.