  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.
