Scale-invariant groups

Details Scale-invariant groups Scale-invariant groups

2008-11-03

arxiv.org/abs/0811.0220v4

Mathematics

Group Theory

25 pages. The paper is reorganized, with more details in several arguments. To appear in Groups, Geometry and Dynamics

Scientific paper

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.

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