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Scale-invariant groups. (English) Zbl 1236.20031
Summary: 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 \(F\wr\mathbb Z\), where \(F\) is any finite Abelian group; the solvable Baumslag-Solitar groups \(\text{BS}(1,m)\); the affine groups \(A\ltimes\mathbb Z^d\), for any \(A\leq\text{GL}(\mathbb Z,d)\). 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.

20E08 Groups acting on trees
20E07 Subgroup theorems; subgroup growth
20E15 Chains and lattices of subgroups, subnormal subgroups
20F05 Generators, relations, and presentations of groups
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
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