Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite group to be scale-invariant if there is a nested sequence of finite index subgroups that are all isomorphic to 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 , where is any finite Abelian group; the solvable Baumslag–Solitar groups ; the affine groups ⋉ , for any . 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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Volodymyr V. Nekrashevych, Gábor Pete, Scale-invariant groups. Groups Geom. Dyn. 5 (2011), no. 1, pp. 139–167DOI 10.4171/GGD/119