# Groups, Geometry, and Dynamics

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**Volume 5, Issue 1, 2011, pp. 139–167**

**DOI: 10.4171/GGD/119**

Published online: 2011-01-19

Scale-invariant groups

Volodymyr V. Nekrashevych^{[1]}and Gábor Pete

^{[2]}(1) Texas A&M University, College Station, United States

(2) University of Toronto, Toronto, Canada

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 $\mathrm{F}\wr \mathbb{Z}$, where $\mathrm{F}$ is any finite Abelian group; the solvable Baumslag–Solitar groups $\mathrm{BS}(1,m)$; the affine groups $A$ ⋉ $\mathbb{Z}^d$, for any $A\leq \mathrm{GL}(\mathbb{Z},d)$. 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.

*Keywords: *Self-similar groups, finite automata, expanding maps, co-Hopfian groups, percolation renormalization, scale invariance, Følner monotilings, Heisenberg group, lamplighter group, affine groups

Nekrashevych Volodymyr, Pete Gábor: Scale-invariant groups. *Groups Geom. Dyn.* 5 (2011), 139-167. doi: 10.4171/GGD/119