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 Gn 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≀Z, where F is any finite Abelian group; the solvable Baumslag–Solitar groups BS(1,m); the affine groups A ⋉ Zd, for any A≤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.
@article{10_4171_ggd_119,
author = {Volodymyr V. Nekrashevych and G\'abor Pete},
title = {Scale-invariant groups},
journal = {Groups, geometry, and dynamics},
pages = {139--167},
year = {2011},
volume = {5},
number = {1},
doi = {10.4171/ggd/119},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/119/}
}
TY - JOUR
AU - Volodymyr V. Nekrashevych
AU - Gábor Pete
TI - Scale-invariant groups
JO - Groups, geometry, and dynamics
PY - 2011
SP - 139
EP - 167
VL - 5
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/119/
DO - 10.4171/ggd/119
ID - 10_4171_ggd_119
ER -
%0 Journal Article
%A Volodymyr V. Nekrashevych
%A Gábor Pete
%T Scale-invariant groups
%J Groups, geometry, and dynamics
%D 2011
%P 139-167
%V 5
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/119/
%R 10.4171/ggd/119
%F 10_4171_ggd_119
