JournalsggdVol. 5, No. 1pp. 139–167

Scale-invariant groups

  • Volodymyr V. Nekrashevych

    Texas A&M University, College Station, United States
  • Gábor Pete

    University of Toronto, Toronto, Canada
Scale-invariant groups cover
Download PDF


Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite group GG to be scale-invariant if there is a nested sequence of finite index subgroups GnG_n that are all isomorphic to GG 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 FZ\mathrm{F}\wr \mathbb{Z}, where F\mathrm{F} is any finite Abelian group; the solvable Baumslag–Solitar groups BS(1,m)\mathrm{BS}(1,m); the affine groups AAZd\mathbb{Z}^d, for any AGL(Z,d)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.

Cite this article

Volodymyr V. Nekrashevych, Gábor Pete, Scale-invariant groups. Groups Geom. Dyn. 5 (2011), no. 1, pp. 139–167

DOI 10.4171/GGD/119