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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.

MSC:
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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References:
[1] D. Aldous and R. Lyons, Processes on unimodular random networks. Electron. J. Probab. 12 (2007), 1454-1508. · Zbl 1131.60003
[2] N. Alon, I. Benjamini, and A. Stacey, Percolation on finite graphs and isoperimetric inequalities. Ann. Probab. 32 (2004), 1727-1745. · Zbl 1046.05071
[3] P. Antal and A. Pisztora, On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 (1996), 1036-1048. · Zbl 0871.60089
[4] A. Bandyopadhyay, J. Steif, and Á. Timár, On the cluster size distribution for percolation on some general graphs. Rev. Mat. Iberoam. 26 (2010), 529-550. · Zbl 1203.60142
[5] D. J. Barsky, G. R. Grimmett, and C. M. Newman, Percolation in half-spaces: equality of critical densities and continuity of the percolation probability. Probab. Theory Related Fields 90 (1991), 111-148. · Zbl 0727.60118
[6] L. Bartholdi, R. Grigorchuk, and V. Nekrashevych, From fractal groups to fractal sets. In Fractals in Graz 2001 , Trends Math., Birkhäuser, Basel 2003, 25-118. · Zbl 1037.20040
[7] L. Bartholdi, R. I. Grigorchuk, and Z. Šuniḱ, Branch groups. In Handbook of algebra , Vol. 3, North-Holland, Amsterdam 2003, 989-1112. · Zbl 1140.20306
[8] L. Bartholdi and Z. Šuniḱ, Some solvable automaton groups. In Topological and asymptotic aspects of group theory , Contemp. Math. 394, Amer. Math. Soc., Providence, RI, 2006, 11-29. · Zbl 1106.20021
[9] G. Baumslag and D. Solitar, Some two-generator one-relator non-Hopfian groups. Bull. Amer. Math. Soc. 68 (1962), 199-201. · Zbl 0108.02702
[10] M. E. B. Bekka and A. Valette, Group cohomology, harmonic functions and the first L2-Betti number. Potential Anal. 6 (1997), 313-326. · Zbl 0882.22013
[11] I. Belegradek, On co-Hopfian nilpotent groups. Bull. London Math. Soc. 35 (2003), 805-811. · Zbl 1042.20022
[12] I. Benjamini, Post on personal website, 2006:
[13] I. Benjamini, R. Lyons, and O. Schramm, Percolation perturbations in potential theory and random walks. In Random walks and discrete potential theory (Cor- tona, 1997), Sympos. Math. 39, Cambridge University Press, Cambridge 1999, 56-84. · Zbl 0958.05121
[14] I. Benjamini and O. Schramm, Percolation beyond Zd , many questions and a few answers. Electron. Comm. Probab. 1 (1996), 71-82. · Zbl 0890.60091
[15] I. Benjamini and O. Schramm, Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 (2001), no. 23. · Zbl 1010.82021
[16] M. Bridson, A. Hinkkanen, and G. Martin, Quasiregular self-mappings of man- ifolds and word hyperbolic groups. Compos. Math. 143 (2007), 1613-1622. · Zbl 1131.20031
[17] A. M. Brunner and S. Sidki, The generation of GL.n; Z/ by finite state automata. Internat. J. Algebra Comput. 8 (1998), 127-139. · Zbl 0923.20023
[18] D. C. Brydges, Lectures on the renormalisation group. In Statistical mechanics , IAS/Park City Math. Ser. 16, Amer. Math. Soc., Providence, RI, 2009, 7-93. · Zbl 1186.82033
[19] D. Chen and Y. Peres, Anchored expansion, percolation and speed. With an · Zbl 1069.60093
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