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Testing Cayley graph densities. (Tester les densités de graphes de Cayley.) (French) Zbl 1191.20025

Summary: We present a computer-assisted analysis of combinatorial properties of the Cayley graphs of certain finitely generated groups: given a group with a finite set of generators, we study the density of the corresponding Cayley graph, that is, the least upper bound for the average vertex degree (= number of adjacent edges) of any finite subgraph. It is known that an \(m\)-generated group is amenable if and only if the density of the corresponding Cayley graph equals \(2m\). We test amenable and non-amenable groups, and also groups for which amenability is unknown. In the latter class we focus on Richard Thompson’s group \(F\).

MSC:

20F05 Generators, relations, and presentations of groups
20F65 Geometric group theory
43A07 Means on groups, semigroups, etc.; amenable groups
20-04 Software, source code, etc. for problems pertaining to group theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
05C42 Density (toughness, etc.)
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References:

[1] Belk, J. M.; Brown, K. S., Forest diagrams for elements of Thompson’s group \({F}\), Internat. J. Algebra Comput., 15 (5-6), 815-850 (2005) · Zbl 1163.20310
[2] Cannon, J. W.; Floyd, W. J.; Parry, W. R., Introductory notes on Richard Thompson’s groups, L’Enseignement Mathématique, 42 (2), 215-256 (1996) · Zbl 0880.20027
[3] Ceccherini-Silberstein, T.; Grigorchuk, R.; de la Harpe, P., Amenability and paradoxal decompositions for pseudogroups and for discrete metric spaces, Proc. Steklov Inst. Math., 224 (1), 57-97 (1999) · Zbl 0968.43002
[4] Guba, V. S., On the properties of the Cayley graph of Richard Thompson’s group \({F}\), Internat. J. Algebra Comput., 14 (5-6), 677-702 (2004) · Zbl 0965.20025
[5] de la Harpe, P., Topics in geometric group theory (2000) · Zbl 1088.20021
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