Arzhantseva, Goulnara N.; Guba, Victor S.; Lustig, Martin; Préaux, Jean-Philippe Testing Cayley graph densities. (Tester les densités de graphes de Cayley.) (French) Zbl 1191.20025 Ann. Math. Blaise Pascal 15, No. 2, 233-286 (2008). 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 \(F\). Cited in 5 Documents 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.) Keywords:Cayley graphs; finitely generated groups; amenability; Thompson group \(F\); computer-assisted group theory PDFBibTeX XMLCite \textit{G. N. Arzhantseva} et al., Ann. Math. Blaise Pascal 15, No. 2, 233--286 (2008; Zbl 1191.20025) Full Text: DOI Numdam EuDML 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 · doi:10.1142/S021819670500261X [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 · doi:10.1142/S021819670400192X [5] de la Harpe, P., Topics in geometric group theory (2000) · Zbl 1088.20021 [6] Lyndon, R. S.; Schupp, P. E., Combinatorial group theory (2001) · Zbl 0997.20037 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.