Testing Cayley graph densities

Goulnara N. Arzhantseva[1]; Victor S. Guba[2]; Martin Lustig[3]; Jean-Philippe Préaux[4]

  • [1] Section de Mathématiques Université de Genève CP 64, 1211 Genève 4 SWITZERLAND
  • [2] Department of Mathematics Vologda State University 6 S. Orlov St., Vologda 160600 RUSSIA
  • [3] LATP, UMR CNRS 6632 Mathématiques Université d’Aix-Marseille III Avenue Escadrille Normandie-Niemen 13397 Marseille cédex 20 FRANCE
  • [4] Centre de recherche de l’Armée de l’air Ecole de l’air 13661 Salon-air FRANCE LATP, UMR CNRS 6632 Université de Provence 39 rue Joliot-Curie 13453 Marseille cédex 13 FRANCE

Annales mathématiques Blaise Pascal (2008)

  • Volume: 15, Issue: 2, page 233-286
  • ISSN: 1259-1734

Abstract

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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 to 2 m . 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 .

How to cite

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Arzhantseva, Goulnara N., et al. "Testing Cayley graph densities." Annales mathématiques Blaise Pascal 15.2 (2008): 233-286. <http://eudml.org/doc/10562>.

@article{Arzhantseva2008,
abstract = {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 to $2m$. 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 $F$.},
affiliation = {Section de Mathématiques Université de Genève CP 64, 1211 Genève 4 SWITZERLAND; Department of Mathematics Vologda State University 6 S. Orlov St., Vologda 160600 RUSSIA; LATP, UMR CNRS 6632 Mathématiques Université d’Aix-Marseille III Avenue Escadrille Normandie-Niemen 13397 Marseille cédex 20 FRANCE; Centre de recherche de l’Armée de l’air Ecole de l’air 13661 Salon-air FRANCE LATP, UMR CNRS 6632 Université de Provence 39 rue Joliot-Curie 13453 Marseille cédex 13 FRANCE},
author = {Arzhantseva, Goulnara N., Guba, Victor S., Lustig, Martin, Préaux, Jean-Philippe},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Amenability; Thompson’s group $F$; computer-assisted analysis; Cayley graphs; finitely generated groups; amenability; Thompson group ; computer-assisted group theory},
language = {eng},
month = {7},
number = {2},
pages = {233-286},
publisher = {Annales mathématiques Blaise Pascal},
title = {Testing Cayley graph densities},
url = {http://eudml.org/doc/10562},
volume = {15},
year = {2008},
}

TY - JOUR
AU - Arzhantseva, Goulnara N.
AU - Guba, Victor S.
AU - Lustig, Martin
AU - Préaux, Jean-Philippe
TI - Testing Cayley graph densities
JO - Annales mathématiques Blaise Pascal
DA - 2008/7//
PB - Annales mathématiques Blaise Pascal
VL - 15
IS - 2
SP - 233
EP - 286
AB - 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 to $2m$. 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 $F$.
LA - eng
KW - Amenability; Thompson’s group $F$; computer-assisted analysis; Cayley graphs; finitely generated groups; amenability; Thompson group ; computer-assisted group theory
UR - http://eudml.org/doc/10562
ER -

References

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  1. J. M. Belk, K. S. Brown, Forest diagrams for elements of Thompson’s group F , Internat. J. Algebra Comput. 15 (5-6) (2005), 815-850 Zbl1163.20310
  2. J. W. Cannon, W. J. Floyd, W. R. Parry, Introductory notes on Richard Thompson’s groups, L’Enseignement Mathématique 42 (2) (1996), 215-256 Zbl0880.20027
  3. T. Ceccherini-Silberstein, R. Grigorchuk, P. de la Harpe, Amenability and paradoxal decompositions for pseudogroups and for discrete metric spaces, Proc. Steklov Inst. Math. 224 (1) (1999), 57-97 Zbl0968.43002MR1721355
  4. V. S. Guba, On the properties of the Cayley graph of Richard Thompson’s group F , Internat. J. Algebra Comput. 14 (5-6) (2004), 677-702 Zbl0965.20025MR1786869
  5. P. de la Harpe, Topics in geometric group theory, (2000), University of Chicago Press, Chicago, IL Zbl1088.20021
  6. R.S. Lyndon, P.E. Schupp, Combinatorial group theory, (2001), Springer-Verlag, Berlin Zbl0997.20037MR1812024

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