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Degree of commutativity of infinite groups

Authors: Yago Antolín, Armando Martino and Enric Ventura
Journal: Proc. Amer. Math. Soc. 145 (2017), 479-485
MSC (2010): Primary 20P05
Published electronically: July 28, 2016
MathSciNet review: 3577854
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Abstract: We prove that, in a finitely generated residually finite group of subexponential growth, the proportion of commuting pairs is positive if and only if the group is virtually abelian. In particular, this covers the case where the group has polynomial growth (i.e., virtually nilpotent groups). We also show that for non-elementary hyperbolic groups, the proportion of commuting pairs is always zero.

References [Enhancements On Off] (What's this?)

  • [1] Y. Antolín and A. Sale, Permute and conjugate: the conjugacy problem in relatively hyperbolic groups. To appear in Bulletin of the LMS.
  • [2] M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486
  • [3] J. Burillo and E. Ventura, Counting primitive elements in free groups, Geom. Dedicata 93 (2002), 143-162. MR 1934695,
  • [4] P. Dani, Genericity of infinite-order elements in hyperbolic groups, preprint.
  • [5] P. X. Gallagher, The number of conjugacy classes in a finite group, Math. Z. 118 (1970), 175-179. MR 0276318
  • [6] M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53-73. MR 623534
  • [7] W. H. Gustafson, What is the probability that two group elements commute?, Amer. Math. Monthly 80 (1973), 1031-1034. MR 0327901
  • [8] C. T. C. Wall, Poincaré complexes. I, Ann. of Math. (2) 86 (1967), 213-245. MR 0217791

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Additional Information

Yago Antolín
Affiliation: Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, Tennessee 37240

Armando Martino
Affiliation: Department of Mathematics, University of Southampton, Highfield, Southampton SO17 1BJ, United Kindgom

Enric Ventura
Affiliation: Departament de Matemàtiques, Universitat Politècnica de Catalunya, Av. Bases de Manresa 61–73, 08242-Manresa, 08034 Barcelona (Catalonia), Spain

Keywords: Abelian groups, degree of commutativity, polynomial growth
Received by editor(s): November 23, 2015
Received by editor(s) in revised form: January 19, 2016, and March 21, 2016
Published electronically: July 28, 2016
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2016 American Mathematical Society

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