Hostname: page-component-76d6cb85b7-lcgwf Total loading time: 0 Render date: 2026-07-14T04:41:36.103Z Has data issue: false hasContentIssue false
  • English
  • Français

The Conjugacy Ratio of Groups

Part of: Probabilistic methods in group theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  22 February 2019

Laura Ciobanu ,
Charles Garnet Cox  and
Armando Martino
Laura Ciobanu
Affiliation:
Heriot-Watt University, Edinburgh, EH14 4AS, UK (l.ciobanu@hw.ac.uk)
Charles Garnet Cox
Affiliation:
University of Bath, BA2 7AY, UK (cpgc20@bath.ac.uk)
Armando Martino
Affiliation:
Mathematical Sciences, University of Southampton, SO17 1BJ, UK (A.Martino@soton.ac.uk)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

In this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is 0 for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups and the lamplighter group.

Keywords

conjugacy growthdegree of commutativitypolynomial growthRAAGshyperbolic groupswreath products

MSC classification

Primary: 20P05: Probabilistic methods in group theory
Secondary: 20F69: Asymptotic properties of groups

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 3 , August 2019 , pp. 895 - 911
Copyright
Copyright © Edinburgh Mathematical Society 2019 

References

1Antolín, Y. and Ciobanu, L., Formal conjugacy growth in acylindrically hyperbolic groups, Int. Math. Res. Not. IMRN 1 (2017), 121–157.Google Scholar
2Antolín, Y., Martino, A. and Ventura, E., Degree of commutativity of infinite groups, Proceedings of the American Mathematical Society (AMS, 2017).Google Scholar
3Bass, H., The degree of polynomial growth of finitely generated nilpotent groups, Proc. Lond. Math. Soc. (3) 25 (1972), 603–614.Google Scholar
4Burillo, J. and Ventura, E., Counting primitive elements in free groups, Geom. Dedicata 93 (2002), 143–162.Google Scholar
5Ciobanu, L., Hermiller, S. and Mercier, V., Conjugacy growth in graph products, preprint, 2017.Google Scholar
6Coornaert, M., Mesures de Patterson–Sullivan dans les espaces hyperboliques au sens de Gromov, Pacific J. Math. 159 (1993), 241–270.Google Scholar
7Cox, C. G., The degree of commutativity and lamplighter groups, Int. J. Algebr. Comput. 28(7) (2018), 1163–1173.Google Scholar
8Crisp, J., Godelle, E. and Wiest, B., Linear time solution to the conjugacy problem in right-angled Artin groups and their subgroups, J. Topol. 2 (2009), 442–460.Google Scholar
9Gallagher, P. X., The number of conjugacy classes in a finite group, Math Z. 118 (1970), 175–179.Google Scholar
10Gustafson, W. H., What is the probability that two group elements commute?, Amer. Math. Monthly 80 (1973), 1031–1034.Google Scholar
11Mercier, V., Conjugacy growth series in some wreath products, preprint (arXiv:1610.07868, 2017).Google Scholar
12Parry, W., Growth series of some wreath products, Trans. Amer. Math. Soc. 331(2) (1992), 751–759.Google Scholar
13Valiunas, M., Rational growth and degree of commutativity of graph products, J. Algebra 522 (2019), 309–331.Google Scholar