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Conjugacy growth series and languages in groups

Authors: Laura Ciobanu and Susan Hermiller
Journal: Trans. Amer. Math. Soc. 366 (2014), 2803-2825
MSC (2010): Primary 20F65, 20E45
Published electronically: November 6, 2013
MathSciNet review: 3165656
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Abstract: In this paper we introduce the geodesic conjugacy language and geodesic conjugacy growth series for a finitely generated group. We study the effects of various group constructions on rationality of both the geodesic conjugacy growth series and spherical conjugacy growth series, as well as on regularity of the geodesic conjugacy language and spherical conjugacy language. In particular, we show that regularity of the geodesic conjugacy language is preserved by the graph product construction, and rationality of the geodesic conjugacy growth series is preserved by both direct and free products.

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  • [1] Marcus Brazil, Calculating growth functions for groups using automata, Computational algebra and number theory (Sydney, 1992) Math. Appl., vol. 325, Kluwer Acad. Publ., Dordrecht, 1995, pp. 1-18. MR 1344918 (96m:20050)
  • [2] Emmanuel Breuillard and Yves de Cornulier, On conjugacy growth for solvable groups, Illinois J. Math. 54 (2010), no. 1, 389-395. MR 2777001 (2012c:20091)
  • [3] I. M. Chiswell, The growth series of a graph product, Bull. London Math. Soc. 26 (1994), no. 3, 268-272. MR 1289045 (95f:20050),
  • [4] L. Ciobanu, and S. Hermiller, Conjugacy growth series and languages in groups, arXiv:1205.3857.
  • [5] John B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York, 1978. MR 503901 (80c:30003)
  • [6] M. Coornaert and G. Knieper, Growth of conjugacy classes in Gromov hyperbolic groups, Geom. Funct. Anal. 12 (2002), no. 3, 464-478. MR 1924369 (2003f:20071),
  • [7] Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869 (2001i:20081)
  • [8] David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1161694 (93i:20036)
  • [9] Rostislav Grigorchuk and Tatiana Nagnibeda, Complete growth functions of hyperbolic groups, Invent. Math. 130 (1997), no. 1, 159-188. MR 1471889 (98i:20038),
  • [10] Victor Guba and Mark Sapir, On the conjugacy growth functions of groups, Illinois J. Math. 54 (2010), no. 1, 301-313. MR 2776997 (2012d:20088)
  • [11] Susan Hermiller and John Meier, Algorithms and geometry for graph products of groups, J. Algebra 171 (1995), no. 1, 230-257. MR 1314099 (96a:20052),
  • [12] Derek F. Holt, Sarah Rees, and Claas E. Röver, Groups with context-free conjugacy problems, Internat. J. Algebra Comput. 21 (2011), no. 1-2, 193-216. MR 2787458 (2012j:20103),
  • [13] John E. Hopcroft and Jeffrey D. Ullman, Introduction to automata theory, languages, and computation, Addison-Wesley Publishing Co., Reading, Mass., 1979. Addison-Wesley Series in Computer Science. MR 645539 (83j:68002)
  • [14] M. Hull, and D. Osin, Conjugacy growth of finitely generated groups, arXiv:1107.1826v2.
  • [15] Joseph Loeffler, John Meier, and James Worthington, Graph products and Cannon pairs, Internat. J. Algebra Comput. 12 (2002), no. 6, 747-754. MR 1949695 (2003k:20059),
  • [16] Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. MR 1812024 (2001i:20064)
  • [17] Avinoam Mann, How groups grow, London Mathematical Society Lecture Note Series, vol. 395, Cambridge University Press, Cambridge, 2012. MR 2894945
  • [18] Walter D. Neumann and Michael Shapiro, Automatic structures, rational growth, and geometrically finite hyperbolic groups, Invent. Math. 120 (1995), no. 2, 259-287. MR 1329042 (96c:20066),
  • [19] Igor Rivin, Some properties of the conjugacy class growth function, Group theory, statistics, and cryptography, Contemp. Math., vol. 360, Amer. Math. Soc., Providence, RI, 2004, pp. 113-117. MR 2105439 (2005h:20073),
  • [20] Igor Rivin, Growth in free groups (and other stories)--twelve years later, Illinois J. Math. 54 (2010), no. 1, 327-370. MR 2776999 (2012h:20055)
  • [21] Charles C. Sims, Computation with finitely presented groups, Encyclopedia of Mathematics and its Applications, vol. 48, Cambridge University Press, Cambridge, 1994. MR 1267733 (95f:20053)
  • [22] Michael Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126 (1996), no. 1, 85-109. MR 1408557 (98d:20033),

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Additional Information

Laura Ciobanu
Affiliation: Department of Mathematics, University of Neuchâtel, Rue Emile-Argand 11, CH-2000 Neuchâtel, Switzerland

Susan Hermiller
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130

Keywords: Conjugacy growth, generating functions, graph products, regular languages
Received by editor(s): May 21, 2012
Received by editor(s) in revised form: October 30, 2012
Published electronically: November 6, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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