Conjugacy growth series and languages in groups
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- by Laura Ciobanu and Susan Hermiller PDF
- Trans. Amer. Math. Soc. 366 (2014), 2803-2825 Request permission
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.References
- 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
- Emmanuel Breuillard and Yves de Cornulier, On conjugacy growth for solvable groups, Illinois J. Math. 54 (2010), no. 1, 389–395. MR 2777001
- I. M. Chiswell, The growth series of a graph product, Bull. London Math. Soc. 26 (1994), no. 3, 268–272. MR 1289045, DOI 10.1112/blms/26.3.268
- L. Ciobanu, and S. Hermiller, Conjugacy growth series and languages in groups, arXiv:1205.3857.
- John B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978. MR 503901
- M. Coornaert and G. Knieper, Growth of conjugacy classes in Gromov hyperbolic groups, Geom. Funct. Anal. 12 (2002), no. 3, 464–478. MR 1924369, DOI 10.1007/s00039-002-8254-8
- Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869
- 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
- Rostislav Grigorchuk and Tatiana Nagnibeda, Complete growth functions of hyperbolic groups, Invent. Math. 130 (1997), no. 1, 159–188. MR 1471889, DOI 10.1007/s002220050181
- Victor Guba and Mark Sapir, On the conjugacy growth functions of groups, Illinois J. Math. 54 (2010), no. 1, 301–313. MR 2776997
- Susan Hermiller and John Meier, Algorithms and geometry for graph products of groups, J. Algebra 171 (1995), no. 1, 230–257. MR 1314099, DOI 10.1006/jabr.1995.1010
- 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, DOI 10.1142/S0218196711006133
- John E. Hopcroft and Jeffrey D. Ullman, Introduction to automata theory, languages, and computation, Addison-Wesley Series in Computer Science, Addison-Wesley Publishing Co., Reading, Mass., 1979. MR 645539
- M. Hull, and D. Osin, Conjugacy growth of finitely generated groups, arXiv:1107.1826v2.
- Joseph Loeffler, John Meier, and James Worthington, Graph products and Cannon pairs, Internat. J. Algebra Comput. 12 (2002), no. 6, 747–754. MR 1949695, DOI 10.1142/S021819670200122X
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. MR 1812024, DOI 10.1007/978-3-642-61896-3
- Avinoam Mann, How groups grow, London Mathematical Society Lecture Note Series, vol. 395, Cambridge University Press, Cambridge, 2012. MR 2894945
- 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, DOI 10.1007/BF01241129
- 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, DOI 10.1090/conm/360/06573
- Igor Rivin, Growth in free groups (and other stories)—twelve years later, Illinois J. Math. 54 (2010), no. 1, 327–370. MR 2776999
- Charles C. Sims, Computation with finitely presented groups, Encyclopedia of Mathematics and its Applications, vol. 48, Cambridge University Press, Cambridge, 1994. MR 1267733, DOI 10.1017/CBO9780511574702
- Michael Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126 (1996), no. 1, 85–109. MR 1408557, DOI 10.1007/s002220050090
Additional Information
- Laura Ciobanu
- Affiliation: Department of Mathematics, University of Neuchâtel, Rue Emile-Argand 11, CH-2000 Neuchâtel, Switzerland
- MR Author ID: 797163
- Email: laura.ciobanu@unine.ch
- Susan Hermiller
- Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130
- MR Author ID: 311019
- Email: smh@math.unl.edu
- Received by editor(s): May 21, 2012
- Received by editor(s) in revised form: October 30, 2012
- Published electronically: November 6, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 2803-2825
- MSC (2010): Primary 20F65, 20E45
- DOI: https://doi.org/10.1090/S0002-9947-2013-06052-7
- MathSciNet review: 3165656