Rational growth in virtually abelian groups. (English) Zbl 1515.20217

Summary: We show that any subgroup of a finitely generated virtually abelian group \(G\) grows rationally relative to \(G\), that the set of right cosets of any subgroup of \(G\) grows rationally, and that the set of conjugacy classes of \(G\) grows rationally. These results hold regardless of the choice of finite weighted generating set for \(G\).


20F65 Geometric group theory
20E45 Conjugacy classes for groups
05E16 Combinatorial aspects of groups and algebras
20F05 Generators, relations, and presentations of groups
20F69 Asymptotic properties of groups
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[1] Y. Antolín and L. Ciobanu, Formal conjugacy growth in acylindrically hyperbolic groups, Int. Math. Res. Not. IMRN 2017 (2017), no. 1, 121-157. · Zbl 1404.20036
[2] I. K. Babenko, Closed geodesics, asymptotic volume and the characteristics of growth of groups, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 4, 675-711, 895. · Zbl 0657.53024
[3] M. Benson, Growth series of finite extensions of \(\mathbf{Z}^n\) are rational, Invent. Math. 73 (1983), no. 2, 251-269. · Zbl 0498.20022
[4] E. Breuillard and Y. de Cornulier, On conjugacy growth for solvable groups, Illinois J. Math. 54 (2010), no. 1, 389-395. · Zbl 1231.20032
[5] J. W. Cannon, The growth of the closed surface groups and compact hyperbolic coxeter groups, Circulated typescript, Univ. Wisconsin, 1980.
[6] J. W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), no. 2, 123-148. · Zbl 0606.57003
[7] L. Ciobanu, S. Hermiller, D. Holt, and S. Rees, Conjugacy languages in groups, Israel J. Math. 211 (2016), no. 1, 311-347. · Zbl 1398.20050
[8] L. Ciobanu, S. Hermiller, and V. Mercier, Formal conjugacy growth in graph products, in preparation 2019. · Zbl 1515.20214
[9] R. Cluckers, J. Gordon, and I. Halupczok, Integrability of oscillatory functions on local fields: Transfer principles, Duke Math. J. 163 (2014), no. 8, 1549-1600. · Zbl 1327.14073
[10] M. Coornaert and G. Knieper, Growth of conjugacy classes in Gromov hyperbolic groups, Geom. Funct. Anal. 12 (2002), no. 3, 464-478. · Zbl 1042.20031
[11] T. C. Davis and A. Y. Olshanskii, Relative subgroup growth and subgroup distortion, Groups Geom. Dyn. 9 (2015), no. 1, 237-273. · Zbl 1365.20019
[12] J. Denef, The rationality of the Poincaré series associated to the \(p\)-adic points on a variety, Invent. Math. 77 (1984), no. 1, 1-23. · Zbl 0537.12011
[13] L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with \(n\) distinct prime factors, Amer. J. Math. 35 (1913), no. 4, 413-422. · JFM 44.0220.02
[14] M. Duchin, Counting in groups: Fine asymptotic geometry, Notices of the Amer. Math. Soc. 63 (2016), no. 8, 871-874. · Zbl 1352.05003
[15] M. Duchin and M. Shapiro, The Heisenberg group is pan-rational, Adv. Math. 346 (2019), 219-263. · Zbl 1504.20043
[16] D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson, and W. P. Thurston, Word Processing in Groups, Jones and Bartlett, Boston, 1992. · Zbl 0764.20017
[17] I. Gekhtman and W. Yang, Counting conjugacy classes in groups with contracting elements, arXiv e-print, arXiv:1810.02969.
[18] M. Gromov, “Hyperbolic groups” in Essays in Group Theory, Math. Sci. Res. Inst. 8, Springer, New York, 1987, pp. 75-263.
[19] F. J. Grunewald, D. Segal, and G. C. Smith, Subgroups of finite index in nilpotent groups, Invent. Math. 93 (1988), no. 1, 185-223. · Zbl 0651.20040
[20] V. Guba and M. Sapir, On the conjugacy growth functions of groups, Illinois J. Math. 54 (2010), no. 1, 301-313. · Zbl 1234.20041
[21] D. F. Holt, “Automatic groups, subgroups and cosets,” The Epstein Birthday Schrift, Geom. Topol. Monogr. 1, Geom. Topol. Coventry, 1998, pp. 249-260. · Zbl 0917.20031
[22] D. A. Klarner, Mathematical crystal growth. I, Discrete Appl. Math. 3 (1981), no. 1, 47-52. · Zbl 0466.05007
[23] F. Liardet, Croissance dans le groupes virtuellement abéliens, Ph.D. dissertation, Université De Genève, 1997.
[24] V. Mercier, Conjugacy growth series of some wreath products, arXiv e-print, arXiv:1610.07868.
[25] M. Presburger, “Über die vollständigkeit eines gewissen systems der arithmetik ganzer zahlen, in welchem die addition als einzige operation hervortritt,” in Comptes-rendus du I Congrès des Mathématiciens des Pays Slaves (1929), 92-101. · JFM 56.0825.04
[26] I. Rivin, “Some properties of the conjugacy class growth function” in Group Theory, Statistics, and Cryptography, Contemp. Math. 360, Amer. Math. Soc., Providence, 2004, pp. 113-117. · Zbl 1071.20504
[27] I. Rivin, Growth in free groups (and other stories)—Twelve years later, Illinois J. Math. 54 (2010), no. 1, 327-370. · Zbl 1225.05128
[28] M. Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126 (1996), no. 1, 85-109. · Zbl 0869.20018
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