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1 - Growth of Groups and Wreath Products

Published online by Cambridge University Press:  20 July 2017

Laurent Bartholdi
Affiliation:
Universität Göttingen, Bunsenstrasse
Tullio Ceccherini-Silberstein
Affiliation:
Università degli Studi del Sannio, Italy
Maura Salvatori
Affiliation:
Università degli Studi di Milano
Ecaterina Sava-Huss
Affiliation:
Cornell University, New York
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Print publication year: 2017

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References

[1] Miklós, Abért, Group laws and free subgroups in topological groups, Bull. London Math. Soc. 37 (2005), no. 4, 525–34, available at arXiv:math.GR/ 0306364. MR2143732.Google Scholar
[2] Stanislav V., Alešin, Finite automata and the Burnside problem for periodic groups, Mat. Zametki 11 (1972), 319–28. MR46#265.Google Scholar
[3] Daniel, Allen, Megan, Cream, Kate, Finlay, John, Meier, and Ranjan, Rohatgi, Complete growth series and products of groups, New York J. Math. 17 (2011), 321–9. MR2811067 (2012h:20091)Google Scholar
[4] Laurent Bartholdi Anna G., Erschler, Growth of permutational extensions, Invent. Math. 189 (2012), no. 2, 431–55, DOI 10.1007/s00222-011-0368-x, available at arXiv:math/1011.5266. MR2947548Google Scholar
[5] Laurent Bartholdi Anna G., Erschler, Imbeddings into groups of intermediate growth, Groups Geom. Dyn. 8 (2014), no. 3, 605–20, DOI 10.4171/GGD/241, available at arXiv:math/1403.5584. MR3267517Google Scholar
[6] Laurent Bartholdi Anna G., Erschler, Groups of given intermediate word growth, Ann. Inst. Fourier 64 (2014), no. 5, 2003–36, available at arXiv:math/1110.3650.Google Scholar
[7] Laurent Bartholdi Anna G., Erschler, Ordering the space of finitely generated groups, Ann. Inst. Fourier 65 (2015), no. 5, 2091–144, DOI 10.5802/aif.2984, available at arXiv:math/1301.4669.Google Scholar
[8] Laurent Bartholdi Anna G., Erschler, Distortion of imbeddings of groups of intermediate growth into metric spaces. Proc. Amer. Math. Soc. 145 (2017), no. 5, 1943–1952.Google Scholar
[9] Michael, Benson, Growth series of finite extensions of Z n are rational, Invent. Math. 73 (1983), 251–69.Google Scholar
[10] Jean, Berstel, Sur les pôles et le quotient de Hadamard de séries Nrationnelles, C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A1079–A1081 (French). MR0285521 (44 #2739).Google Scholar
[11] Garrett, Birkhoff, Extensions of Jentzsch's theorem, Trans. Amer. Math. Soc. 85 (1957), 219–27. MR0087058 (19,296a)Google Scholar
[12] Jérémie, Brieussel, Growth behaviors in the range erα Afr. Mat. 25 (2014), no. 4, 1143–1163.Google Scholar
[13] William S., Burnside, On an unsettled question in the theory of discontinuous groups, Quart. J. Pure Appl. Math. 33 (1902), 230–8.Google Scholar
[14] Fritz, Carlson, Ǖber Potenzreihen mit ganzzahligen Koeffizienten, Math. Z. 9 (1921), 1–13, DOI 10.1007/BF01378331 (German).Google Scholar
[15] Ian M., Chiswell, Euler characteristics of groups, Math. Z. 147 (1976), no. 1, 1–11. MR0396785 (53 #645)Google Scholar
[16] Václáv, Chvatal David Sankoff, Longest common subsequences of two random sequences, J. Appl. Probability 12 (1975), 306–15. MR0405531 (53 #9324)Google Scholar
[17] Yves de, Cornulier, Finitely presented wreath products and double coset decompositions, Geom. Dedicata 122 (2006), 89–108, DOI 10.1007/ s10711-006-9061-4. MR2295543 (2008e:20040)Google Scholar
[18] Moon, Duchin and Michael, Shapiro, Rational growth in the Heisenberg group (2014), preprint, available at arXiv:1411.4201.
[19] Vadim A., Efremovich, The proximity geometry of Riemannian manifolds, Uspekhi Mat. Nauk 8 (1953), 189.Google Scholar
[20] Alex, Eskin, Shahar, Mozes, and Hee, Oh, Uniform exponential growth for linear groups, Int. Math. Res. Not. 31 (2002), 1675–83. MR1916 428Google Scholar
[21] Pierre, Fatou, Sur les séeries entiéeres à coefficients entiers, C. R. Acad. Sci., Paris 138 (1904), 342–4 (French).Google Scholar
[22] Mihály, Fekete, Ǖber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z. 17 (1923), no. 1, 228–49 (German). MR1544613Google Scholar
[23] William J., Floyd and Steven P., Plotnick, Growth functions on Fuchsian groups and the Euler characteristic, Invent. Math. 88 (1987), no. 1, 1–29.Google Scholar
[24] Michael R., Garey, David S., Johnson, and Larry J., Stockmeyer, Some simplified NP-complete graph problems, Theoret. Comput. Sci. 1 (1976), no. 3, 237–67. MR0411240 (53 #14978)Google Scholar
[25] Michael R., Garey and David S., Johnson, The rectilinear Steiner tree problem is NP-complete, SIAM J. Appl. Math. 32 (1977), no. 4, 826–34. MR0443426 (56 #1796)Google Scholar
[26] Evgueniĭ S., Golod, On nil-algebras and finitely approximable p-groups, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 273–6.Google Scholar
[27] Frank, Gray, Pulse code communication, 1953. US Patent 2,632,058.
[28] Rostislav I., Grigorchuk, On Burnside's problem on periodic groups, 14 (1980), no. 1, 53–54. English translation: Functional Anal. Appl.14 (1980), 41–3. MR81m:20045
[29] Rostislav I., Grigorchuk, On the Milnor problem of group growth, Dokl. Akad. Nauk SSSR 271 (1983), no. 1, 30–3. MR85g:20042Google Scholar
[30] Rostislav I., Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, 48 (1984), no. 5, 939–985. English translation: Math. USSR-Izv. 25 (1985), no. 2, 259–300. MR86h:20041
[31] Rostislav I., Grigorchuk, On the Hilbert-Poincaré series of graded algebras that are associated with groups, 180 (1989), no. 2, 207–25, 304. English translation: Math. USSR-Sb. 66 (1990), no. 1, 211–29. MR90j:20063
[32] Rostislav I., Grigorchuk, Growth functions, rewriting systems and Euler characteristic, Mat. Zametki 58 (1995), no. 5, 653–68, 798. MR97d:20031aGoogle Scholar
[33] Rostislav I., Grigorchuk and Tatiana, Nagnibeda, Complete growth functions of hyperbolic groups, Invent. Math. 130 (1997), no. 1, 159–88. MR98i:20038Google Scholar
[34] Mikhael L., Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73.Google Scholar
[35] Mikhael L., Gromov, Structures métriques pour les variétés riemanniennes, CEDIC, Paris, 1981. Edited by J., Lafontaine and P., Pansu.
[36] Mikhael L., Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263, DOI 10.1007/978-1- 4613-9586-7 3. MR919829 (89e:20070)
[37] Geoffrey H., Hardy and Srinivasa, Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. S2-17, no. 1, 75, DOI 10.1112/plms/s2-17.1.75. MR1575586
[38] Pierre de la, Harpe, Uniform growth in groups of exponential growth, Geom. Dedicata 95 (2002), 1–17.Google Scholar
[39] Graham, Higman, Bernard H., Neumann, and Hanna, Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247–54. MR0032641 (11,322d)Google Scholar
[40] Stephen A., Jennings, The structure of the group ring of a p-group over a modular field, Trans. Amer. Math. Soc. 50 (1941), 175–85.Google Scholar
[41] Stephen A., Jennings, The group ring of a class of infinite nilpotent groups, Canad. J. Math. 7 (1955), 169–87.Google Scholar
[42] David L., Johnson, Rational growth of wreath products, Groups—St. Andrews 1989, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 160, Cambridge University Press, 1991, pp. 309–15, DOI 10.1017/CBO9780511 661846.005. MR1123986 (92g:20045)
[43] Camille, Jordan, Traité des substitutions et des équations algébriques, Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics], Éditions Jacques Gabay, Sceaux, 1989 (French). Reprint of the 1870 original. MR1188877 (94c:01039)
[44] Lev A., Kaloujnine, La structure des p-groupes de Sylow des groupes symétriques finis, Ann. École Norm. Sup. (3) 65 (1948), 239–76.Google Scholar
[45] Lev A., Kaloujnine and Marc, Krasner, Le produit complet des groupes de permutations et le problème d'extension des groupes, C. R. Acad. Sci. Paris 227 (1948), 806–8 (French). MR0027758 (10,351e)Google Scholar
[46] Michael E., Kapovich, Arithmetic aspects of self-similar groups, Groups Geom. Dyn. 6 (2012), no. 4, 737–54, DOI 10.4171/GGD/172. MR2996409Google Scholar
[47] David A., Klarner, Mathematical crystal growth. I, Discrete Appl. Math. 3 (1981), no. 1, 47–52. MR604265 (82e:05016)Google Scholar
[48] David A., Klarner, Mathematical crystal growth. II, Discrete Appl. Math. 3 (1981), no. 2, 113–17. MR607910 (83a:05018)Google Scholar
[49] Malik, Koubi, Croissance uniforme dans les groupes hyperboliques, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 5, 1441–53. MR99m:20080Google Scholar
[50] Hans Ulrich, Krause, Gruppenstruktur und Gruppenbild, Thesis, Eidgen össische Technische Hochschule, Zürich, 1953. MR15,99b
[51] Jacques, Lewin, The growth function of a graph group, Comm. Algebra 17 (1989), no. 5, 1187–91, DOI 10.1080/00927878908823782. MR993397 (90c:20040)Google Scholar
[52] Fabrice, Liardet, Croissance des groupes virtuellement abéliens, PhD Thesis, 1996.
[53] Avinoam, Mann, How groups grow. London Mathematical Society Lecture Note Series, 395. Cambridge University Press, Cambridge, 2012. x+199 pp. ISBN: 978-1-107-65750-2
[54] Grigori A., Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR1090825 (92h:22021)
[55] John W., Milnor, Growth of finitely generated solvable groups, J. Differential Geom. 2 (1968), 447–9.Google Scholar
[56] John W., Milnor, A note on curvature and fundamental group, J. Differential Geom. 2 (1968), 1–7.Google Scholar
[57] John W., Milnor, Problem 5603, Amer. Math. Monthly 75 (1968), 685–6.Google Scholar
[58] John W., Milnor and John C., Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–64. MR0174052 (30 #4259)CrossRefGoogle Scholar
[59] Melvyn B., Nathanson, Asymptotic density and the asymptotics of partition functions, Acta Math. Hungar. 87 (2000), no. 3, 179–95. MR2001b:11091Google Scholar
[60] Volodymyr V., Nekrashevych, A minimal Cantor set in the space of 3-generated groups, Geom. Dedicata 124 (2007), 153–90. MR2318543 (2008d:20075)Google Scholar
[61] Bernard H., Neumann and Hanna, Neumann, Embedding theorems for groups, J. London Math. Soc. 34 (1959), 465–79. MR0163968 (29 #1267)Google Scholar
[62] Petʹr S., Novikov, On algorithmic unsolvability of the problem of identity, Doklady Akad. Nauk SSSR (N.S.) 85 (1952), 709–12 (Russian). MR0052436 (14,618h)Google Scholar
[63] Alexander Yu., Olʹshanskiĭ and Denis V., Osin, A quasi-isometric embedding theorem for groups, Duke Math. J. 162 (2013), no. 9, 1621–48, DOI 10.1215/00127094-2266251. MR3079257Google Scholar
[64] Denis V., Osin, The entropy of solvable groups, Ergodic Theory Dynam. Systems 23 (2003), no. 3, 907–18. MR1992 670Google Scholar
[65] Walter R., Parry, Growth series of some wreath products, Trans. Amer. Math. Soc. 331 (June 1992), no. 2, 751–9.Google Scholar
[66] Richard E., Phillips, Embedding methods for periodic groups, Proc. London Math. Soc. (3) 35 (1977), no. 2, 238–56. MR0498874 (58 #16896)Google Scholar
[67] Alexander V., Rozhkov, Lower central series of a group of tree automorphisms, Mat. Zametki 60 (1996), no. 2, 225–37, 319.Google Scholar
[68] Arto, Salomaa and Matti, Soittola, Automata-theoretic aspects of formal power series, Springer-Verlag, 1978.
[69] Andrea, Sambusetti, Minimal growth of non-Hopfian free products, C. R. Acad. Sci. Paris Séer. I Math. 329 (1999), no. 11, 943–6. MR2000j:20056Google Scholar
[70] Leonard L., Scott, Representations in characteristic p, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 319–31. MR604599 (82e:20052)
[71] Jean-Pierre, Serre, Trees, Springer-Verlag, Berlin, 1980. Translated from the French by John Stillwell. MR82c:20083
[72] Yehuda, Shalom and Terence, Tao, A finitary version of Gromov's polynomial growth theorem, Geom. Funct. Anal. 20 (2010), no. 6, 1502–47, DOI 10.1007/s00039-010-0096-1. MR2739001 (2011m:20100)Google Scholar
[73] Neville F., Smythe, Growth functions and Euler series, Invent. Math. 77 (1984), 517–31.Google Scholar
[74] John R., Stallings, Centerless groups—an algebraic formulation of Gottlieb's theorem, Topology 4 (1965), 129–134. MR0202807 (34 #2666)Google Scholar
[75] Michael, Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126 (1996), no. 1, 85–109. MR98d:20033Google Scholar
[76] Albert S., Švarc, A volume invariant of coverings, Dokl. Akad. Nauk SSSR 105 (1955), 32–4 (Russian).Google Scholar
[77] Moss E., Sweedler, Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, New York, 1969. MR0252485 (40 #5705)
[78] Ludwig, Sylow, Théorèmes sur les groupes de substitutions, Math. Ann. 5 (1872), 584–94, DOI 10.1007/BF01442913.Google Scholar
[79] Jacques, Tits, Free subgroups in linear groups, J. Algebra 20 (1972), 250–70.Google Scholar
[80] John S., Wilson, Embedding theorems for residually finite groups, Math. Z. 174 (1980), no. 2, 149–57, DOI 10.1007/BF01293535. MR592912 (81m:20041)Google Scholar
[81] John S., Wilson, On exponential and uniformly exponential growth for groups, Invent. Math. 155 (2004), no. 2, 287–303.Google Scholar
[82] John S., Wilson, The gap in the growth of residually soluble groups, Bull. Lond. Math. Soc. 43 (2011), no. 3, 576–82, DOI 10.1112/blms/bdq124. MR2820146 (2012f:20105)Google Scholar
[83] Wolfgang, Woess, Lamplighters, Diestel–Leader graphs, random walks, and harmonic functions, Combin. Probab. Comput. 14 (2005), no. 3, 415–33, DOI 10.1017/S0963548304006443. MR2138121 (2006d:60021)Google Scholar
[84] Joseph A., Wolf, Growth of finitely generated solvable groups and curvature of Riemanniann manifolds, J. Differential Geom. 2 (1968), 421–46.Google Scholar

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