Book Review
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Book Information:
Author: Avinoam Mann
Title: How groups grow
Additional book information: London Mathematical Society Lecture Note Series, Vol. 395, Cambridge University Press, Cambridge, 2012, ix+199 pp., ISBN 978-1-107-65750-2
- Laurent Bartholdi, The growth of Grigorchuk’s torsion group, Internat. Math. Res. Notices 20 (1998), 1049–1054. MR 1656258, DOI https://doi.org/10.1155/S1073792898000622
- Laurent Bartholdi, Lower bounds on the growth of a group acting on the binary rooted tree, Internat. J. Algebra Comput. 11 (2001), no. 1, 73–88. MR 1818662, DOI https://doi.org/10.1142/S0218196701000395
- Laurent Bartholdi, A Wilson group of non-uniformly exponential growth, C. R. Math. Acad. Sci. Paris 336 (2003), no. 7, 549–554 (English, with English and French summaries). MR 1981466, DOI https://doi.org/10.1016/S1631-073X%2803%2900131-6
- H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3) 25 (1972), 603–614. MR 379672, DOI https://doi.org/10.1112/plms/s3-25.4.603
- Laurent Bartholdi and Anna Erschler, Groups of given intermediate word growth, arXiv:math/1110.3650v2, 2011.
- Laurent Bartholdi and Anna Erschler, Growth of permutational extensions, Invent. Math. 189 (2012), no. 2, 431–455. MR 2947548, DOI https://doi.org/10.1007/s00222-011-0368-x
- M. Benson, Growth series of finite extensions of ${\bf Z}^{n}$ are rational, Invent. Math. 73 (1983), no. 2, 251–269. MR 714092, DOI https://doi.org/10.1007/BF01394026
- Kai-Uwe Bux and Rodrigo Pérez, On the growth of iterated monodromy groups, Topological and asymptotic aspects of group theory, Contemp. Math., vol. 394, Amer. Math. Soc., Providence, RI, 2006, pp. 61–76. MR 2216706, DOI https://doi.org/10.1090/conm/394/07434
- J. Brieussel, Growth behaviours in the range $e^{(r^\alpha )}$, (preprint, arXiv:math/1107.1632), 2011.
- James W. Cannon, The combinatorial structure of cocompact discrete hyperbolic groups, Geom. Dedicata 16 (1984), no. 2, 123–148. MR 758901, DOI https://doi.org/10.1007/BF00146825
- Christophe Champetier, L’espace des groupes de type fini, Topology 39 (2000), no. 4, 657–680 (French, with English summary). MR 1760424, DOI https://doi.org/10.1016/S0040-9383%2898%2900063-9
- J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$ groups, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 61, Cambridge University Press, Cambridge, 1999. MR 1720368
- Jacek Fabrykowski and Narain Gupta, On groups with sub-exponential growth functions. II, J. Indian Math. Soc. (N.S.) 56 (1991), no. 1-4, 217–228. MR 1153150
- Rostislav I. Grigorchuk, Milnor’s problem on the growth of groups, Sov. Math., Dokl. 28 (1983), 23–26.
- R. I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 5, 939–985 (Russian). MR 764305
- R. I. Grigorchuk, On the Hilbert-Poincaré series of graded algebras that are associated with groups, Mat. Sb. 180 (1989), no. 2, 207–225, 304 (Russian); English transl., Math. USSR-Sb. 66 (1990), no. 1, 211–229. MR 993455, DOI https://doi.org/10.1070/SM1990v066n01ABEH002083
- Rostislav I. Grigorchuk, On growth in group theory, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 325–338. MR 1159221
- R. Grigorchuk, On the gap conjecture concerning group growth, Bulletin of Mathematical Sciences 2 (2012).
- Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
- Mikhael Gromov, Structures métriques pour les variétés riemanniennes, Textes Mathématiques [Mathematical Texts], vol. 1, CEDIC, Paris, 1981 (French). Edited by J. Lafontaine and P. Pansu. MR 682063
- Yves Guivarc’h, Groupes de Lie à croissance polynomiale, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A237–A239 (French). MR 272943
- Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869
- Bruce Kleiner, A new proof of Gromov’s theorem on groups of polynomial growth, J. Amer. Math. Soc. 23 (2010), no. 3, 815–829. MR 2629989, DOI https://doi.org/10.1090/S0894-0347-09-00658-4
- Martin Kassabov and Igor Pak, Groups of oscillating intermediate growth, (preprint arxiv:math/1108.0262), 2011.
- Yu. G. Leonov, On a lower bound for the growth function of the Grigorchuk group, Mat. Zametki 67 (2000), no. 3, 475–477 (Russian); English transl., Math. Notes 67 (2000), no. 3-4, 403–405. MR 1779480, DOI https://doi.org/10.1007/BF02676677
- Alexander Lubotzky and Avinoam Mann, On groups of polynomial subgroup growth, Invent. Math. 104 (1991), no. 3, 521–533. MR 1106747, DOI https://doi.org/10.1007/BF01245088
- Avinoam Mann, How groups grow, London Mathematical Society Lecture Note Series, vol. 395, Cambridge University Press, Cambridge, 2012. MR 2894945
- John Milnor, Growth of finitely generated solvable groups, J. Differential Geometry 2 (1968), 447–449. MR 244899
- L. Carlitz, A. Wilansky, John Milnor, R. A. Struble, Neal Felsinger, J. M. S. Simoes, E. A. Power, R. E. Shafer, and R. E. Maas, Problems and Solutions: Advanced Problems: 5600-5609, Amer. Math. Monthly 75 (1968), no. 6, 685–687. MR 1534960, DOI https://doi.org/10.2307/2313822
- Volodymyr Nekrashevych, A group of non-uniform exponential growth locally isomorphic to ${\rm IMG}(z^2+i)$, Trans. Amer. Math. Soc. 362 (2010), no. 1, 389–398. MR 2550156, DOI https://doi.org/10.1090/S0002-9947-09-04825-9
- D. V. Osin, The entropy of solvable groups, Ergodic Theory Dynam. Systems 23 (2003), no. 3, 907–918. MR 1992670, DOI https://doi.org/10.1017/S0143385702000937
- D. V. Osin, Algebraic entropy of elementary amenable groups, Geom. Dedicata 107 (2004), 133–151. MR 2110759, DOI https://doi.org/10.1007/s10711-003-3497-6
- Michael Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175–199. MR 240824, DOI https://doi.org/10.2307/2373276
- Michael Shub, Expanding maps, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 273–276. MR 0266251
- Michael Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126 (1996), no. 1, 85–109. MR 1408557, DOI https://doi.org/10.1007/s002220050090
- A. S. Švarc, A volume invariant of coverings, Dokl. Akad. Nauk SSSR (N.S.) 105 (1955), 32–34 (Russian). MR 0075634
- L. van den Dries and A. J. Wilkie, Gromov’s theorem on groups of polynomial growth and elementary logic, J. Algebra 89 (1984), no. 2, 349–374. MR 751150, DOI https://doi.org/10.1016/0021-8693%2884%2990223-0
- John S. Wilson, Further groups that do not have uniformly exponential growth, J. Algebra 279 (2004), no. 1, 292–301. MR 2078400, DOI https://doi.org/10.1016/j.jalgebra.2004.01.002
- John S. Wilson, On exponential growth and uniformly exponential growth for groups, Invent. Math. 155 (2004), no. 2, 287–303. MR 2031429, DOI https://doi.org/10.1007/s00222-003-0321-8
- Joseph A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geometry 2 (1968), 421–446. MR 248688
Review Information:
Reviewer: V. Nekrashevych
Affiliation: Texas A & M University
Journal: Bull. Amer. Math. Soc. 50 (2013), 495-502
DOI: https://doi.org/10.1090/S0273-0979-2013-01406-X
Published electronically: April 4, 2013
Review copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.