Book Review
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3049875
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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 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 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 10.1016/S1631-073X(03)00131-6
H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3) 25 (1972), 603–614. MR 379672, DOI 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 10.1007/s00222-011-0368-x
M. Benson, Growth series of finite extensions of $\textbf {Z}^{n}$ are rational, Invent. Math. 73 (1983), no. 2, 251–269. MR 714092, DOI 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 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 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 10.1016/S0040-9383(98)00063-9
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, DOI 10.1017/CBO9780511470882
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 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 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 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 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 10.2307/2313822
Volodymyr Nekrashevych, A group of non-uniform exponential growth locally isomorphic to $\textrm {IMG}(z^2+i)$, Trans. Amer. Math. Soc. 362 (2010), no. 1, 389–398. MR 2550156, DOI 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 10.1017/S0143385702000937
D. V. Osin, Algebraic entropy of elementary amenable groups, Geom. Dedicata 107 (2004), 133–151. MR 2110759, DOI 10.1007/s10711-003-3497-6
Michael Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175–199. MR 240824, DOI 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 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 10.1016/0021-8693(84)90223-0
John S. Wilson, Further groups that do not have uniformly exponential growth, J. Algebra 279 (2004), no. 1, 292–301. MR 2078400, DOI 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 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
References
- Laurent Bartholdi, The growth of Grigorchuk’s torsion group, Internat. Math. Res. Notices 20 (1998), 1049–1054. MR 1656258 (99i:20049), DOI 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 (2001m:20044), DOI 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 (2004c:20051), DOI 10.1016/S1631-073X(03)00131-6
- H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3) 25 (1972), 603–614. MR 0379672 (52 \#577)
- 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 10.1007/s00222-011-0368-x
- M. Benson, Growth series of finite extensions of $\textbf {Z}^{n}$ are rational, Invent. Math. 73 (1983), no. 2, 251–269. MR 714092 (85e:20026), DOI 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 (2006m:20062), DOI 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 (86j:20032), DOI 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 (2001i:20084), DOI 10.1016/S0040-9383(98)00063-9
- 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 (2000m:20039), DOI 10.1017/CBO9780511470882
- 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 (93g:20053)
- 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 (86h:20041)
- 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 (90j:20063)
- Rostislav I. Grigorchuk, On growth in group theory, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 325–338. MR 1159221 (93e:20001)
- 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 (83b:53041)
- 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 (85e:53051)
- Yves Guivarc’h, Groupes de Lie à croissance polynomiale, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A237–A239 (French). MR 0272943 (42 \#7824)
- Pierre de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2000. MR 1786869 (2001i:20081)
- 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 (2012d:20089), DOI 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 (2001f:20062), DOI 10.1007/BF02676677
- Alexander Lubotzky and Avinoam Mann, On groups of polynomial subgroup growth, Invent. Math. 104 (1991), no. 3, 521–533. MR 1106747 (92d:20038), DOI 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 0244899 (39 \#6212)
- 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 10.2307/2313822
- Volodymyr Nekrashevych, A group of non-uniform exponential growth locally isomorphic to $\textrm {IMG}(z^2+i)$, Trans. Amer. Math. Soc. 362 (2010), no. 1, 389–398. MR 2550156 (2010m:20067), DOI 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 (2004f:20065), DOI 10.1017/S0143385702000937
- D. V. Osin, Algebraic entropy of elementary amenable groups, Geom. Dedicata 107 (2004), 133–151. MR 2110759 (2006b:37010), DOI 10.1007/s10711-003-3497-6
- Michael Shub, Endomorphisms of compact differentiable manifolds, Amer. J. Math. 91 (1969), 175–199. MR 0240824 (39 \#2169)
- 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 (42 \#1158)
- Michael Stoll, Rational and transcendental growth series for the higher Heisenberg groups, Invent. Math. 126 (1996), no. 1, 85–109. MR 1408557 (98d:20033), DOI 10.1007/s002220050090
- A. S. Švarc, A volume invariant of coverings, Dokl. Akad. Nauk SSSR (N.S.) 105 (1955), 32–34 (Russian). MR 0075634 (17,781d)
- 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 (85k:20101), DOI 10.1016/0021-8693(84)90223-0
- John S. Wilson, Further groups that do not have uniformly exponential growth, J. Algebra 279 (2004), no. 1, 292–301. MR 2078400 (2005e:20066), DOI 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 (2004k:20085), DOI 10.1007/s00222-003-0321-8
- Joseph A. Wolf, Growth of finitely generated solvable groups and curvature of Riemanniann manifolds, J. Differential Geometry 2 (1968), 421–446. MR 0248688 (40 \#1939)
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.
