Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Books
  3. How Groups Grow
  • Cited by 10
Publisher:
Cambridge University Press
Online publication date:
January 2012
Print publication year:
2011
Online ISBN:
9781139095129
DOI:
https://doi.org/10.1017/CBO9781139095129
Subjects:
Mathematics (general), Mathematics, Algebra
Series:
London Mathematical Society Lecture Note Series (395)
Information

Book description

Growth of groups is an innovative new branch of group theory. This is the first book to introduce the subject from scratch. It begins with basic definitions and culminates in the seminal results of Gromov and Grigorchuk and more. The proof of Gromov's theorem on groups of polynomial growth is given in full, with the theory of asymptotic cones developed on the way. Grigorchuk's first and general groups are described, as well as the proof that they have intermediate growth, with explicit bounds, and their relationship to automorphisms of regular trees and finite automata. Also discussed are generating functions, groups of polynomial growth of low degrees, infinitely generated groups of local polynomial growth, the relation of intermediate growth to amenability and residual finiteness, and conjugacy class growth. This book is valuable reading for researchers, from graduate students onward, working in contemporary group theory.

Reviews

'How Groups Grow is an excellent introduction to growth of groups for everybody interested in this subject. It also touches a variety of adjacent subjects (such as amenability, isoperimetric inequalities, groups generated by automata, etc.) It is written in a very accessible style, with very clear exposition of all main results.'

V. Nekrashevych Source: Bulletin of the American Mathematical Society

Refine List

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

Contents

Contents
References
References
[Ad 75]. Adian, S.I., The Burnside Problem and Identities in Groups, Nauka, 1975 (Russian; English translation, Springer, 1979).
[Al 72]. Aleshin, S.V., Finite automata and the Burnside problem for periodic groups, Mat. Zametki 11 (1972), 319–328 (Russian); English translation in Math. Notes 11 (1972).
[Al 91]. Alonso, J.M., Growth functions of amalgams. In Arboreal Group Theory, Springer, New York 1991, 1–34.
[Al 02]. Alperin, R.C., Uniform growth of polycyclic groups, Geo. Ded. 92 (2002), 105–113.
[AO 96]. Arzhantseva, G.N. and Olshanskii, A.U., Generality of the class of groups in which subgroups with a lesser number of generators are free, Mat. Zametki. 59 (1996), 489–496, 638 (In Russian; English translation in Math. Notes59 (1996), 350–355).
[BS 92]. Babai, L. and Szegedy, M., Local expansion of symmetric graphs, Combinatorics, Probability and Computing 1 (1992), 1–11.
[BM 07]. Bajorska, B. and Macedonska, O., A note on groups of intermediate growth, Comm. Alg. 35(12) (2007), 4112–4115.
[Ba 98]. Bartholdi, L., The growth of Grigorchuk's torsion group, Int. Math. Research Notices 20 (1998), 1049–1054.
[Ba 01]. Bartholdi, L., Lower bounds on the growth of a group acting on the binary rooted tree, Int. J. Alg. Comp. 11 (2001), 73–88.
[Ba 03]. Bartholdi, L., A Wilson group of non-uniformly exponential growth, C. R. Math. Acad. Sci. Paris 336 (2003), 549–554.
[BE 10]. Bartholdi, L. and Erschler, A., Growth of permutational extensions, arXiv preprint [math.Gr] 1011.5266, November 2010 (18 pages).
[BV 05]. Bartholdi, L. and Virag, B., Amenability via random walks, Duke Math. J. 130 (2005), 39–56.
[Bs 72]. Bass, H., The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. 25 (1972), 603–614.
[Bu 01]. Baumagin, I., On small cancellation k-generator groups with (k – 1)-generator subgroups all free, Internat. J. Alg. Comp. 11 (2001), 507–524.
[Be 83] Benson, M., Growth series of finite extensions of Zn are rational, Inv. Math. 73 (1983), 251–269.
[Be 87]. Benson, M., On the rational growth of virtually nilpotent groups. In Combinatorial Group Theory and Topology, 185–196, Ann. Math. Studies 111, Princeton 1987.
[Bi 07]. Bieri, R., Deficiency and the geometric invariants of a group, J. Pure App. Alg. 208 (2007), 951–959.
[BS 78]. Bieri, R. and Strebel, R., Almost finitely presented groups, Comm. Math. Helvetici. 53 (1978), 258–278.
[Br 07]. Breuillard, E., On uniform exponential growth for solvable groups, Pure Appl. Math. Q. 3 (2007).
[BC 10]. Breuillard, E. and Cornulier, Y., On conjugacy growth for solvable groups, Ill. J. Math. 54 (2010), 389–395.
[BCLM 11]. Breuillard, E., Cornulier, Y., Lubotzky, A., and Meiri, C., On conjugacy growth of linear groups, arXiv preprint [math.Gr]1106.4773 (21 pages).
[BG 08]. Breuillard, E. and Gelander, T., Uniform independence in linear groups, Inv. Math. 173 (2008), 225–263.
[Br 05]. Bridson, M.R., On the growth of groups of automorphisms, Int. J. Alg. Comp. 15 (2005), 869–874.
[Br 09]. Brieussel, J., Amenability and non-uniform growth of some directed automorphism groups of a rooted tree, Math. Z. 263 (2009), 265–293.
[Br 11]. Brieussel, J., Growth behaviors in the range, arXiv preprint [math.GR] 1107.1632.
[Bu 99]. Bucher, M., Croissance de groupes et produits libres avec amalgamation, diploma thesis, Geneva 1999, see http://www.math.kth.se./mickar/ (18 pages).
[Bu 09]. Button, J.O., Uniform exponential growth of semidirect HNN extensions, preprint (January 2010: 22 pages).
[Ch 94a]. Chiswell, I.M., The growth series of a graph product, Bull. London Math. Soc. 26 (1994), 268–272.
[Ch 94b]. Chiswell, I.M., The growth series of HNN extensions, Comm. Alg. 22 (1994), 2969–2981.
[Ch 80]. Chou, C., Elementary amenable groups, Ill. J. Math. 24 (1980), 396–407.
[Co 07]. Collins, M.J., On Jordan's theorem for complex linear groups, J. Group Th. 10 (2007), 411–423.
[Co 08]. Collins, M.J., Modular analogues of Jordan's theorem for finite linear groups, J. Reine Angew. Math. (Crelle's) 624 (2008), 143–171.
[Co 05]. Coornaert, M., Asymptotic growth of conjugacy classes in finitely generated free groups, Int. J. Alg. Comp. 15 (2005), 887–892.
[CK 02]. Coornaert, M. and Knieper, G., Growth of conjugacy classes in Gromov hyperbolic groups, Geo. Func. Ana. 12 (2002), 464–478.
[CK 04]. Coornaert, M. and Knieper, G., An upper bound for the growth of conjugacy classes in torsion-free word hyperbolic groups, Int. J. Alg. Comp. 14 (2004), 395–401.
[CSC 93]. Coulhon, T. and Saloff-Coste, L., Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamericana 9 (1993), 293–314.
[CR 62]. Curtis, C.W. and Reiner, I., Representation Theory of Finite Groups and Associative Algebras, Interscience, New York 1962.
[DG 11]. Dahmani, F. and Guirardel, V., The isomorphism problem for all hyperbolic groups, Geom. Func. Anal. 21 (2011), 223–300.
[DDMS 99]. Dixon, J.D., Sautoy, M.P.F., Mann, A., and Segel, D., Analytic pro-p groups, 2nd edn., Cambridge University Press, Cambridge 1999.
[Dr 02]. Drutu, C., Quasi-isometry invariants and asymptotic cones, Int. J. Alg. Comp. 12 (2002), 99–135.
[Dy 00]. Dyubina, A., Instability of the virtual solvability and the property of being virtually torsion-free for quasi-isometric groups, Int. Math. Res. Notices 21 (2000), 1097–1101.
[ECHLPT 92]. Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., and Thurston, W.P., Word Processing in Groups, Jones and Bartlett, Boston 1992.
[Er 04]. Erschler, A., Not residually finite groups of intermediate growth, commensurability and non-geometricity, J. Alg. 272 (2004), 154–172.
[EMO 05]. Eskin, A., Mozes, S., and Oh, H., On uniform exponential growth for linear groups, Inv. Math. 160 (2005), 1–30.
[Fe 95]. Feit, W., The orders of finite linear groups, preprint.
[Fe 97]. Feit, W., Finite linear groups and theorems of Minkowski and Schur, Proc. Amer. Math. Soc. 125 (1997), 1259–1262.
[FP 87]. Floyd, W.J. and Plotnick, S.P., Growth functions on Fuchsian groups and the Euler characteristic, Inv. Math. 88 (1987), 1–29.
[Fr 97]. Friedland, S., The maximal orders of finite subgroups in GLn(ℚ), Proc. Amer. Math. Soc. 125 (1997), 3519–3526.
[FS 08]. Freden, E.M. and Schofield, J., The growth series for Higman: 3, J. Group Th. 11 (2008), 277–298.
[GH 90]. Ghys, E. and Harpe, P. (editors), Sur les Groupes Hyperboliques d'après Mikhael Gromov, Birkhauser, Boston 1990.
[Gi 99]. Gill, C.P., Growth series of stem products of cyclic groups, Int. J. Alg. Comp. 9 (1999), 1–30.
[Go 64]. Golod, E.S., On nil-algebras and finitely approximable p-groups, Izv. Akad. Nauk SSSR, Ser. Mat. 28 (1964), 273–276 (In Russian; English translation in Transl. Amer. Math. Soc. (2)48 (1965), 103–106).
[Gri 80]. Grigorchuk, R.I., Burnside's problem on periodic groups, Fun. Anal. App. 14 (1980), 41–43.
[Gri 84]. Grigorchuk, R.I., Degrees of growth of finitely generated groups, and the theory of invariant means, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), 939–985 (In Russian; English translation in Math. USSR Izv.25 (1985), 259–300).
[Gri 85]. Grigorchuk, R.I., On the growth degrees of p-groups and torsionfree groups, Mat. Sb. 126 (1985), 194–214 (In Russian; English translation in Math. USSR Sbornik54 (1986), 185–205.
[Gri 99]. Grigorchuk, R.I., On the system of defining relations and the Schur multiplier of periodic groups defined by finite automata. In Groups St Andrews 1997 in Bath I, Cambridge University Press, Cambridge 1999, 290–317.
[GH 01]. Grigorchuk, R.I. and Harpe, P., One-relator groups of exponential growth have uniformly exponential growth, Mat. Zametki 69 (2001), 628–630 (In Russian; English translation in Math. Notes69 575–577).
[Gro 81]. Gromov, M., Groups of polynomial growth and expanding maps, Publ. Math. IHES 53 (1981), 53–73.
[GS 84]. Grunewald, F. and Segal, D., Reflections on the classification of torsion-free nilpotent groups, Group Theory: Essays for Philip Hall121–158, Academic Press, London 1984.
[GSS 82]. Gruenwald, F., Segal, D., and Sterling, L.S., Nilpotent Groups of Hirsch Length Six, Math. Z. 179 (1982), 219–235.
[GS 10]. Guba, V.S. and Sapir, M.V., On the conjugacy growth functions of groups, Ill. J. Math. 54 (2010), 301–313.
[Ha 54]. Hall, P., Finiteness conditions for soluble groups, Proc. London Math. Soc. (3) 4 (1954), 419–436.
[Hr 00]. Harpe, P., Topics in Geometric Group Theory, University of Chicago Press, Chicago 2000.
[HB 00]. Harpe, P. and Bucher, M., Free products with amalgamation and HNN-extensions of uniformly exponential growth, Mat. Zametki 67 (2000), 811–815 (In Russian; English translation in Math. Notes 67 (2000), 686–689).
[Ho 63]. Horejs, J., Transformations defined by finite automata, Problems in Cybernetics 9 (1963), 23–26 (Russian).
[Hu 11]. Hull, M., Conjugacy growth in polycyclic groups, Arch. Math. 96 (2011), 131–134.
[HO 11]. Hull, M. and Osin, D., Conjugacy growth of finitely generated groups, arXiv preprint [math.GR] 1107.1826.
[Hu 67]. Huppert, B., Endliche Gruppen I, Springer, New York 1967.
[HW 42]. Hurewicz, W. and Wallman, H., Dimension Theory, Princeton University Press, Princeton 1942.
[IS 87]. Imrich, W. and Seifert, N., A bound for groups of linear growth, Arch. Math. (Basel) 48 (1987), 100–104.
[Is 76]. Isaacs, I.M., Character Theory of Finite Groups, Academic Press, San Diego 1976.
[JKS 95]. Johnson, D.L., Kim, A.C., and Song, H.J., The growth of the trefoil group. In Groups Korea 94, de Gruyter, Berlin (1995), 157–161.
[Jo 91]. Johnson, D.L., Rational growth of wreath products. In Groups St Andrews 1989 II, Cambridge University Press, Cambridge (1991), 309–315.
[Ju 71]. Justin, J., Groupes et semi-groupes à croissants linéare, C. R. Acad. Sci. Paris Ser A–B 273 (1971), A212–A214.
[Ka 95]. Kaplansky, I., Lie Algebras and Locally Compact Groups, 2nd edn., University of Chicago Press, Chicago 1995.
[Kl 10]. Kleiner, B., A new proof of Gromov's theorem on groups of polynomial growth, J. Amer. Math. Soc. 23 (2010), 815–829.
[Ko 98]. Koubi, M., Croissance uniforme dans les groupes hyperboliques, Ann. Inst. Fourier 48 (1998), 1441–1453.
[Ku 56]. Kurosh, A.G., The Theory of Groups, vol. 2, 2nd edn. (English translation by Hirsch, K.A.), Chelsea, New York 1956.
[Le 91]. Lewin, J., The growth function of some free products of groups, Comm. Alg. 19 (1991), 2405–2418.
[Le 00]. Leonov, Yu.G., On a lower bound for the growth function of Grigorchuk's group, Math. Zametki 67 (2000), 475–477 (Russian); English translation in Math. Notes 67 (2000), 403–405.
[LP 98]. Larsen, M. and Pink, R., Finite Subgroups of Algebraic Groups, J. Amer. Math. Soc. 24 (2011), 1105–1158.
[LS 03]. Lubotzky, A. and Segal, D., Subgroup Growth, Birkhäuser, Basel 2003.
[LS 77]. Lyndon, R.C. and Schupp, P.E., Combinatorial Group Theory, Springer, Berlin 1977.
[LPV 08]. Lyons, R., Pichot, M., and Vassout, S., Uniform non-amenability, cost, and the first l2-Betti number, Groups Geom. Dyn. 2 (2008), 595–617.
[Ma 07]. Mann, A., Growth conditions in infinitely generated groups, Groups, Geometry, and Dynamics 1 (2007), 613–622.
[Ma 11]. Mann, A., The growth of free products, J. Alg. 326 (Karl W. Gruenberg memorial issue) (2011), 208–217.
[Mi 68]. Milnor, J., Growth of finitely generated solvable groups, J. Diff. Geo. 2 (1968), 447–449.
[Mi 87]. Minkowski, H., Collected Works I, 212–218.
[MZ 55]. Montgomery, D. and Zippin, L., Topological Transformation Groups, Interscience, New York 1955.
[MP 01]. Muchnik, R. and Pak, I., On growth of Grigorchuk's groups, Int. J. Alg. Comp. 11 (2001), 1–17.
[Ol 91]. Olshanskii, A. Yu., Geometry of Defining Relations in Groups, Kluwer, Dordrecht 1991.
[Ol 92]. Ol'shanskii, A. Yu., Almost every group is hyperbolic, Int. J. Alg. Comp. 2 (1992), 1–17.
[Os 03]. Osin, D.V., The entropy of solvable groups, Erg. Th. Dyn. Sys. 23 (2003), 907–918.
[Os 04]. Osin, D.V., Algebraic entropy of elementary amenable groups, Geo. Ded. 107 (2004), 133–151.
[Pa 83]. Pansu, P., Croissance des boules et des géodésiques fermées dans les nilvariétés, Erg. Th. Dyn. Sys. 3 (1983), 415–445.
[Pa 92]. Parry, W., Growth series of some wreath products, Trans. Amer. Math. Soc. 331 (1992), 751–759.
[Pi 00]. Pittet, Ch., The isoperimetric profile of homogeneous Riemannian manifolds, J. Diff. Geo. 54 (2000), 255–302.
[Re 98]. Remmert, R., Classical Topics in Complex Function Theory, Springer, New York 1998.
[Ri 82]. Rips, E., Subgroups of small cancellation groups, Bull. London Math. Soc. 14 (1982), 45–47.
[Ri 10]. Rivin, I., Growth in free groups (and other stories) – twelve years later, Ill. J. Math. 54 (2010), 327–370.
[Ro 96]. Robinson, D.J.S., A Course in the Theory of Groups, 2nd edn., Springer, New York 1996.
[Ro 95]. Rotman, J.J., Introduction to the Theory of Groups, 4th edn., Springer, New York 1995.
[Sc 11]. Scott, R., Rationality and reciprocity for the greedy normal form of a Coxeter groups, Trans. Amer. Math. Soc. 363 (2011), 385–415.
[Se 83]. Segal, D., Polycyclic Groups, Cambridge University Press, Cambridge 1983.
[Se 95]. Sela, Z., The isomorphism problem for hyperbolic groups, I., Ann. Math. (2) 141 (1995), 217–283.
[Se 80]. Serre, J.P., Trees, Springer-Verlag, Berlin, 1980.
[Sh 94]. Shapiro, M., Growth of a PSL2R manifold group, Math. Nach. 167 (1994), 279–312.
[Sh 98]. Shalom, Y., The growth of linear groups, J. Alg. 199 (1998), 169–174.
[SW 92]. Shalen, P.B. and Wagreich, P., Growth rates, ℤp-homology, and volumes of hyperbolic 3-manifolds, Trans. Amer. Math. Soc. 331 (1992), 895–917.
[Sl]. Sloane's Online Encyclopedia of Integer Sequences, http://www.research.att.com/ njas/sequences/Seis.html.
[So 06]. Soifer, I., Properties of growth functions of Fuchsian groups, M.Sc. thesis, Hebrew University, Jerusalem 2006.
[St 96]. Stoll, M., Rational and transcendental growth series for the higher Heisenberg groups, Inv. Math. 126 (1996), 85–109.
[St 98]. Stoll, M., On the asymptotics of the growth of 2-step nilpotent groups, J. London Math. Soc. 58 (1998), 38–48.
[Su 79]. Sushchanskii, V.I., Periodic p-groups of permutations and the unrestricted Burnside problem (in Russian), Dokl. Akad. Nauk SSSR 247 (1979), 557–561.
[Ta 10]. Tao, T., A proof of Gromov's theorem (a blog entry) http://terrytao.wordpress.com/2010/02/18/a-proof-of-gromovs-theorem/
[Te 07]. Tessera, R., Volume of spheres in doubling metric measured spaces and in groups of polynomial growth, Bull. Soc. Math. France 135 (2007), 47–64.
[Ti 39]. Titchmarch, E.C., The Theory of Functions, 2nd edn., Oxford University Press, Oxford 1939 (reprinted 1952).
[Ti 72]. Tits, J., Free subgroups in linear groups, J. Alg. 20 (1972), 250–270.
[TJ 74]. Tyrer-Jones, J.M., Direct products and the Hopf property, J. Austral. Math. Soc. 17 (1974), 174–196.
[VdDW 84(1)]. Dries, L. and Wilkie, A.J., On Gromov's theorem concerning groups of polynomial growth and elementary logic, J. Alg. 89 (1984), 349–374.
[VdDW 84(2)]. Dries, L. and Wilkie, A.J., An effective bound for groups of linear growth, Arch. Math. (Basel) 42 (1984), 391–396.
[We 73]. Wehrfritz, B.A.F., Infinite Linear Groups, Springer, Berlin 1973.
[Wi 04(1)]. Wilson, J.S., On exponential growth and uniformly exponential growth for groups, Inv. Math. 155 (2004), 287–303.
[Wi 04(2)]. Wilson, J.S., Further groups that do not have uniformly exponential growth, J. Alg. 279 (2004), 292–301.
[Wi 10]. Wilson, J.S., Free subgroups in groups with few relators, Enseign. Math. (2) 56 (2010), 173–185.
[Wi 11]. Wilson, J.S., The gap in the growth of residually soluble groups, Bull. London Math. Soc. 43 (2011), 576–582.
[Wo 68]. Wolf, J.A., Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Diff. Geo. 2 (1968), 421–446.
[Wo 97]. Worthington, R.L., The growth series of Hwr(ℤ × Z2), Arch. Math. 68 (1997), 110–121.
[Xi 07]. Xi, X., Growth of relatively hyperbolic groups, Proc. Amer. Math. Soc. 135 (2007), 695–704.
[Z 00]. zuk, Andrzej, On an isoperimetric inequality for infinite finitely generated groups, Topology 39 (2000), 947–956.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Book summary page views

Total views: 0 *
Loading metrics...

* Views captured on Cambridge Core between #date#. This data will be updated every 24 hours.

Usage data cannot currently be displayed.