Groups of oscillating intermediate growth. (English) Zbl 1283.20027

Kassabov, Martin; Pak, Igor

Ann. Math. (2) 177, No. 3, 1113-1145 (2013).

For a finitely generated group \(\Gamma\), its growth function \(\gamma_\Gamma(n)\) is given by the number of elements in \(\Gamma\) with word length \(\leq n\). Answering a famous question of J. Milnor [Am. Math. Mon. 75, 685-686 (1968)], R. I. Grigorchuk constructed the first example of a group of intermediate growth [Sov. Math., Dokl. 28, 23-26 (1983); translation from Dokl. Akad. Nauk SSSR 271, 30-33 (1983; Zbl 0547.20025)], i.e. with \(\gamma_\Gamma(n)\) growing strictly faster than any polynomial and strictly slower than the exponential function. During the following thirty years the study of groups of intermediate growth turned into a prominent subject. The present paper is an important new contribution to this field.
The main result can be roughly described as follows. Let \(f_1\), \(f_2\), \(g_1\), \(g_2\) be monotone increasing subexponential integer functions that satisfy \[ f_1\succeq f_2\succeq g_1\succeq g_2=\gamma_{G_\omega},\tag{*} \] where \(G_\omega\) is a Grigorchuk group of intermediate growth with the usual set of generators. Then under some further technical conditions strengthening (*), there exists a finitely generated group \(\Gamma\) with growth function \(h(n)=\gamma_\Gamma(n)\) such that \(g_2(n)<h(n)<f_1(n)\) and \(h(n)\) takes values in the intervals \([g_2(n),g_1(n)]\) and \([f_2(n),f_1(n)]\) infinitely often. – The proof of this theorem is based on the beautiful idea of approximating the Grigorchuk group by an infinite sequence of finite groups \(G_N=\text{PSL}_N(\mathbb Z_N)\) and then using expanding properties of \(G_N\) to analyze the growth of the group \(\Gamma=\otimes G_N\) (we refer to the paper for the definition of the product that is used here).
Other important results about the growth functions of groups of intermediate growth were obtained recently by L. Bartholdi and A. Erschler [Invent. Math. 189, No. 2, 431-455 (2012; Zbl 1286.20025)] and J. Brieussel [preprint arXiv 1107.1632].

Reviewer: Mikhail Belolipetsky (Rio de Janeiro)

MSC:

20F05	Generators, relations, and presentations of groups
20F69	Asymptotic properties of groups
20E08	Groups acting on trees

Keywords:

Grigorchuk groups; groups of intermediate growth; growth functions; finitely generated groups

Citations:

Zbl 0547.20025; Zbl 1286.20025

References:

[1]	L. Bartholdi, ”The growth of Grigorchuk’s torsion group,” Internat. Math. Res. Notices, iss. 20, pp. 1049-1054, 1998. · Zbl 0942.20027
[2]	L. Bartholdi, ”Lower bounds on the growth of a group acting on the binary rooted tree,” Internat. J. Algebra Comput., vol. 11, iss. 1, pp. 73-88, 2001. · Zbl 1028.20025
[3]	L. Bartholdi and A. G. Erschler, Growth of permutational extensions. · Zbl 1286.20025
[4]	L. Bartholdi and A. G. Erschler, Groups of given intermediate word growth. · Zbl 1317.20043
[5]	L. Bartholdi, R. I. Grigorchuk, and Z. vSuni’k, ”Branch groups,” in Handbook of Algebra, Vol. 3, Amsterdam: North-Holland, 2003, pp. 989-1112. · Zbl 1140.20306
[6]	I. Benjamini and O. Schramm, ”Recurrence of distributional limits of finite planar graphs,” Electron. J. Probab., vol. 6, p. 23, 2001. · Zbl 1010.82021
[7]	J. Bourgain and A. Gamburd, ”Uniform expansion bounds for Cayley graphs of \({ SL}_2(\mathbb F_p)\),” Ann. of Math., vol. 167, iss. 2, pp. 625-642, 2008. · Zbl 1216.20042
[8]	E. Breuillard and A. Gamburd, ”Strong uniform expansion in \({ SL}(2,p)\),” Geom. Funct. Anal., vol. 20, iss. 5, pp. 1201-1209, 2010. · Zbl 1253.20051
[9]	J. Brieussel, Growth of certain groups of automorphisms of rooted trees, 2008.
[10]	J. Brieussel, ”Amenability and non-uniform growth of some directed automorphism groups of a rooted tree,” Math. Z., vol. 263, iss. 2, pp. 265-293, 2009. · Zbl 1227.43001
[11]	J. Brieussel, Growth behaviors in the range \(e^{r^\alpha}\).
[12]	J. Brieussel, M. Kassabov, and I. Pak, The oscillating rate of escape of random walks on groups, 2013.
[13]	D. A. Craven, ”Lower bounds for representation growth,” J. Group Theory, vol. 13, iss. 6, pp. 873-890, 2010. · Zbl 1211.20008
[14]	A. Erschler, ”Boundary behavior for groups of subexponential growth,” Ann. of Math., vol. 160, iss. 3, pp. 1183-1210, 2004. · Zbl 1089.20025
[15]	A. Erschler, ”On the degrees of growth of finitely generated groups,” Funktsional. Anal. i Prilozhen., vol. 39, iss. 4, pp. 86-89, 2005. · Zbl 1122.20016
[16]	R. I. Grigorchuk, ”On Burnside’s problem on periodic groups,” Funktsional. Anal. i Prilozhen., vol. 14, iss. 1, pp. 53-54, 1980. · Zbl 0595.20029
[17]	R. I. Grigorchuk, ”On the Milnor problem of group growth,” Soviet Math. Dokl., vol. 28, pp. 23-26, 1983. · Zbl 0547.20025
[18]	R. I. Grigorchuk, ”Degrees of growth of finitely generated groups, and the theory of invariant means,” Math. USSR-Izv., vol. 25, pp. 259-300, 1985. · Zbl 0583.20023
[19]	R. I. Grigorchuk, ”Degrees of growth of \(p\)-groups and torsion-free groups,” Math. USSR-Sb., vol. 54, pp. 185-205, 1986. · Zbl 0583.20024
[20]	R. I. Grigorchuk, ”Solved and unsolved problems around one group,” in Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, Basel: Birkhäuser, 2005, vol. 248, pp. 117-218. · Zbl 1165.20021
[21]	R. I. Grigorchuk, ”Some topics of dynamics of group actions on rooted trees,” Proc. Steklov Inst. Math., vol. 273, pp. 1-118, 2011. · Zbl 1268.20027
[22]	R. I. Grigorchuk, Milnor’s problem on the growth of groups and its consequences, 2011. · Zbl 1325.20034
[23]	R. I. Grigorchuk and P. de la Harpe, ”On problems related to growth, entropy, and spectrum in group theory,” J. Dynam. Control Systems, vol. 3, iss. 1, pp. 51-89, 1997. · Zbl 0949.20033
[24]	R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskiui, ”Automata, dynamical systems, and groups,” Tr. Mat. Inst. Steklova, vol. 231, iss. Din. Sist., Avtom. i Beskon. Gruppy, pp. 134-214, 2000. · Zbl 1155.37311
[25]	R. I. Grigorchuk and I. Pak, ”Groups of intermediate growth: an introduction,” Enseign. Math., vol. 54, iss. 3-4, pp. 251-272, 2008. · Zbl 1204.20049
[26]	M. Gromov, ”Hyperbolic groups,” in Essays in Group Theory, New York: Springer-Verlag, 1987, vol. 8, pp. 75-263. · Zbl 0634.20015
[27]	P. de la Harpe, Topics in Geometric Group Theory, Chicago, IL: University of Chicago Press, 2000. · Zbl 0965.20025
[28]	P. de la Harpe, ”Uniform growth in groups of exponential growth,” in Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II, 2002, pp. 1-17. · Zbl 1025.20027
[29]	S. Hoory, N. Linial, and A. Wigderson, ”Expander graphs and their applications,” Bull. Amer. Math. Soc., vol. 43, iss. 4, pp. 439-561, 2006. · Zbl 1147.68608
[30]	A. Jaikin-Zapirain, ”Zeta function of representations of compact \(p\)-adic analytic groups,” J. Amer. Math. Soc., vol. 19, iss. 1, pp. 91-118, 2006. · Zbl 1092.20023
[31]	M. Kassabov and N. Nikolov, ”Cartesian products as profinite completions,” Int. Math. Res. Not., vol. 2006, p. I, 2006. · Zbl 1129.20019
[32]	M. Larsen and A. Lubotzky, ”Representation growth of linear groups,” J. Eur. Math. Soc. \((\)JEMS\()\), vol. 10, iss. 2, pp. 351-390, 2008. · Zbl 1142.22006
[33]	Y. G. Leonov, ”On a lower bound for the growth function of the Grigorchuk group,” Mat. Zametki, vol. 67, iss. 3, pp. 475-477, 2000. · Zbl 0984.20020
[34]	A. Lubotzky, Discrete Groups, Expanding Graphs and Invariant Measures, Basel: Birkhäuser, 1994, vol. 125. · Zbl 0826.22012
[35]	A. Lubotzky, ”Subgroup growth,” in Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Basel, 1995, pp. 309-317. · Zbl 0857.20013
[36]	A. Lubotzky and D. Segal, Subgroup Growth, Basel: Birkhäuser, 2003, vol. 212. · Zbl 1071.20033
[37]	R. Muchnik and I. Pak, ”On growth of Grigorchuk groups,” Internat. J. Algebra Comput., vol. 11, iss. 1, pp. 1-17, 2001. · Zbl 1024.20031
[38]	V. Nekrashevych, Self-Similar Groups, Providence, RI: Amer. Math. Soc., 2005, vol. 117. · Zbl 1087.20032
[39]	B. H. Neumann, ”Some remarks on infinite groups,” J. London Math. Soc., vol. 12, pp. 120-127, 1937. · Zbl 0016.29501
[40]	P. M. Neumann, ”Some questions of Edjvet and Pride about infinite groups,” Illinois J. Math., vol. 30, iss. 2, pp. 301-316, 1986. · Zbl 0598.20029
[41]	V. G. Pestov, ”Hyperlinear and sofic groups: a brief guide,” Bull. Symbolic Logic, vol. 14, iss. 4, pp. 449-480, 2008. · Zbl 1206.20048
[42]	L. Pyber, ”Old groups can learn new tricks,” in Groups, Combinatorics & Geometry, World Sci. Publ., River Edge, NJ, 2003, pp. 243-255. · Zbl 1046.20024
[43]	L. Pyber, ”Groups of intermediate subgroup growth and a problem of Grothendieck,” Duke Math. J., vol. 121, iss. 1, pp. 169-188, 2004. · Zbl 1057.20019
[44]	S. M. Ruiz, ”A result on prime numbers,” Math. Gazette, vol. 81, pp. 269-270, 1997.
[45]	D. Segal, ”The finite images of finitely generated groups,” Proc. London Math. Soc., vol. 82, iss. 3, pp. 597-613, 2001. · Zbl 1022.20011
[46]	Y. Shalom and T. Tao, ”A finitary version of Gromov’s polynomial growth theorem,” Geom. Funct. Anal., vol. 20, iss. 6, pp. 1502-1547, 2010. · Zbl 1262.20044
[47]	V. A. Ufnarovskij, ”Combinatorial and asymptotic methods in algebra,” in Algebra, VI, New York: Springer-Verlag, 1995, vol. 57, pp. 1-196. · Zbl 0826.16001
[48]	C. Voll, ”Functional equations for zeta functions of groups and rings,” Ann. of Math., vol. 172, iss. 2, pp. 1181-1218, 2010. · Zbl 1314.11057
[49]	J. S. Wilson, ”On exponential growth and uniformly exponential growth for groups,” Invent. Math., vol. 155, iss. 2, pp. 287-303, 2004. · Zbl 1065.20054
[50]	W. Woess, Random Walks on Infinite Graphs and Groups, Cambridge: Cambridge Univ. Press, 2000, vol. 138. · Zbl 0951.60002