Hostname: page-component-7479d7b7d-q6k6v Total loading time: 0 Render date: 2024-07-10T20:54:14.635Z Has data issue: false hasContentIssue false
  • English
  • Français

Subnormal subgroups in free groups, their growth and cogrowth

Published online by Cambridge University Press:  27 February 2017

A. YU. OLSHANSKII
A. YU. OLSHANSKII*
Affiliation:
Department of Mathematics, Vanderbilt University, Nashville 37240, U.S.A., and Moscow State University, Moscow 119991, Russia. e-mail: alexander.olshanskiy@vanderbilt.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, the author (1) compare subnormal closures of finite sets in a free group F; (2) obtains the limit for the series of subnormal closures of a single element in F; (3) proves that the exponential growth rate (exp.g.r.) $\lim_{n\to \infty}\sqrt[n]{g_H(n)}$, where gH(n) is the growth function of a subgroup H with respect to a finite free basis of F, exists for any subgroup H of the free group F; (4) gives sharp estimates from below for the exp.g.r. of subnormal subgroups in free groups; and (5) finds the cogrowth of the subnormal closures of free generators.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 3 , November 2017 , pp. 499 - 531
Copyright
Copyright © Cambridge Philosophical Society 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[AGV] Abert, M., Glasner, V. and Virag, B. The measurable Kesten theorem. arXiv1111.2080v2Google Scholar
[BO] Bahturin, Yu. A. and Olshanskii, A. Yu. Actions of maximal growth. Proc. London Math. Soc. 101 (2010), no.1, 27–72.CrossRefGoogle Scholar
[BO1] Bahturin, Yu. A. and Olshanskii, A. Yu. Growth of subalgebras and subideals in free Lie algebras. J. Algebra 422 (2015), 277–305.Google Scholar
[DO] Davis, T.C. and Olshanskii, A. Yu. Relative subgroup growth and subgroup distortion. Groups, Geometry and Dynamics 9 (2015), no.1, 237–273.CrossRefGoogle Scholar
[F] Feller, W. An Introduction to Probability Theory and Its Applications, volume 2, 2nd ed. (Wiley & Sons, Inc. 1971).Google Scholar
[Fo] Fox, R.H. Free differential calculus. I, Derivation in free group rings. Ann. of Math. (2), 57 (1953), no. 3, 547–560.Google Scholar
[GM] Godsil, C.D. and Mohar, B. Walk generating functions and special measures of infinite graphs. Linear Algebra Appl. 107 (1988), 191–206.Google Scholar
[G] Greenleaf, F. Invariant Means on Topological Groups and Their Applications (van Nostrand, N.Y. 1967).Google Scholar
[Gr] Grigorchuk, R. Symmetric random walks on discrete groups. In Multicomponent Random Systems (Nauka, Moscow 1978) (in Russian), 132–152.Google Scholar
[GH] Grigorchuk, R. and de la Harpe, P. On problems related to growth, entropy and spectrum in Group Theory. J. Dynam. Control Systems 3 (1997), no. 1, 51–89.CrossRefGoogle Scholar
[GKN] Grigorchuk, R., Kaymanovich, V. and Smirnova-Nagnibeda, T. Ergodic properties of boundary actions and Nielsen-Schreier theory. Adv. Math. 230 (2012), no. 3, 1340–1380.CrossRefGoogle Scholar
[I] Iossif, G. Return probabilities for correlated random walks. J. Appl. Prob. 23 (1986), 201–207.Google Scholar
[KSS] Kapovich, I., Shpilrain, V. and Schupp, P. Generic properties of Whitehead's Algorithm and isomorphism rigidity of random one-relator groups. Pacific. J. Math. 223 (2006), 113–140.Google Scholar
[K] Knuth, D. Big omicron, big omega and big theta. ACM SIGACT News 8 (1976), no. 2, 18–24.CrossRefGoogle Scholar
[SL] Lennox, J.C. and Stonehewer, S.E. Subnormal Subgroups of Groups (Clarendon Press 1987).Google Scholar
[LS] Lyndon, R.C. and Schupp, P.E. Combinatorial Group Theory (Springer–Verlag 2001).Google Scholar
[MKS] Magnus, W., Karrass, A. and Solitar, D. Combinatorial Group Theory (Wiley & Sons, Inc. NY-London-Sydney 1966).Google Scholar
[O] Olshanskii, A. Yu. The Geometry of Defining Relations in Groups (Nauka, Moscow, 1989) (in Russian); (English traslation by Kluwer Publications 1991).Google Scholar
[S] Sambusetti, A. Growth tightness of free and amalgamated products. Ann. Sci. Ecole Norm. Sup. 4 Series, 35 (2002), 477–488.Google Scholar
[Sa] Sawyer, S. Isotropic random walks in a tree. Z. Wahrsch. verw. Gebiete 42 (1978), no. 4, 279–292.CrossRefGoogle Scholar
[Sh] Shmel'kin, A. Wreath products and varieties of groups (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 29 (1) (1965), 149–170.Google Scholar
[W] Weiss, G. Aspects and Applications of the Random Walk (North-Holand, Amsterdam-London-NY-Tokyo 1994).Google Scholar