Dimension and randomness in groups acting on rooted trees

Abért, Miklós; Virág, Bálint

doi:10.1090/S0894-0347-04-00467-9

Authors: Miklós Abért and Bálint Virág
Journal: J. Amer. Math. Soc. 18 (2005), 157-192
MSC (2000): Primary 20E08, 60J80, 37C20; Secondary 20F69, 20E18, 20B27, 28A78
DOI: https://doi.org/10.1090/S0894-0347-04-00467-9
Published electronically: September 2, 2004
MathSciNet review: 2114819
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We explore the structure of the $p$-adic automorphism group $Y$ of the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turán. We initiate the study of a general dimension theory of groups acting on rooted trees. We describe the relationship between dimension and other properties of groups such as solvability, existence of dense free subgroups and the normal subgroup structure. We show that subgroups of $W$ generated by three random elements are full dimensional and that there exist finitely generated subgroups of arbitrary dimension. Specifically, our results solve an open problem of Shalev and answer a question of Sidki.

A. G. Abercrombie, Subgroups and subrings of profinite rings, Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 2, 209–222. MR 1281541, DOI https://doi.org/10.1017/S0305004100072522

[In prep]av02 M. Abért and B. Virág. Polynomials of $p$-trees. In preparation.

Krishna B. Athreya and Peter E. Ney, Branching processes, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 196. MR 0373040
Y. Barnea and M. Larsen, A non-abelian free pro-$p$ group is not linear over a local field, J. Algebra 214 (1999), no. 1, 338–341. MR 1684856, DOI https://doi.org/10.1006/jabr.1998.7682
Yiftach Barnea and Aner Shalev, Hausdorff dimension, pro-$p$ groups, and Kac-Moody algebras, Trans. Amer. Math. Soc. 349 (1997), no. 12, 5073–5091. MR 1422889, DOI https://doi.org/10.1090/S0002-9947-97-01918-1
Y. Barnea, A. Shalev, and E. I. Zelmanov, Graded subalgebras of affine Kac-Moody algebras, Israel J. Math. 104 (1998), 321–334. MR 1622319, DOI https://doi.org/10.1007/BF02897069
Laurent Bartholdi and Rostislav I. Grigorchuk, Lie methods in growth of groups and groups of finite width, Computational and geometric aspects of modern algebra (Edinburgh, 1998) London Math. Soc. Lecture Note Ser., vol. 275, Cambridge Univ. Press, Cambridge, 2000, pp. 1–27. MR 1776763, DOI https://doi.org/10.1017/CBO9780511600609.002
Meenaxi Bhattacharjee, The ubiquity of free subgroups in certain inverse limits of groups, J. Algebra 172 (1995), no. 1, 134–146. MR 1320624, DOI https://doi.org/10.1006/jabr.1995.1053
J. D. Biggins, Chernoff’s theorem in the branching random walk, J. Appl. Probability 14 (1977), no. 3, 630–636. MR 464415, DOI https://doi.org/10.1017/s0021900200025900
Nigel Boston, $p$-adic Galois representations and pro-$p$ Galois groups, New horizons in pro-$p$ groups, Progr. Math., vol. 184, Birkhäuser Boston, Boston, MA, 2000, pp. 329–348. MR 1765126
F. M. Dekking and B. Host, Limit distributions for minimal displacement of branching random walks, Probab. Theory Related Fields 90 (1991), no. 3, 403–426. MR 1133373, DOI https://doi.org/10.1007/BF01193752
John D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969), 199–205. MR 251758, DOI https://doi.org/10.1007/BF01110210
J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$-groups, London Mathematical Society Lecture Note Series, vol. 157, Cambridge University Press, Cambridge, 1991. MR 1152800
Marcus du Sautoy, Dan Segal, and Aner Shalev (eds.), New horizons in pro-$p$ groups, Progress in Mathematics, vol. 184, Birkhäuser Boston, Inc., Boston, MA, 2000. MR 1765115
P. Erdős and P. Turán, On some problems of a statistical group-theory. I, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 4 (1965), 175–186 (1965). MR 184994, DOI https://doi.org/10.1007/BF00536750
Steven N. Evans, Eigenvalues of random wreath products, Electron. J. Probab. 7 (2002), no. 9, 15. MR 1902842, DOI https://doi.org/10.1214/EJP.v7-108
Piotr W. Gawron, Volodymyr V. Nekrashevych, and Vitaly I. Sushchansky, Conjugation in tree automorphism groups, Internat. J. Algebra Comput. 11 (2001), no. 5, 529–547. MR 1869230, DOI https://doi.org/10.1142/S021819670100070X
R. I. Grigorchuk, Just infinite branch groups, New horizons in pro-$p$ groups, Progr. Math., vol. 184, Birkhäuser Boston, Boston, MA, 2000, pp. 121–179. MR 1765119
R. I. Grigorchuk, W. N. Herfort, and P. A. Zalesskii, The profinite completion of certain torsion $p$-groups, Algebra (Moscow, 1998) de Gruyter, Berlin, 2000, pp. 113–123. MR 1754662
R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskiĭ, Automata, dynamical systems, and groups, Tr. Mat. Inst. Steklova 231 (2000), no. Din. Sist., Avtom. i Beskon. Gruppy, 134–214 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 4(231) (2000), 128–203. MR 1841755
J. M. Hammersley, Postulates for subadditive processes, Ann. Probability 2 (1974), 652–680. MR 370721, DOI https://doi.org/10.1214/aop/1176996611
L. G. Kovács and M. F. Newman, Generating transitive permutation groups, Quart. J. Math. Oxford Ser. (2) 39 (1988), no. 155, 361–372. MR 957277, DOI https://doi.org/10.1093/qmath/39.3.361
M. F. Newman, Csaba Schneider, and Aner Shalev, The entropy of graded algebras, J. Algebra 223 (2000), no. 1, 85–100. MR 1738253, DOI https://doi.org/10.1006/jabr.1999.8053
P. P. Pálfy and M. Szalay, On a problem of P. Turán concerning Sylow subgroups, Studies in pure mathematics, Birkhäuser, Basel, 1983, pp. 531–542. MR 820249

[2001]puchta2 J.-C. Puchta. Unpublished.

[2003]puchta J.-C. Puchta. The order of elements of $p$-sylow subgroups of the symmetric group. Preprint.

László Pyber and Aner Shalev, Residual properties of groups and probabilistic methods, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 4, 275–278 (English, with English and French summaries). MR 1854764, DOI https://doi.org/10.1016/S0764-4442%2801%2902044-4
Aner Shalev, Probabilistic group theory, Groups St. Andrews 1997 in Bath, II, London Math. Soc. Lecture Note Ser., vol. 261, Cambridge Univ. Press, Cambridge, 1999, pp. 648–678. MR 1676661, DOI https://doi.org/10.1017/CBO9780511666148.028
Aner Shalev, Lie methods in the theory of pro-$p$ groups, New horizons in pro-$p$ groups, Progr. Math., vol. 184, Birkhäuser Boston, Boston, MA, 2000, pp. 1–54. MR 1765116
Said Sidki, Automorphisms of one-rooted trees: growth, circuit structure, and acyclicity, J. Math. Sci. (New York) 100 (2000), no. 1, 1925–1943. Algebra, 12. MR 1774362, DOI https://doi.org/10.1007/BF02677504

[2001]sidki01 S. Sidki. Oxford University Algebra Seminar Talk, October 30, 2001

G. A. Soifer and T. N. Venkataramana, Finitely generated profinitely dense free groups in higher rank semi-simple groups, Transform. Groups 5 (2000), no. 1, 93–100. MR 1745714, DOI https://doi.org/10.1007/BF01237181

[1984]sushchansky V. Sushchansky. Isometry groups of the Baire $p$-spaces. Dop. AN URSR, 8:27–30. (in Ukranian).

J. S. Wilson, Groups with every proper quotient finite, Proc. Cambridge Philos. Soc. 69 (1971), 373–391. MR 274575, DOI https://doi.org/10.1017/s0305004100046818
John S. Wilson, On just infinite abstract and profinite groups, New horizons in pro-$p$ groups, Progr. Math., vol. 184, Birkhäuser Boston, Boston, MA, 2000, pp. 181–203. MR 1765120
P. A. Zalesskii, Profinite groups admitting just infinite quotients, Monatsh. Math. 135 (2002), no. 2, 167–171. MR 1894095, DOI https://doi.org/10.1007/s006050200013
E. Zelmanov, On groups satisfying the Golod-Shafarevich condition, New horizons in pro-$p$ groups, Progr. Math., vol. 184, Birkhäuser Boston, Boston, MA, 2000, pp. 223–232. MR 1765122