Finitely generated subgroups of branch groups and subdirect products of just infinite groups

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Abstract

The aim of this paper is to describe the structure of finitely generated subgroups of a familyof branch groups containing the first Grigorchuk group and the Gupta–Sidki $3$-group. We thenuse this to show that all the groups in this family are subgroup separable (LERF).These results are obtained as a corollary of a more general structural statement on subdirectproducts of just infinite groups.

About the authors

Rostislav Ivanovich Grigorchuk

Department of Mathematics, Texas A&M University

Email: grigorch@math.tamu.edu

Paul-Henry Leemann

Université de Neuchâtel

Tat'yana Valer'evna Nagnibeda

University of Geneva; Saint Petersburg State University

Email: tatiana.smirnova-nagnibeda@unige.ch

References

  1. R. I. Grigorchuk, W. N. Herfort, P. A. Zalesskii, “The profinite completion of certain torsion $p$-groups”, Algebra, Moscow, 1998, de Gruyter, Berlin, 2000, 113–123
  2. Р. И. Григорчук, “Ветвящиеся группы”, Матем. заметки, 67:6 (2000), 852–858
  3. R. I. Grigorchuk, “Just infinite branch groups”, New horizons in pro-$p$ groups, Progr. Math., 184, Birkhäuser Boston, Boston, MA, 2000, 121–179
  4. V. Nekrashevych, Self-similar groups, Math. Surveys Monogr., 117, Amer. Math. Soc., Providence, RI, 2005, xii+231 pp.
  5. Р. И. Григорчук, В. В. Некрашевич, И. И. Сущанский, “Автоматы, динамические системы и группы”, Динамические системы, автоматы и бесконечные группы, Сборник статей, Труды МИАН, 231, Наука, МАИК «Наука/Интерпериодика», М., 2000, 134–214
  6. Р. И. Григорчук, “К проблеме Милнора о групповом росте”, Докл. АН СССР, 271:1 (1983), 30–33
  7. P. de la Harpe, Topics in geometric group theory, Chicago Lectures in Math., Univ. of Chicago Press, Chicago, IL, 2000, vi+310 pp.
  8. R. Grigorchuk, “Solved and unsolved problems around one group”, Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math., 248, Birkhäuser, Basel, 2005, 117–218
  9. T. Ceccherini-Silberstein, F. Scarabotti, F. Tolli, “The top of the lattice of normal subgroups of the Grigorchuk group”, J. Algebra, 246:1 (2001), 292–310
  10. P.-H. Leemann, On subgroups and Schreier graphs of finitely generated groups, PhD thesis, № 4973, Univ. Genève, Genève, 2016, xii+155 pp.
  11. Е. Л. Первова, “Всюду плотные подгруппы одной группы автоморфизмов дерева”, Динамические системы, автоматы и бесконечные группы, Сборник статей, Труды МИАН, 231, Наука, МАИК «Наука/Интерпериодика», М., 2000, 356–367
  12. P.-H. Leemann, Weakly maximal subgroups of branch groups
  13. R. Grigorchuk, T. Nagnibeda, “On subgroup structure of a 3-generated 2-group of intermediate growth”, in rep. No. 28/2008 ‘Profinite and asymptotic group theory’ (Oberwolfach, 2008), Oberwolfach Rep., 5 (2008), 1568–1571
  14. R. I. Grigorchuk, J. S. Wilson, “A structural property concerning abstract commensurability of subgroups”, J. London Math. Soc. (2), 68:3 (2003), 671–682
  15. A. Garrido, “Abstract commensurability and the Gupta–Sidki group”, Groups Geom. Dyn., 10:2 (2016), 523–543
  16. L. Bartholdi, R. I. Grigorchuk, “On the spectrum of Hecke type operators related to some fractal groups”, Динамические системы, автоматы и бесконечные группы, Сборник статей, Труды МИАН, 231, Наука, МАИК «Наука/Интерпериодика», М., 2000, 5–45
  17. Kh. Bou-Rabee, P.-H. Leemann, T. Nagnibeda, “Weakly maximal subgroups in regular branch groups”, J. Algebra, 455 (2016), 347–357
  18. R. Skipper, Ph. Wesolek, “On the Cantor–Bendixson rank of the Grigorchuk group and the Gupta–Sidki 3 group”, J. Algebra, 555 (2020), 386–405
  19. Р. И. Григорчук, “Степени роста конечно порожденных групп и теория инвариантных средних”, Изв. АН СССР. Сер. матем., 48:5 (1984), 939–985
  20. N. Gupta, S. Sidki, “On the Burnside problem for periodic groups”, Math. Z., 182:3 (1983), 385–388
  21. N. Gupta, S. Sidki, “Some infinite $p$-groups”, Алгебра и логика, 22:5 (1983), 584–589
  22. L. Bartholdi, R. I. Grigorchuk, “On parabolic subgroups and Hecke algebras of some fractal groups”, Serdica Math. J., 28:1 (2002), 47–90
  23. D. Francoeur, P.-H. Leemann, Subgroup induction property for branch groups
  24. M. Hall, Jr., “Coset representations in free groups”, Trans. Amer. Math. Soc., 67:2 (1949), 421–432
  25. P. Scott, “Subgroups of surface groups are almost geometric”, J. London Math. Soc. (2), 17:3 (1978), 555–565
  26. A. M. Brunner, R. G. Burns, D. Solitar, “The subgroup separability of free products of two free groups with cyclic amalgamation”, Contributions to group theory, Contemp. Math., 33, Amer. Math. Soc., Providence, RI, 1984, 90–115
  27. H. Wilton, “Hall's theorem for limit groups”, Geom. Funct. Anal., 18:1 (2008), 271–303
  28. L. Ribes, P. A. Zalesskii, “On the profinite topology on a free group”, Bull. London Math. Soc., 25:1 (1993), 37–43
  29. D. F. Holt, W. Plesken, Perfect groups, With an appendix by W. Hanrath, Oxford Math. Monogr., The Clarendon Press, Oxford Univ. Press, New York, 1989, xii+364 pp.
  30. D. McCarthy, “Infinite groups whose proper quotient groups are finite. I”, Comm. Pure Appl. Math., 21:6 (1968), 545–562
  31. H. Fitting, “Beiträge zur theorie der gruppen endlicher ordnung”, Jahresber. Dtsch. Math.-Ver., 48 (1938), 77–141
  32. L. Bartholdi, R. I. Grigorchuk, Z. Šunik, “Branch groups”, Handbook of algebra, v. 3, Elsevier/North-Holland, Amsterdam, 2003, 989–1112
  33. E. Goursat, “Sur les substitutions orthogonales et les divisions regulières de l'espace”, Ann. Sci. Ecole Norm. Sup. (3), 6 (1889), 9–102
  34. L. Bartholdi, O. Siegenthaler, P. Zalesskii, “The congruence subgroup problem for branch groups”, Israel J. Math., 187 (2012), 419–450

Copyright (c) 2021 Grigorchuk R.I., Leemann P., Nagnibeda T.V.

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