Abstract
We answer a question of Bartholdi, Siegenthaler and Zalesskii, showing that the congruence subgroup problem for branch groups is independent of the branch action on a tree. We prove that the congruence topology of a branch group is determined by the group, specifically, by its structure graph, an object first introduced by Wilson. We also give a more natural definition of this graph.
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L. Bartholdi, R. I. Grigorchuk and Z. Šunik, Branch roups, in Handbook of Algebra, Vol. 3, North-Holland, Amsterdam, 2003, pp. 989–1112.
L. Bartholdi, O. Siegenthaler and P. Zalesskii, The congruence subgroup problem for branch groups, Israel Journal of Mathematics 187 (2012), 419–450.
H. Bass, M. Lazard and J.-P. Serre, Sous-groupes d’indice fini dans SL(n,Z), Bulletin of the American Mathematical Society 70 (1964), 385–392.
K.-U. Bux, M. V. Ershov and A. S. Rapinchuk, The congruence subgroup property for Aut F2: a group-theoretic proof of Asada’s theorem, Groups, Geometry, and Dynamics 5 (2011), 327–353.
A. Garrido and J. S. Wilson, On subgroups of finite index in branch groups, Journal of Algebra 397 (2014), 32–38.
R. I. Grigorchuk, Just infinite branch groups, in New Horizons in pro-p Groups, Progress in Mathematics, Vol. 184, Birkhäuser Boston, Boston, MA, 2000, pp. 121–179.
R. I. Grigorchuk and J. S. Wilson, The uniqueness of the actions of certain branch groups on rooted trees, Geometriae Dedicata 100 (2003), 103–116.
P. D. Hardy, Aspects of abstract and profinite group theory, Ph.D. thesis, University of Birmingham, 2002.
J. L. Mennicke, Finite factor groups of the unimodular group, Annals of Mathematics 81 (1965), 31–37.
M. S. Raghunathan, The congruence subgroup problem, Indian Academy of Sciences. Proceedings. Mathematical Sciences 114 (2004), 299–308.
D. J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, Vol. 80, Springer, New York, 1996.
J. S. Wilson, Groups with every proper quotient finite, Proceedings of the Cambridge Philosophical Society 69 (1971), 373–391.
J. S. Wilson, On just infinite abstract and profinite groups, in New Horizons in prop Groups, Progress in Mathematics, Vol. 184, Birkhäuser Boston, Boston, MA, 2000, pp. 181–203.
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This research was supported by Fundación La Caixa, Spain. I would like to thank my supervisor Prof. John S. Wilson for his help in the preparation of this paper and the referee for useful comments.
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Garrido, A. On the congruence subgroup problem for branch groups. Isr. J. Math. 216, 1–13 (2016). https://doi.org/10.1007/s11856-016-1402-x
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DOI: https://doi.org/10.1007/s11856-016-1402-x