Duke Mathematical Journal

A product for permutation groups and topological groups

Simon M. Smith

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We introduce a new product for permutation groups. It takes as input two permutation groups, M and N and produces an infinite group MN which carries many of the permutational properties of M. Under mild conditions on M and N the group MN is simple.

As a permutational product, its most significant property is the following: MN is primitive if and only if M is primitive but not regular, and N is transitive. Despite this remarkable similarity with the wreath product in product action, MN and MWrN are thoroughly dissimilar.

The product provides a general way to build exotic examples of nondiscrete, simple, totally disconnected, locally compact, compactly generated topological groups from discrete groups.

We use this to obtain the first construction of uncountably many pairwise nonisomorphic simple topological groups that are totally disconnected, locally compact, compactly generated, and nondiscrete. The groups we construct all contain the same compact open subgroup. The analogous result for discrete groups was proved in 1953 by Ruth Camm.

To build the product, we describe a group U(M,N) that acts on a biregular tree T. This group has a natural universal property and is a generalization of the iconic universal group construction of Marc Burger and Shahar Mozes for locally finite regular trees.

Article information

Duke Math. J., Volume 166, Number 15 (2017), 2965-2999.

Received: 12 April 2017
Revised: 1 May 2017
First available in Project Euclid: 9 August 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22D05: General properties and structure of locally compact groups
Secondary: 20B07: General theory for infinite groups 20E08: Groups acting on trees [See also 20F65]

infinite permutation groups primitive permutation groups groups acting on trees totally disconnected locally compact groups totally compact groups locally compact simple groups


Smith, Simon M. A product for permutation groups and topological groups. Duke Math. J. 166 (2017), no. 15, 2965--2999. doi:10.1215/00127094-2017-0022. https://projecteuclid.org/euclid.dmj/1502244254

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  • [1] C. Banks, M. Elder, and G. A. Willis, Simple groups of automorphisms of trees determined by their actions on finite subtrees, J. Group Theory 18 (2015), 235–261.
  • [2] Y. Barnea, M. Ershov, and T. Weigel, Abstract commensurators of profinite groups, Trans. Amer. Math. Soc. 363 (2011), 5381–5417.
  • [3] B. Bekka, P. de la Harpe, and A. Valette, Kazhdan’s Property T, New Math. Monogr. 11, Cambridge Univ. Press, Cambridge, 2008.
  • [4] M. Burger and S. Mozes, Groups acting on trees: From local to global structure, Publ. Math. Inst. Hautes Études Sci. 92 (2000), 113–150.
  • [5] R. Camm, Simple free products, J. Lond. Math. Soc. 28 (1953), 66–76.
  • [6] P.-E. Caprace and T. De Medts, Simple locally compact groups acting on trees and their germs of automorphisms, Transformation Groups 16 (2011), 375–411.
  • [7] P.-E. Caprace, C. D. Reid, and G. A. Willis, Locally normal subgroups of totally disconnected groups, Part II, Compactly generated simple groups, Forum Math. Sigma 5 (2017), e12.
  • [8] J. D. Dixon and B. Mortimer, Permutation Groups, Grad. Texts in Math. 163, Springer, New York, 1996.
  • [9] D. M. Evans, A note on automorphism groups of countably infinite structures, Arch. Math. 49 (1987), 479–483.
  • [10] D. M. Evans, An infinite highly arc-transitive digraph, European J. Combin. 18 (1997), 281–286.
  • [11] D. G. Higman, Intersection matrices for finite permutation groups, J. Algebra 6 (1967), 22–42.
  • [12] M. W. Liebeck, C. E. Praeger, and J. Saxl, On the O’Nan–Scott theorem for finite primitive permutation groups, J. Austral. Math. Soc. 44 (1988), 389–396.
  • [13] R. G. Möller, Structure theory of totally disconnected locally compact groups via graphs and permutations, Canad. J. Math. 54 (2002), 795–827.
  • [14] R. G. Möller and J. Vonk, Normal subgroups of groups acting on trees and automorphism groups of graphs, J. Group Theory 15 (2012), 831–850.
  • [15] V. N. Obraztsov, Embedding into groups with well-described lattices of subgroups, Bull. Aust. Math. Soc. 54 (1996), 221–240.
  • [16] A. Y. Ol’shanskiĭ, Geometry of Defining Relations in Groups, Math. Appl. (Soviet Ser.) 70, Kluwer, Dordrecht, 1991.
  • [17] D. Osin, Small cancellations over relatively hyperbolic groups and embedding theorems, Ann. of Math. 172 (2010), 1–39.
  • [18] J.-P. Serre, Trees, Springer Monogr. in Math., Springer, Berlin, 2003.
  • [19] S. M. Smith, Infinite primitive directed graphs, J. Algebr. Comb. 31 (2010), 131–141.
  • [20] S. M. Smith, Subdegree growth rates of infinite primitive permutation groups, J. Lond. Math. Soc. 82 (2010), 526–548.
  • [21] J. Tits, “Sur le groupe des automorphismes d’un arbre” in Essays on Topology and Related Topics: Mémoires dédiés à Georges de Rham, Springer, Berlin/New York, 1970, 188–211.
  • [22] G. A. Willis, Compact open subgroups in simple totally disconnected groups, J. Algebra 312 (2007), 405–417.
  • [23] W. Woess, Topological groups and infinite graphs, Discrete Math. 95 (1991), 373–384.