A product for permutation groups and topological groups

Smith, Simon (2017) A product for permutation groups and topological groups. Duke Mathematical Journal, 166 (15). pp. 2965-2999. ISSN 0012-7094

Full content URL: https://doi.org/10.1215/00127094-2017-0022

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A product for permutation groups and topological groups
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Abstract

We introduce a new product for permutation groups. It takes as input two permutation groups, M and N, and produces an infinite group M [X] N which carries many of the permutational properties of M. Under mild conditions on M and N the group M [X] N is simple. As a permutational product, its most significant property is the following: M [X] N 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, M [X] N and M Wr N are thoroughly dissimilar. The product provides a general way to build exotic examples of non-discrete, simple, totally disconnected, locally compact, compactly generated topological groups from discrete groups. We use this to solve a well-known open problem from topological group theory, by obtaining the first construction of uncountably many pairwise non-isomorphic simple topological groups that are totally disconnected, locally compact, compactly generated and non-discrete. The groups we construct all contain the same compact open subgroup. To build the product, we describe a group U(M,N) that acts on an edge-transitive biregular tree T. This group has a natural universal property and is analogous to the iconic universal group construction of M. Burger and S. Mozes for locally finite regular trees.

Keywords:Universal groups, Groups acting on trees, Infinite permutation groups, Primitive permutation groups, Locally compact groups
Subjects:G Mathematical and Computer Sciences > G110 Pure Mathematics
Divisions:College of Science > School of Mathematics and Physics
ID Code:27788
Deposited On:04 Jul 2017 11:19

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