Translator Disclaimer
15 October 2017 A product for permutation groups and topological groups
Simon M. Smith
Duke Math. J. 166(15): 2965-2999 (15 October 2017). DOI: 10.1215/00127094-2017-0022


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.


Download Citation

Simon M. Smith. "A product for permutation groups and topological groups." Duke Math. J. 166 (15) 2965 - 2999, 15 October 2017.


Received: 12 April 2017; Revised: 1 May 2017; Published: 15 October 2017
First available in Project Euclid: 9 August 2017

zbMATH: 1380.20029
MathSciNet: MR3712169
Digital Object Identifier: 10.1215/00127094-2017-0022

Primary: 22D05
Secondary: 20B07, 20E08

Rights: Copyright © 2017 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.166 • No. 15 • 15 October 2017
Back to Top