## Duke Mathematical Journal

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

### A product for permutation groups and topological groups

#### Abstract

We introduce a new product for permutation groups. It takes as input two permutation groups, $M$ and $N$ and produces an infinite group $M\u22a0N$ which carries many of the permutational properties of $M$. Under mild conditions on $M$ and $N$ the group $M\u22a0N$ is simple.

As a permutational product, its most significant property is the following: $M\u22a0N$ 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, $M\u22a0N$ 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 $\mathcal{U}(M,N)$ 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

**Source**

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

**Dates**

Received: 12 April 2017

Revised: 1 May 2017

First available in Project Euclid: 9 August 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.dmj/1502244254

**Digital Object Identifier**

doi:10.1215/00127094-2017-0022

**Mathematical Reviews number (MathSciNet)**

MR3712169

**Zentralblatt MATH identifier**

1380.20029

**Subjects**

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]

**Keywords**

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

#### Citation

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