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 $M⊠N$ which carries many of the permutational properties of $M$. Under mild conditions on $M$ and $N$ the group $M⊠N$ is simple.

As a permutational product, its most significant property is the following: $M⊠N$ 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, $M⊠N$ 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.