A product for permutation groups and topological groups

TL;DR

This paper introduces a new product for permutation groups that generates simple, totally disconnected, locally compact topological groups, solving a longstanding open problem by constructing uncountably many such non-discrete groups.

Contribution

It defines a novel permutation group product, explores its properties, and uses it to construct uncountably many non-discrete simple topological groups with specific features.

Findings

The product preserves many permutational properties of the original groups.

Under mild conditions, the product group is simple.

Constructs uncountably many non-isomorphic simple 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 [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.