An exotic group as limit of finite special linear groups

TL;DR

This paper studies a Polish group formed as the limit of finite special linear groups, revealing its topological simplicity, extreme amenability, and lack of non-trivial unitary representations, contributing to the understanding of infinite-dimensional group limits.

Contribution

It introduces a new example of a Polish group as a limit of finite groups with unique topological and representation-theoretic properties.

Findings

The group is topologically simple modulo its center.

It is extremely amenable.

It admits no non-trivial strongly continuous unitary representations.

Abstract

We consider the Polish group obtained as the rank-completion of an inductive limit of finite special linear groups. This Polish group is topologically simple modulo its center, it is extremely amenable and has no non-trivial strongly continuous unitary representation on a Hilbert space.