# Two special subgroups of the universal sofic group

## TL;DR

This paper introduces a specific subgroup of the universal sofic group, characterized as the normalizer of a separable abelian subalgebra, and demonstrates its universality for conjugating all sofic representations.

## Contribution

It defines a new subgroup of the universal sofic group and proves its universality for conjugating all sofic representations.

## Key findings

- The subgroup is obtained as an extension by automorphisms on a standard probability space.
- Each sofic representation can be conjugated inside this subgroup.
- The subgroup is the normalizer of a separable abelian subalgebra.

## Abstract

We define a subgroup of the universal sofic group, obtained as the normaliser of a separable abelian subalgebra. This subgroup can be obtained as an extension by the group of automorphisms on a standard probability space. We show that each sofic representation can be conjugated inside this subgroup.