# Profinite completions of Burnside-type quotients of surface groups

## TL;DR

This paper demonstrates that certain surface group quotients have complex profinite completions that are not virtually prosolvable and constructs numerous finite simple characteristic quotients using quantum representations.

## Contribution

It introduces a novel approach using quantum representations to analyze profinite completions and constructs infinitely many finite simple characteristic quotients of surface groups.

## Key findings

- Profinite completions of Burnside-type surface group quotients are not virtually prosolvable.
- Constructed infinitely many finite simple characteristic quotients of surface groups.
- Quantum representations are effective tools for studying surface group quotients.

## Abstract

Using quantum representations of mapping class groups we prove that profinite completions of Burnside-type surface group quotients are not virtually prosolvable, in general. Further, we construct infinitely many finite simple characteristic quotients of surface groups.