# On the smallest non-trivial quotients of mapping class groups

## TL;DR

This paper proves that the smallest non-trivial quotient of the mapping class group for genus at least 3 surfaces is a specific symplectic group, confirming a conjecture and extending previous representation results.

## Contribution

It establishes the minimal non-trivial quotient of the mapping class group as a symplectic group and generalizes representation results to any field.

## Key findings

- Confirmed the conjecture that the smallest non-trivial quotient is Sp_{2g}(2)
- Extended Korkmaz's results to projective representations over arbitrary fields
- Provided new insights into the structure of mapping class groups

## Abstract

We prove that the smallest non-trivial quotient of the mapping class group of a connected orientable surface of genus at least 3 without punctures is $\mathrm{Sp}_{2g}(2)$, thus confirming a conjecture of Zimmermann. In the process, we generalise Korkmaz's results on $\mathbb{C}$-linear representations of mapping class groups to projective representations over any field.