# Overcommuting pairs in groups and 3-manifolds bounding them

## TL;DR

This paper introduces the concepts of overcommutation and overcommutation length in groups, exploring their connection to 3-manifold fundamental groups and Heegaard genus, with examples and bounds in SL_2.

## Contribution

It defines overcommutation and overcommutation length, linking these to 3-manifold groups and providing bounds in SL_2 related to the Steinberg relation.

## Key findings

- Overcommutation length bounds in SL_2 are established.
- Examples include affine group translations.
- Overcommutation relates to 3-manifold Heegaard genus.

## Abstract

We introduce the notions of overcommutation and overcommutation length in groups, and show that these concepts are closely related to representations of the fundamental groups of 3-manifold and their Heegaard genus. We give many examples including translations in the affine group of the line and provide upper bounds for the overcommutation length in SL_2, related to the Steinberg relation.