Minimal generating sets of groups of Kim-Manturov

TL;DR

This paper investigates groups introduced by Kim and Manturov related to surface triangulations and configurations, providing a minimal generating set and analyzing their abelianization to better understand their structure.

Contribution

It offers the first explicit minimal generating set for these groups and explores related groups to shed light on their algebraic properties.

Findings

Established a minimal generating set for Kim-Manturov groups.

Determined the abelianization of these groups.

Introduced related groups to analyze structural properties.

Abstract

We consider a series of groups defined by Kim and Manturov. These groups have their background in triangulations of a surface and configurations of points, lines or circles on the surface. They are expected to have relationships to many geometric objects. In this paper, we give a minimal generating set of the group and determine the abelianization. We also introduce some related groups which might be helpful to understand the structure of the original groups.