Artin Relations in the Mapping Class Group

TL;DR

This paper demonstrates the existence of elements in the mapping class group satisfying Artin relations of various lengths, expanding understanding of algebraic relations in these groups.

Contribution

It introduces new methods to find Artin relations of arbitrary length in the mapping class group, including direct computations and alternative approaches using Artin group theory.

Findings

Existence of elements satisfying Artin relations of all even lengths greater than 6

Existence of elements satisfying Artin relations of all odd lengths greater than 1

Two new methods for constructing Artin relations of various lengths in the mapping class group

Abstract

For every integer l bigger than one, we find elements x and y in the mapping class group of an appropriate orientable surface S, satisfying the Artin relation of length l. That is, xyx... = yxy..., where each side of the equality contains l terms. By direct computations, we first find elements x and y in Mod(S) satisfying Artin relations of every even length bigger than 6, and every odd length bigger than 1. Then using the theory of Artin groups, we give two more alternative ways for finding Artin relations in Mod(S). The first provides Artin relations of every length greater than 3, while the second produces Artin relations of every even length greater than 4.