Artin Relations in the Mapping Class Group
Jamil Mortada

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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Algebraic Geometry and Number Theory
