Presentations of mapping class groups and an application to cluster algebras from surfaces

TL;DR

This paper provides new presentations of mapping class groups of marked surfaces stabilizing boundaries and applies these to determine the automorphism groups of cluster algebras from such surfaces, including special cases like the 4-punctured sphere.

Contribution

It introduces boundary-stabilizing presentations of mapping class groups and applies them to explicitly describe cluster automorphism groups for a broader class of surfaces.

Findings

Presented boundary-stabilizing mapping class groups for any genus.

Derived cluster automorphism groups for feasible surfaces.

Characterized automorphism group for the 4-punctured sphere.

Abstract

In this paper, we give presentations of the mapping class groups of marked surfaces stabilizing boundaries for any genus. Note that in the existing works, the mapping class groups of marked surfaces were the isotopy classes of homeomorphisms fixing boundaries pointwise. The condition for stabilizing boundaries of mapping class groups makes the requirement for mapping class groups to fix boundaries pointwise to be unnecessary. As an application of presentations of the mapping class groups of marked surfaces stabilizing boundaries, we obtain the presentation of the cluster automorphism group of a cluster algebra from a feasible surface $(S,M) $. Lastly, for the case (1) 4-punctured sphere, the cluster automorphism group of a cluster algebra from the surface is characterized. Since cluster automorphism groups of cluster algebras from those surfaces were given in \cite{ASS} in the cases (2) the once-punctured 4-gon and (3) the twice-punctured digon, we indeed give presentations of cluster automorphism groups of cluster algebras from surfaces which are not feasible.