Symplectic structures on right-angled Artin groups: between the mapping class group and the symplectic group

TL;DR

This paper introduces a new family of groups called mapping class groups over graphs, which generalize classical groups like the mapping class group and symplectic group, and proves their finite generation.

Contribution

It defines these novel groups based on labeled graphs and establishes their finite generation, extending known results to a broader class of algebraic structures.

Findings

The groups are finitely generated.

Generalization of Magnus's theorem.

Includes classical groups as special cases.

Abstract

We define a family of groups that include the mapping class group of a genus g surface with one boundary component and the integral symplectic group Sp(2g,Z). We then prove that these groups are finitely generated. These groups, which we call mapping class groups over graphs, are indexed over labeled simplicial graphs with 2g vertices. The mapping class group over the graph Gamma is defined to be a subgroup of the automorphism group of the right-angled Artin group A_Gamma of Gamma. We also prove that the kernel of the map Aut A_Gamma to Aut H_1(A_Gamma) is finitely generated, generalizing a theorem of Magnus.