On an action of the braid group B_{2g+2} on the free group F_{2g}

TL;DR

This paper constructs a new action of the braid group B_{2g+2} on the free group F_{2g} that extends previous work and induces a homomorphism into the symplectic group, with a detailed case study for g=2.

Contribution

It introduces a novel braid group action on free groups and characterizes the induced homomorphism into the symplectic group, including kernel determination for g=2.

Findings

Homomorphism from B_{2g+2} to Sp_{2g}(Z) is established.

For g=2, the homomorphism is surjective.

Kernel of the homomorphism for g=2 is explicitly determined.

Abstract

We construct an action of the braid group B_{2g+2} on the free group F_{2g} extending an action of B_4 on F_2 introduced earlier by Reutenauer and the author. Our action induces a homomorphism from B_{2g+2} into the symplectic modular group Sp_{2g}(Z). In the special case g=2 we show that the latter homomorphism is surjective and determine its kernel, thus obtaining a braid-like presentation of Sp_4(Z).