# Braid groups in handlebodies and corresponding Hecke algebras

## TL;DR

This paper investigates the structure of braid groups in handlebodies, showing their kernels form semi-direct products of free groups, and introduces a new algebra analogous to Hecke algebras associated with these braid groups.

## Contribution

It characterizes the kernel of the handlebody braid group homomorphism and introduces a new algebraic structure related to these braid groups.

## Key findings

- Kernel of the handlebody braid group homomorphism is a semi-direct product of free groups
- Introduces an algebra $H_{g,n}(q)$ analogous to the Hecke algebra
- Provides structural insights into handlebody braid groups

## Abstract

In this paper we study the kernel of the homomorphism $B_{g,n} \to B_n$ of the braid group $B_{g,n}$ in the handlebody $\mathcal{H}_g$ to the braid group $B_n$. We prove that this kernel is a semi-direct product of free groups. Also, we introduce an algebra $H_{g,n}(q)$, which is some analog of the Hecke algebra $H_n(q)$, constructed by the braid group $B_n$.