  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 $B_n$.
