On braid groups in handlebodies (Q1288229)

Let \(B_{n+g}\) be the braid group with \(n+g\) strands and let \(B_n^g\) be its subgroup consisting of all braids with the first \(g\) strands unbraided. The elements of \(B_n^g\) can be naturally mapped in a handlebody of genus \(g\). Let \(\sigma_1,\dots,\sigma_{g+n-1}\) be the standard generators of \(B_{n+g}\) and let \(\tau_k=\sigma_g\sigma_{g-1}\cdots\sigma_{k+1}\sigma_k^2\sigma_{k+1}^{-1}\cdots\sigma_{g-1}^{-1}\sigma_g^{-1}\), \(k=1,2,\dots,g\). Theorem. The group \(B_n^g\) is generated by \(\sigma_{g+1},\dots,\sigma_{g+n-1},\tau_1,\dots,\tau_g\) and has the following defining relations: \[ \begin{alignedat}{2} \sigma_i\sigma_j&=\sigma_j\sigma_i,\quad | i-j|>1,&\quad \sigma_i\sigma_{i+1}\sigma_i&=\sigma_{i+1}\sigma_i\sigma_{i+1},\\ \tau_k\sigma_i&=\sigma_i\tau_k, \quad k\geq 1,\;i\geq 2,&\quad \tau_k\sigma_1\tau_k\sigma_1&=\sigma_1\tau_k\sigma_1\tau_k,\quad k=1,2,\dots,g,\end{alignedat} \] \[ \tau_k\sigma_1^{-1}\tau_{k+l}\sigma_1=\sigma_1^{-1}\tau_{k+l}\sigma_1\tau_k,\quad k=1,2,\dots,g-1;\;l=1,2,\dots,g-k. \] These defining relations were indicated without any proof by \textit{A. B. Sossinskij} [in Quantum groups, Proc. Workshops, Euler Int. Math. Inst. Leningrad/USSR 1990, Lect. Notes Math. 1510, 354-362 (1992; Zbl 0765.57009)].