Minimal generating sets and the abelianization for the quasitoric braid group (Q6648723)

Let \(B_{n}=\langle \sigma_{1}, \ldots, \sigma_{n} \rangle\) be the classical braid group of \(n \geq 1\) strands and let \(\Psi\) be the surjective homomorphism defined by \(\Psi: B_{n} \rightarrow S_{n}\), \(\sigma_{i} \mapsto (i, i+1)\) if \(i<n\) and \(\sigma_{n}=(n,1)\). The pure braid group \(PB_{n}=\ker \Psi\) is the kernel of \(\Psi\). Let \(QB_{n}\) be the subset of \(B_{n}\) which consists of \(n\)-quasitoric braids, then \(QB_{n}\) is a subgroup of \(B_{n}\) and \(PB_{n} \leq QB_{n}\) ([\textit{V. O. Manturov}, Eur. J. Comb. 23, No. 2, 207--212 (2002; Zbl 0991.57007)]).\N\NThe main result in the paper under review is Theorem 1.1: (1) For \(n \geq 3\) odd, \(QB_{n}\) is generated by \((n+1)/2\) elements; (2) for \(n \geq 4\) even, \(QB_{n}\) is generated by \((n+2)/2\) elements.\N\NThe minimalities of the generating sets are obtained from a lower bound by the number of generators for the abelianization. The author also proves that in case (1) \(H_{1}(QB_{n}, \mathbb{Z}) \simeq \mathbb{Z}^{\frac{n+1}{2}} \boldsymbol{\oplus} \mathbb{Z}_{n}\) and in case (2) \(H_{1}(QB_{n}, \mathbb{Z}) \simeq \mathbb{Z}^{\frac{n}{2}} \boldsymbol{\oplus} \mathbb{Z}_{\frac{n}{2}}\).