On the normalizers of some classes of subgroups in braid groups. (Q1889501)

The authors describe the normalizers of some classes of subgroups in braid groups. In particular, they consider the subgroup \(\Pi=\langle\sigma^4_1,\sigma^4_2\rangle\times\langle\sigma_4\sigma_3 \sigma_2\sigma^2_1\sigma_2\sigma_3\sigma_4\rangle\subset R_5\) (\(R_5\) denotes the subgroup of colored 5-braids). It is known that \(\Pi\) is the direct product of free groups of rank 2. The authors find the normalizer \(N_{R_5}(\Pi)\) in \(R_5\). They also show that if \(H=H_1\times H_2\) is a finitely generated subgroup of \(\Pi\), \(H_1\) and \(H_2\) are not cyclic, then \(N_{K_5}(H)\subset N_{R_5}(\Pi)\).