On the normalizers of some classes of subgroups in braid groups. (English. Russian original) Zbl 1078.20037

Bezverkhnij, V. N.; Dobrynina, I. V.

Math. Notes 74, No. 1, 18-29 (2003); translation from Mat. Zametki 74, No. 1, 19-31 (2003).

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 \(N_{K_5}(H)\subset N_{R_5}(\Pi)\).

Reviewer: Andrei M. Akimenkov (Krasnogorsk)

MSC:

20F36	Braid groups; Artin groups
20E07	Subgroup theorems; subgroup growth

Keywords:

braid groups; subgroups of finite index; normalizers; finitely generated subgroups