Brunnian planar braids and simplicial groups (Q6614203)

The twin group or the planar braid group \(T_{n}\) on \(n \geq 2\) strands, is a right-angled Coxeter group generated by \(n-1\) involutions that admit only far commutativity relations. A presentation for \(T_{n}\) is\N\[\NT_{n}=\big \langle s_{1},s_{2}, \ldots s_{n-1} \; \big | \; s^{2}_{i}=1 \mbox{ and } s_{i}s_{j}=s_{j}s_{i} \mbox{ for } |i-j| \geq 2 \, \big \rangle.\N\]\NTwin groups are planar analogues of Artin braid groups and play a crucial role in the Alexander-Markov correspondence for the isotopy classes of immersed circles on the 2-sphere without triple and higher intersections. These groups admit diagrammatic representations, leading to maps obtained by the addition and deletion of strands.\N\NIn the paper under review, the authors explore Brunnian twin groups, which are subgroups of twin groups composed of twins that become trivial when any of their strands are deleted. They establish that Brunnian twin groups consisting of more than two strands are free groups. Furthermore, they provide a necessary and sufficient condition for a Brunnian doodle on the \(2\)-sphere to be the closure of a Brunnian twin. Also, they study two generalizations of Brunnian twins, namely, \(k\)-decomposable twins and Cohen twins (see [\textit{F. R. Cohen}, Contemp. Math. 188, 49--55 (1995; Zbl 0849.55015); \textit{F. R. Cohen} and \textit{J. Wu}, Q. J. Math. 62, No. 4, 891--921 (2011; Zbl 1252.55005)]) and prove some interesting structural results about these groups. Finally they investigate a simplicial structure on pure twin groups that admits a simplicial homomorphism from Milnor's construction of the simplicial \(2\)-sphere.