A family of pseudo-Anosov braids with large conjugacy invariant sets. (Q2840286)

Conjugacy invariant sets are relevant to the solution of the conjugacy problem in braid groups. In fact, all the known algorithms for checking whether two braids are conjugate or not, are based on the enumeration of such an invariant set for one of them.NEWLINENEWLINE Here, the authors show that in the case of pseudo-Anosov braids (essentially the only non-trivial case), the size of the smallest conjugacy invariant set of a braid \(\beta\) can grow exponentially in the index of \(\beta\), even if the canonical length of \(\beta\) is kept bounded. This result has the obvious implication on the complexity of the above-mentioned algorithms and improves an analogous result by \textit{A. V. Prasolov} [Topology Appl. 160, No. 14, 1918-1956 (2013; Zbl 1287.55003)], where no restriction on the length \(\beta\) was considered.NEWLINENEWLINE Namely, the main theorem of the paper is the following: for any \(n\geq 14\) and \(k\geq 2\) there exists a family \(\mathcal R_{k,n}\) of rigid pseudo-Anosov braids of index \(n\) and canonical length \(k\), whose minimal conjugacy invariant set consists of \(k\cdot 2^{\lfloor(n-2)/2\rfloor-3}\) rigid pseudo-Anosov conjugates.