Conjugacy in Garside groups I: cyclings, powers and rigidity

Abstract

In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where ‘rigid’ means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown that, given $X$ in a Garside group, if some power $X^m$ is conjugate to a rigid element, then $m$ can be bounded above by $\Vert \Delta {\Vert }^3$. In the particular case of braid groups $\{B_n;n\in N\}$, this implies that a pseudo-Anosov braid has a small power whose ultra summit set consists of rigid elements. This solves one of the problems in the way of a polynomial solution to the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in braid groups. In addition to proving the rigidity theorem, it will be shown how this paper fits into the authors’ program for finding a polynomial algorithm to the CDP/CSP, and what remains to be done.