The cyclic sliding operation in Garside groups

TL;DR

This paper introduces cyclic sliding, a new operation in Garside groups that simplifies conjugacy algorithms and has better theoretical properties than previous methods like cycling and decycling.

Contribution

The paper proposes cyclic sliding as a more natural and effective alternative to cycling and decycling for conjugacy problems in Garside groups.

Findings

Cyclic sliding simplifies conjugacy algorithms.

It has advantageous theoretical properties.

Optimal rigid conjugates are obtained via iterated cyclic sliding.

Abstract

We present a new operation to be performed on elements in a Garside group, called cyclic sliding, which is introduced to replace the well known cycling and decycling operations. Cyclic sliding appears to be a more natural choice, simplifying the algorithms concerning conjugacy in Garside groups and having nicer theoretical properties. We show, in particular, that if a super summit element has conjugates which are 'rigid' (that is, which have a certain particularly simple structure), then the optimal way of obtaining such a rigid conjugate through conjugation by positive elements is given by iterated cyclic sliding.