Conjugacy invariants and rigidity in Garside groups: a uniformity phenomenon

Abstract

Consider an element~$x$ of a Garside group which is rigid in the sense of Garside-theory. Let $SC(x)$ be the set of rigid conjugates of~$x$ -- this is a well-known characteristic subset of the conjugacy class of~$x$. We present computational evidence that the sequence $( |SC(x^n)| )_{n\in\mathbb N}$ is not only bounded, but in fact periodic, and that there is a bound on the length of the period which depends only on the underlying group and its Garside structure. We prove this result in the special case of the circular Garside groups, including the 2-generator Artin groups with their classical and dual structures (where we prove that the sequence is always constant), and in the case of the dual $4$-strand braid group.