Conjugacy in Garside groups II: Structure of the ultra summit set

Joan S. Birman, Volker Gebhardt, Juan Gonzalez-Meneses

Published 2006-06-26Version 1

This paper is the second in a series in which the authors study the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in Garside groups. The ultra summit set USS(X) of an element X in a Garside group G is a finite set of elements in G, introduced by the second author, which is a complete invariant of the conjugacy class of X in G. A fundamental question, if one wishes to find bounds on the size of USS(X), is to understand its structure. In this paper we introduce two new operations on elements of USS(X), called `partial cycling' and `partial twisted decycling', and prove that if Y and Z belong to USS(X), then Y and Z are related by sequences of partial cyclings and partial twisted decyclings. These operations are a concrete way to understand the minimal simple elements which result from the convexity theorem in the mentioned paper by the second author. Using partial cycling and partial twisted decycling, we investigate the structure of a directed graph \Gamma_X which is determined by USS(X), and show that \Gamma_X can be decomposed into `black' and `grey' subgraphs. There are applications which relate to the program, outlined in the first paper in this series, for finding a polynomial solution to the CDP/CSP in the case of braids. A different application is to give a new algorithm for solving the CDP/CSP in Garside groups which is faster than all other known algorithms, even though its theoretical complexity is the same as that given by the second author. There are also applications to the theory of reductive groups.