# Conjugacy in Garside groups II: structure of the ultra summit set

### Joan S. Birman

Barnard College, Columbia University, New York, United States### Volker Gebhardt

University of Western Sydney, Australia### Juan González-Meneses

Universidad de Sevilla, Spain

## Abstract

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, 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 Y ∈ USS(X), called ‘partial cycling’ and ‘partial twisted decycling’, and prove that if Y, Z ∈ 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 whose existence follows from the convexity property of ultra summit sets.

Using partial cycling and partial twisted decycling, we investigate the structure of a directed graph ΓX determined by USS(X), and show that ΓX can be decomposed into ‘black’ and ‘grey’ subgraphs. There are applications relating to the authors’ program for finding a polynomial solution to the CDP/CSP in the case of braids, which is outlined in the first paper of this series. 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 of the established algorithm using ultra summit sets. There are also applications to the theory of reductive groups.

## Cite this article

Joan S. Birman, Volker Gebhardt, Juan González-Meneses, Conjugacy in Garside groups II: structure of the ultra summit set. Groups Geom. Dyn. 2 (2008), no. 1, pp. 13–61

DOI 10.4171/GGD/30