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

By J. Birman, Volker Gebhardt, Juan González-Meneses. 2006

52 citations recorded by the source corpus.

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 2 USS.X/, called 'partial cycling' and 'partial twisted decycling', and prove that if Y; Z 2 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 graphX determined by USS.X/, and show thatX 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3a2955e34af24101a656524c110556606016d4b0

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In Geometry and Topology

1. Juan González-Meneses
2. Volker Gebhardt
3. J. Birman
4. O. Plamenevskaya

In Mathematical Physics

1. Juan González-Meneses
2. Volker Gebhardt
3. J. Birman
4. O. Plamenevskaya

Citations in the corpus

• Conjugacy in Garside groups I: cyclings, powers and rigidity (2006, background): “By the above result, m 9 ], and we will possibly be able to bound the size of USS.X m/. Moreover, since pseudo-Anosov braids have unique roots, if one solves the CDP/CSP for X m and Y m, finding some conjugating element Z, then Z is also a conjugating element”
• Conjugacy in Garside groups III: Periodic braids (2006, unknown): “Although we refer to [ 6 ] for a detailed description of Garside structures, we recall here that such a structure in a group G is given by a lattice order on its elements, together with a distinguished element of G, called the Garside element, which is usually denoted by Δ. This partial order and th”