Conjugacy in braid groups and the LKB representation, and Bessis-Garside groups of rank 3

• Main Author: Coles, Ben
• Published: University of Warwick 2017
• Subjects: 512; QA Mathematics
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.720511

In the first part of this thesis, we give a survey of the conjugacy problem in the braid group, describing the solution provided by Garside theory, and outlining the progress that has been made towards a polynomial time solution in recent years using refinements of Garside's solution, and the Thurston-Nielsen classification of braids, which reduces the problem to the case of pseudo-Anosov braids. Using the faithful Lawrence-Krammer-Bigelow representation of the braid groups, we consider how the eigenspaces of pseudo-Anosov braids can under certain conditions yield invariants of their conjugacy class and thus lead us towards a polynomial time solution of the conjugacy problem. In the second part we introduce Bessis-Garside groups, a generalisation of the methods used by Bessis in his papers on dual braid monoids. We consider the groups given by taking the quotient of the free group by the orbits of its generators under the action of some subgroup of the braid group, and find that in many cases this construction can give us a group with a Garside structure. By means of introduction we review the simple rank 2 case, and summarise examples of such groups already known to admit Garside structures, in particular due to the work of Digne. We then go on to give all those of such groups which can be found as quotients of affine and spherical Artin groups of rank 3. We show that all such groups may be given a cycle presentation, or equivalently may be given as labelled-oriented-graph presented groups, and give conditions on such presentations that are equivalent to the group admitting a `dual' Garside structure. Restricting by the cycle lengths occurring in such presentations we give all Bessis-Garside groups of rank 3 which have all cycles length at most 4, and discuss the case of Bessis-Garside groups with uniform cycle length.