On the Simultaneous Conjugacy Problem in Garside groups

We solve the simultaneous conjugacy problem in Garside groups by means of an effectively computable invariant. This invariant generalizes the one-dimensional notion of super summit set of a conjugacy class. One key ingredient of our solution is the introduction of a provable high-dimensional version of the Birman--Ko--Lee cycling theorem. The complexity of this solution is a small degree polynomial in the cardinalities of our generalized super summit sets and the input parameters. Computer experiments suggest that the cardinality of this invariant, for a list of order $N$ independent elements of Artin's braid group $B_N$, is generically close to 1.