Andrew Sale

For a finitely generated group, the conjugacy problem asks if there is an algorithm which determines whether a pair of elements are conjugate or not. Related to this is the question of understanding, for a given pair of conjugate elements, the family of conjugators between them. We focus in particular on estimating the minimal length of such an element, leading to the definition of the conjugacy length function of a group. This naturally extends beyond the reach of the conjugacy problem itself and can be asked of any group which admits a left-invariant metric. We will describe results for semisimple Lie groups, in particular for pairs of hyperbolic or unipotent elements, and discuss the issue of extending results from the ambient groups to their lattices.