Martin Bridson (Oxford)

Title:

CAT(0) spaces, fixed points, and finite shadows of infinite groups

Abstract:

I shall begin by recalling some classical results concerning the structure of groups that act by isometries on complete CAT(0) spaces. I shall then sketch how CAT(0) spaces have played an important role in exploring the extent to which residually-finite groups are uniquely determined by their set of finite quotients. To conclude, I will explain why, for each positive integer d, there exist pairs of non-isomorphic, finitely-presented, residually-finite groups with the same profinite completion such that one acts on a tree without a fixed point, while the other has a fixed point whenever it acts by semisimple isometries on a complete CAT(0) space of dimension at most d.