Elements of finite order in automorphism groups of homogeneous structures

We study properties of the automorphism groups of Fräıssé limits of classes with certain strong amalgamation properties, including classes with the free amalgamation property and classes of metric spaces. We discuss conditions on a Fräıssé class K that imply that the automorphism group GK of its limit admits generic elements of order n for all n and show that, for many such K, any element of GK is a product of 4 conjugates of the generic element of order n for all n ≥ 2. Our constructions enable us to compute the Borel complexity of the relation of conjugacy between automorphisms of the Henson graphs, and to obtain some new results about the structure of the isometry group of the Urysohn space and the Urysohn sphere.