ar X iv : 1 70 5 . 01 88 8 v 2 [ m at h . L O ] 1 8 Ju l 2 01 7 COHERENT EXTENSION OF PARTIAL AUTOMORPHISMS , FREE AMALGAMATION , AND AUTOMORPHISM GROUPS

We give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fräıssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fräıssé limit of any Fräıssé class that is the class of all F-free structures (in the Herwig–Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language, all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.