Title: Coherent extension of partial automorphisms, free amalgamation, and automorphism groups,
Abstract: 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 Fraisse classes. We
deduce from these results that the isometry group of the rational Urysohn
space, the automorphism group of the Fraisse limit of any Fraisse class that is
the class of all $\mathcal{F}$-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.