On dense, locally finite subgroups of the automorphism group of certain homogeneous structures

Abstract

Let be a countable structure such that each finite partial isomorphism of it can be extended to an automorphism. Evans asked if the age (set of finite substructures) of satisfies Hrushovski's extension property, then is it true that the automorphism group of contains a dense, locally finite subgroup? In order to investigate this question, in the previous decades a coherent variant of Hrushovski's extension property has been introduced and studied. Among other results, we provide equivalent conditions for the existence of a dense, locally finite subgroup of in terms of a (new) variant of the coherent extension property. We also compare our notion with other coherent extension properties.