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

Abstract

Let $A$ 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 $A$ satisfies Hrushovski's extension property, then is it true that the automorphism group $Aut\left(A\right)$ of $A$ 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 $Aut\left(A\right)$ in terms of a (new) variant of the coherent extension property. We also compare our notion with other coherent extension properties.