Locally $\aleph_0$-categorical theories and locally Roelcke precompact groups

Abstract

It is well-known that Polish Roelcke precompact groups are the groups that can be represented as automorphism groups of $\aleph_0$-categorical structures in continuous logic and that there is a precise correspondence between properties of the group and properties of the structure. The goal of this paper is to extend this correspondence to the classes of locally Roelcke precompact groups and locally $\aleph_0$-categorical structures, the latter of which we define here. We characterise locally Roelcke precompact groups in terms of their isometric actions. We define locally $\aleph_0$-categorical theories and structures, prove an appropriate version of the Ryll-Nardzewski theorem, and identify the Polish locally Roelcke precompact groups as the automorphism groups of such structures. In all locally $\aleph_0$-categorical structures, there is a definable metric, which we call localising, and which captures the coarse geometric structure of the corresponding automorphism group. We show that two locally $\aleph_0$-categorical structures are bi-interpretable if and only if their automorphism groups are isomorphic. Finally, we show that (the unit ball of) a Banach space is $\aleph_0$-categorical if and only if the corresponding affine space is locally $\aleph_0$-categorical (as a metric space).