Neostability in countable homogeneous metric spaces

Given a countable, totally ordered commutative monoid R=(R,⊕,≤,0), with least element 0, there is a countable, universal and ultrahomogeneous metric space UR with distances in R. We refer to this space as the R-Urysohn space, and consider the theory of UR in a binary relational language of distance inequalities. This setting encompasses many classical structures of varying model theoretic complexity, including the rational Urysohn space, the free nth roots of the complete graph (e.g. the random graph when n=2), and theories of refining equivalence relations (viewed as ultrametric spaces). We characterize model theoretic properties of Th(UR) by algebraic properties of R, many of which are first-order in the language of ordered monoids. This includes stability, simplicity, and Shelah's SOPn-hierarchy. Using the submonoid of idempotents in R, we also characterize superstability, supersimplicity, and weak elimination of imaginaries. Finally, we give necessary conditions for elimination of hyperimaginaries, which further develops previous work of Casanovas and Wagner.