Extending partial isometries of generalized metric spaces

TL;DR

This paper extends the theory of partial isometries in generalized metric spaces with distances in ordered monoids, establishing the Hrushovski extension property for various classes, which leads to results on automorphism groups with ample generics.

Contribution

It proves the Hrushovski extension property for classes of finite generalized metric spaces over semi-archimedean monoids and applies this to automorphism groups of Urysohn spaces and graphs.

Findings

Hrushovski property holds for finite generalized metric spaces over semi-archimedean monoids.

Automorphism groups of certain Urysohn spaces have ample generics.

Automorphism groups of universal graphs omitting odd cycles also have ample generics.

Abstract

We consider generalized metric spaces taking distances in an arbitrary ordered commutative monoid, and investigate when a class $\mathcal{K}$ of finite generalized metric spaces satisfies the Hrushovski extension property: for any $A\in\mathcal{K}$ there is some $B\in\mathcal{K}$ such that $A$ is a subspace of $B$ and any partial isometry of $A$ extends to a total isometry of $B$. Our main result is the Hrushovski property for the class of finite generalized metric spaces over a semi-archimedean monoid $\mathcal{R}$. When $\mathcal{R}$ is also countable, this can be used to show that the isometry group of the Urysohn space over $\mathcal{R}$ has ample generics. Finally, we prove the Hrushovski property for classes of integer distance metric spaces omitting triangles of uniformly bounded odd perimeter. As a corollary, given odd $n\geq 3$, we obtain ample generics for the automorphism group of the universal, existentially closed graph omitting cycles of odd length bounded by $n$.