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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 $B$. We prove the Hrushovski property for the class of finite generalized metric spaces over a semi-archimedean monoid $\mathcal {R}$. When $\mathcal {R}$ is also countable, we use this 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 metric 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 $n$.