Model Theory of R-trees

Abstract

We show the theory of pointed $\mathbb{R}$-trees with radius at most $r$ is axiomatizable in a suitable continuous signature. We identify the model companion $\rbRT_r$ of this theory and study its properties. In particular, the model companion is complete and has quantifier elimination; it is stable but not superstable. We identify its independence relation and find built-in canonical bases for non-algebraic types. Among the models of $\rbRT_r$ are $\mathbb{R}$-trees that arise naturally in geometric group theory. In every infinite cardinal, we construct the maximum possible number of pairwise non-isomorphic models of $\rbRT_r$; indeed, the models we construct are pairwise non-homeomorphic. We give detailed information about the type spaces of $\rbRT_r$. Among other things, we show that the space of $2$-types over the empty set is nonseparable. Also, we characterize the principal types of finite tuples (over the empty set) and use this information to conclude that $\rbRT_r$ has no atomic model.

Content is the same as the published version except that a small problem in the proof of Lemma 7.7 has been fixed

References in corpus (2)

• Modular functionals and perturbations of Nakano spaces
• Model theory and metric convergence I: Metastability and dominated convergence

Cited by in corpus (1)

• Topometric characterization of type spaces in continuous logic