Model Theory of R-trees

TL;DR

This paper develops a model-theoretic framework for pointed R-trees with bounded radius, establishing axiomatization, properties of the model companion, and analyzing the diversity and structure of models within this class.

Contribution

It introduces the theory of pointed R-trees with bounded radius, identifies its model companion with key properties, and explores the structure and diversity of models, including non-isomorphic and non-homeomorphic models.

Findings

The theory of pointed R-trees with radius at most r is axiomatizable.

The model companion is complete, stable, with quantifier elimination.

There are many non-isomorphic, non-homeomorphic models, and the space of types is nonseparable.

Abstract

We show the theory of pointed $\R$-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 $\R$-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.