Model Theory of R-trees

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 rbRTr 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 rbRTr 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 rbRTr; indeed, the models we construct are pairwise non-homeomorphic. We give detailed information about the type spaces of rbRTr. 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 rbRTr has no atomic model.