Model Theory of R-trees

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.