Submission date: 2 October 2018

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.

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Full text arXiv 1810.00242: pdf, ps.