# Model theory of R-trees

@article{Carlisle2020ModelTO, title={Model theory of R-trees}, author={Sylvia Carlisle and C. Ward Henson}, journal={J. Log. Anal.}, year={2020}, volume={12} }

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…

## Topics from this paper

## 2 Citations

Topometric characterization of type spaces in continuous logic

- Mathematics
- 2021

We show that a topometric space X is topometrically isomorphic to a type space of some continuous first-order theory if and only if X is compact and has an open metric (i.e., satisfies that {p : d(p,…

## References

SHOWING 1-10 OF 29 REFERENCES

On d-finiteness in continuous structures

- Mathematics
- 2007

We observe that certain classical results of first order model theory fail in the context of continuous first order logic. We argue that this happens since finite tuples in a continuous structure may…

Gromov's theorem on groups of polynomial growth and elementary logic

- Mathematics
- 1984

In the fall of 1980 the authors attended Professor Tits’ course at Yale University in which he gave an account of Gromov’s beautiful proof that every finitely generated group of polynomial growth has…

Universal Spaces for R-Trees

- Mathematics
- 1992

R-trees arise naturally in the study of groups of isometries of hyperboIic space. An R-tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. It…

Explicit constructions of universal R-trees and asymptotic geometry of hyperbolic spaces

- Mathematics
- 1999

We present some explicit constructions of universal R-trees with applications to the asymptotic geometry of hyperbolic spaces. In particular, we show that any asymptotic cone of a complete simply…

Continuous first order logic for unbounded metric structures

- Mathematics
- 2008

We present an adaptation of continuous first order logic to unbounded metric structures. This has the advantage of being closer in spirit to C. Ward Henson's logic for Banach space structures than…

Model Theory with Applications to Algebra and Analysis: Model theory for metric structures

- Mathematics
- 2008

A metric structure is a many-sorted structure with each sort a metric space, which for convenience is assumed to have finite diameter. Additionally there are functions (of several variables) between…

Model theory and metric convergence I: Metastability and dominated convergence

- Mathematics
- 2016

We study Tao's finitary viewpoint of convergence in metric spaces, as captured by the notion of metastability. We adopt the perspective of continuous model theory. We show that, in essence,…

Metric Spaces of Non-Positive Curvature

- Mathematics
- 1999

This book describes the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by…

Modular functionals and perturbations of Nakano spaces

- Mathematics
- 2009

We settle several questions regarding the model theory of Nakano spaces left open by the PhD thesis of Pedro Poitevin (11). We start by studying isometric Banach lattice embeddings of Nakano spaces,…

Model-theoretic independence in the banach lattices Lp(µ)

- Mathematics
- 2009

We study model-theoretic stability and independence in Banach lattices of the form Lp(X, U, µ), where 1 ≤ p < ∞. We characterize non-dividing using concepts from analysis and show that canonical…