×

Model theory of \(\mathbb{R}\)-trees. (English) Zbl 1485.03138

Summary: We show the theory of pointed \(\mathbb{R}\)-trees with radius at most \(r\) is axiomatizable in a suitable continuous signature. We identify the model companion rb\(\mathbb{R}\)T\(_r\) 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 rb\(\mathbb{R}\)T\(_r\) are \(\mathbb{R}\)-trees that arise naturally in geometric group theory. In every infinite cardinal, we construct the maximum possible number of pairwise non-isomorphic models of rb\(\mathbb{R}\)T\(_r\); indeed, the models we construct are pairwise non-homeomorphic. We give detailed information about the type spaces of rb\(\mathbb{R}\)T\(_r\). 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 rb\(\mathbb{R}\)T\(_r\) has no atomic model.

MSC:

03C65 Models of other mathematical theories
03C60 Model-theoretic algebra
03C10 Quantifier elimination, model completeness, and related topics
05C05 Trees
20F67 Hyperbolic groups and nonpositively curved groups
54E35 Metric spaces, metrizability
54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] I Ben Yaacov,Continuous first order logic for unbounded metric structures, J. Math. Logic 8 (2008) 197-223;https://dx.doi.org/10.1142/S0219061308000737 · Zbl 1191.03026
[2] I Ben YaacovModular functionals and perturbations of Nakano spaces, J. Logic and Analysis 1 (2009) paper 1, 42 pages;https://dx.doi.org/10.4115/jla.2009.1.1 · Zbl 1163.03021
[3] I Ben Yaacov,A Berenstein,C W Henson,Model-theoretic independence in the Banach lattices Lp(µ), Isr. J. Math. 183 (2011) 285-320;https://dx.doi.org/10. 1007/s11856-011-0050-4 · Zbl 1232.46020
[4] I Ben Yaacov,A Berenstein,C W Henson,A Usvyatsov,Model theory for metric structures, in “Model Theory with Applications to Algebra and Analysis, Vol II”, (eds Z Chatzidakis, D Macpherson, A Pillay, and A Wilkie), Lecture Notes series of the London Mathematical Society No 350, Cambridge University Press (2008) 315-427; https://dx.doi.org/10.1017/CBO9780511735219.011 · Zbl 1152.03006
[5] I Ben Yaacov,A Usvyatsov,On d-finiteness in continuous structures, Fund. Math. 194 (2007) 67-88;https://dx.doi.org/10.4064/fm194-1-4 · Zbl 1121.03045
[6] I Ben Yaacov,A Usvyatsov,Continuous first order logic and local stability, Trans. Amer. Math. Soc. 362 (2010) 5213-5259;https://dx.doi.org/10.4115/10.1090/S00029947-10-04837-3 · Zbl 1200.03024
[7] M R Bridson,A Haefliger,Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften 319 Springer-Verlag (1999);https://dx.doi.org/ 10.1007/978-3-662-12494-9 · Zbl 0988.53001
[8] I Chiswell,Introduction toΛ-trees, World Scientific (2001);https://dx.doi.org/10. 1142/4495 · Zbl 1004.20014
[9] L van den Dries,A Wilkie,Gromov’s theorem on groups of polynomial growth and elementary logic, J. Algebra 89 (1984) 349-374;https://dx.doi.org/10.1016/00218693(84)90223-0 · Zbl 0552.20017
[10] C Drutu,Quasi-isometry invariants and asymptotic cones, Int J Algebra and Computation 12 (2002) 99-135;https://dx.doi.org/10.1142/S0218196702000948 · Zbl 1010.20029
[11] E Due ˜nez,J Iovino,Model theory and metric convergence I: Metastability and dominated convergence, in “Beyond First Order Model Theory”, (ed J Iovino), CRC Press (2017) 131-187;https://dx.doi.org/10.1201/9781315368078 · Zbl 1433.03096
[12] A G Dyubina,I Polterovich,The structure of hyperbolic spaces at infinity(Russian), Uspekhi Mat. Nauk 53 (1998) 239-240; translation in Russian Math. Surveys 53 (1998) 1093-1094;https://dx.doi.org/10.1070/rm1998v053n05ABEH000080 · Zbl 0968.53029
[13] A G Dyubina,I Polterovich,Explicit constructions of universalR-trees and asymptotic geometry of hyperbolic spaces, Bull. London Math. Soc. 33 (2001) 727-734; https://dx.doi.org/10.1112/S002460930100844X · Zbl 1058.53035
[14] W Hodges,Model Theory, Cambridge University Press (1993);https://dx.doi.org/10. 1017/CBO9780511551574 · Zbl 0789.03031
[15] J Roe,Lectures on Coarse Geometry, University Lecture Series 31, Amer. · Zbl 1042.53027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.