Submission date: 24 August 2020

Abstract:

In this paper we develop a bridge between model theory, geometric topology, and geometric group theory. We consider a surface Σ of finite type and its curve graph C(Σ), and we investigate the first-order theory of the curve graph in the language of graph theory. We prove that the theory of the curve graph is ω-stable, give bounds on its Morley rank, and show that it has quantifier elimination with respect to the class of ∃-formulae. We also show that many of the complexes which are naturally associated to a surface are interpretable in the curve graph, which proves that these complexes are all ω-stable and admit certain a priori bounds on their Morley ranks. We are able to use Morley rank to prove that several complexes are not bi-interpretable with the curve graph. As a consequence of quantifier elimination, we show that algebraic intersection number is not definable in the first order theory of the curve graph.

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Full text arXiv 2008.10490: pdf, ps.