The model theory of the curve graph

TL;DR

This paper explores the model-theoretic properties of the curve graph of surfaces, establishing its stability, interpretability of related complexes, and a rigidity phenomenon linking surface homeomorphism to graph interpretability.

Contribution

It introduces the first model-theoretic analysis of the curve graph, proving $ ext{ω}$-stability, quantifier elimination, and a novel interpretation rigidity result.

Findings

The theory of the curve graph is $ ext{ω}$-stable and has quantifier elimination.

Many complexes associated with surfaces are interpretable in the curve graph and have bounded Morley ranks.

Interpretation rigidity implies that mutually interpretable curve graphs correspond to homeomorphic surfaces.

Abstract

In this paper we develop a bridge between model theory, geometric topology, and geometric group theory. In particular, we investigate the Ivanov Metaconjecture from the point of view of model theory, and more broadly we seek to answer the general question: why does the curve graph of a surface play such a central role in the study of surfaces and mapping class groups? More specifically, we consider a surface $\Sigma$ of finite type and its curve graph $\mathcal C(\Sigma)$, and we investigate its first-order theory in the language of graph theory. Crucially, $\mathcal C(\Sigma)$ is bi-interpretable with a certain object called the augmented Cayley graph of the mapping class group of the surface. We use this bi-interpretation to prove that the theory of the curve graph is $\omega$--stable, to compute its Morley rank, and to show that it has quantifier elimination with respect to the class of $\forall\exists$--formulae. We also show that many of the complexes which are naturally associated to a surface are interpretable in $\mathcal C(\Sigma)$. This shows that these complexes are all $\omega$--stable and admit certain a priori bounds on their Morley ranks. We are able to use Morley ranks to prove that various 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. Finally, we prove that the curve graph of a surface enjoys a novel phenomenon that we call interpretation rigidity. That is, if surfaces $\Sigma_1$ and $\Sigma_2$ admits curve graphs that are mutually interpretable, then $\Sigma_1$ and $\Sigma_2$ are homeomorphic to each other. Along the way, numerous technical results are obtained.