Logic Seminar - Valentina Disarlo

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Speaker: Valentina Disarlo (The University of Manchester) Title: The model theory of the curve graph Abstract: The curve graph of a finite-type surface is a countable graph whose vertices are isotopy classes of essential simple closed curves, with adjacency determined by disjointness. Although it arises naturally in low-dimensional topology and geometric group theory, it also provides a remarkably rigid first-order structure. Ivanov's classical theorem identifies the automorphism group of the curve graph with the extended mapping class group, and his subsequent metaconjecture predicts analogous rigidity for a large class of naturally associated complexes made of adjacency relations between topological objects on a surface. In joint work with Thomas Koberda and Javier de la Nuez González, we present the first study of the curve graph from the point of view of model theory. I will discuss the first-order theory of the curve graph, with particular emphasis on definability and stability phenomena. In particular, we prove that the curve graph is w-stable and has a relative form of quantifier elimination. I will then present a model-theoretic framework for the Ivanov Metaconjecture and for comparing the curve graph with other complexes associated to a surface.