Quasi-morphisms on surface diffeomorphism groups

By Jonathan Bowden, Sebastian Hensel, Richard C. H. Webb. Journal of The American Mathematical Society, 2019

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We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its fragmentation norm is unbounded, answering a question of Burago-Ivanov-Polterovich. To do this, we introduce an analogue of the curve graph from the theory of mapping class groups. We show that it is hyperbolic and that the natural group action by isometries satisfies the criterion of Bestvina-Fujiwara.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1281cd955b9bd5ef79b5d1f572424cf764d2a58f