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AbstractAbstract: We prove that almost all arc complexes do not admit a CAT(0) metric with finitely many shapes, in particular any finite-index subgroup of the mapping class group does not preserve such a metric on the arc complex. We also show the analogous statement for all but finitely many disc complexes of handlebodies and free splitting complexes of free groups. The obstruction is combinatorial. These complexes are all hyperbolic and contractible but despite this we show that they satisfy no combinatorial isoperimetric inequality: for any n there is a loop of length 4 that only bounds discs consisting of at least n triangles. On the other hand we show that the curve complexes satisfy a linear combinatorial isoperimetric inequality, which answers a question of Andrew Putman.
CitationGeometric and Functional Analysis, volume 30, issue 5, page 1439-1463
PublisherSpringer International Publishing
DescriptionFrom Springer Nature via Jisc Publications Router
History: received 2020-04-26, rev-recd 2020-09-15, accepted 2020-09-16, pub-print 2020-10, registration 2020-10-01, pub-electronic 2020-10-26, online 2020-10-26
Publication status: Published
Funder: University of Manchester