Contractible, hyperbolic but non-CAT(0) complexes
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GAFA Geometric And Functional Analysis
CONTRACTIBLE, HYPERBOLIC BUT NON-CAT(0) COMPLEXES Richard C. H. Webb
Abstract. 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.
1 Introduction In general the mapping class group Mod(Sg,p ) cannot act properly by semisimple isometries on a complete CAT(0) space [KL96, BH99, Bri10], in particular, it is not a CAT(0) group. However, the Teichm¨ uller space with the Weil–Petersson metric is CAT(0) [Tro86, Wol86, Wol87], furthermore so is its completion [BH99, Corollary II.3.11], and the mapping class group acts on this by semisimple isometries [DW03]. When g ≥ 3, Bridson [Bri10] used this action to show that any non-trivial homomorphism Mod(Sg,p ) → Mod(Σ), must send Dehn twists to roots of multitwists. This result was then used by Aramayona–Souto [AS12] to classify—under topological assumptions on Sg,p and Σ—all non-trivial homomorphisms from Mod(Sg,p ) to Mod(Σ). Despite the fact that the mapping class group is not CAT(0) in general, the study of its algebra has been enhanced by its action on CAT(0) spaces. Perhaps the most striking application of actions on CAT(0) spaces has been provided by CAT(0) cube complexes and their role in the proof of the virtual Haken conjecture, see [Ago13]. Not only did this uncover a difficult topological consequence but there were many exciting algebraic consequences for the fundamental groups of closed, hyperbolic 3-manifolds such as largeness, LERF, linearity over Z,
R. C. H. WEBB
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Figure 1: A natural triangulation of the suspension of R produces a non-locally compact, 2dimensional simplicial complex that satisfies no combinatorial isoperimetric inequality: there are loops of combinatorial length four that require arbitrarily many triangles to deform them to a point. From an appropriate embedding in the euclidean plane it can be endowed with a complete CAT(0) metric.
bi-orderability, conjugacy separability, see for example [AFW15] and the references therein. One theme of this paper concerns the problem of finding CAT(0) metrics on a given space. The spaces of interest in this paper are locally infinite complexes, such as the arc complex of a surface and the free splitting complex of a free group. We say that a complex K (equipped with a metric) has finitely many shapes if there are only finitely many isom
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