Metric Cotype

We introduce the notion of metric cotype, a property of metric spaces related to a property of normed spaces, called Rademacher cotype. Apart from settling a long standing open problem in metric geometry, this property is used to prove the following dichotomy: A family of metric spaces F is either almost universal (i.e., contains any finite metric space with any distortion > 1), or there exists α > 0, and arbitrarily large n-point metrics whose distortion when embedded in any member of F is at least Ω((log n)^α). The same property is also used to prove strong non-embeddability theorems of L_q into L_p, when q > max{2,p}. Finally we use metric cotype to obtain a new type of isoperimetric inequality on the discrete torus

Pants decompositions of random surfaces

Our goal is to show, in two different contexts, that "random" surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus $g$ for which any pants decomposition requires curves of total length at least $g^{7/6 - \epsilon}$. Moreover, we prove that this bound holds for most metrics in the moduli space of hyperbolic metrics equipped with the Weil-Petersson volume form. We then consider surfaces obtained by randomly gluing euclidean triangles (with unit side length) together and show that these surfaces have the same property.Comment: 16 pages, 4 figure