We study the subgroup structure of discrete groups which share cohomological properties which resemble non-negative curvature. Examples include all Gromov hyperbolic groups. A subgroup H ⊂ G is called s-normal, if H∩H is infinite for all g ∈ G. We provide strong restrictions on the possible s-normal subgroups of a Gromov hyperbolic group, or more generally a ’negatively curved’ group. Another r...

This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These GromovWitten type invariants depend on a Calabi-Yau A∞ category, which plays the role of the target in ordinary Gromov-Witten theory. When we use an appropriate A∞ version of the derived category of coherent sheaves on a Calabi-Yau variety, this constru...

We give exact and approximation algorithms for computing the Gromov hyperbolicity of an n-point discrete metric space. We observe that computing the Gromov hyperbolicity from a fixed base-point reduces to a (max,min) matrix product. Hence, using the (max,min) matrix product algorithm by Duan and Pettie, the fixed base-point hyperbolicity can be determined in O(n) time. It follows that the Gromo...

In this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is Gromov hyperbolic. As a particular consequence we obtain that Euclidean space is a borderline case for Gromov hyperbolicity in terms of the isoperimetric function. We prove similar results for the linear filling radius inequality. Our theorems strengthen and gene...

This research announcement discusses our results on Gromov-Witten theory of root gerbes. A complete calculation of genus 0 Gromov-Witten theory of μr-root gerbes over a smooth base scheme is obtained by a direct analysis of virtual fundamental classes. Our result verifies the genus 0 part of the so-called decomposition conjecture which compares Gromov-Witten theory of étale gerbes with that of ...

We prove that the system of Gromov-Witten invariants of the product of two varieties is equal to the tensor product of the systems of Gromov-Witten invariants of the two factors.

We consider Gromov–Thurston examples of negatively curved nmanifolds which do not admit metrics of constant sectional curvature. We show that for each n ≥ 4 some of the Gromov–Thurston manifolds admit strictly convex real–projective structures.

We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case by describing a metric on the set of rooted complete locally compact length spaces endowed with a locally finite measure. We prove that this space with the ex...

This is the second in a sequence of papers in which we construct a quantum version of the Kirwan map from the equivariant quantum cohomology QHG(X) of a smooth polarized complex projective variety X with the action of a connected complex reductive group G to the orbifold quantum cohomology QH(X//G) of its geometric invariant theory quotient X//G, and prove that it intertwines the genus zero gau...

We show that (equivariant) K-theoretic 3-point Gromov-Witten invariants of genus zero on a Grassmann variety are equal to triple intersections computed in the ordinary (equivariant) K-theory of a two-step flag manifold, thus generalizing an earlier result of Buch, Kresch, and Tamvakis. In the process we show that the Gromov-Witten variety of curves passing through 3 general points is irreducibl...