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Nonpositively Curved Surfaces are Loewner
We show that every closed nonpositively curved surface satisfies Loewner’s systolic inequality. The proof relies on a combination of the Gauss–Bonnet...
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Sharp Reverse Isoperimetric Inequalities in Nonpositively Curved Cones
We prove a pair of sharp reverse isoperimetric inequalities for domains in nonpositively curved surfaces: (1) metric disks centered at the vertex of...
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Polyhedral Preliminaries
This chapter deals with cell complexes (also called “polyhedra”) with metrics of piecewise constant curvature. In other words, each cell is required...
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Some remarks on the simplicial volume of nonpositively curved manifolds
We show that any closed manifold with a metric of nonpositive curvature that admits either a single point rank condition or a single point curvature...
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Overview
For over 170 years, the connection between group theory and the fields of geometry and topology has held central in mathematics—the connection is...
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Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces
We prove a quantitative version of the classical Tits’ alternative for discrete groups acting on packed Gromov-hyperbolic spaces supporting a convex...
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Hyperbolization
“Hyperbolization” refers to various functorial constructions of Gromov for converting a cell complex J into an NPC polyhedron...
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VH-ification
We show that various groups act freely on the product of two trees. In particular this holds for word-hyperbolic elementarily free groups. To Rouska....
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The Liouville current of holomorphic quadratic differential metrics
In this paper we study the Liouville current of flat cone metrics coming from a holomorphic quadratic differential. Anja Bankovic and Christopher J....
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Geometry in the Twentieth Century: A Return to Euclid – The Work of Herbert Busemann
This is an exposition of the work of Herbert Busemann (1905–1994), seen as a return to the geometry of Ancient Greece. The importance of this work,...
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Ends of finite volume, nonpositively curved manifolds
We study complete, finite volume n -manifolds M of bounded nonpositive sectional curvature. A classical theorem of M. Gromov says that if such M has negative...
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Diameter estimates for surfaces in conformally flat spaces
The aim of this paper is to give an upper bound for the intrinsic diameter of a surface with boundary immersed in a conformally flat three...
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Rectifiability of the Singular Set of Harmonic Maps into Buildings
In Gromov and Schoen’s (Publications Mathématiques de l’Institut des Hautes Études Scientifiques 76(1):165–246, 1992), a notion of energy-minimizing...
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Geometry in the Twentieth Century: A Return to Euclid – The Work of Herbert Busemann
This is an exposition of the work of Herbert Busemann (1905–1994), seen as a return to the geometry of Ancient Greece. The importance of this work,...
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Haken 3-Manifolds in Small Covers
The authors prove that a 3-dimensional small cover M is a Haken manifold if and only if M is aspherical or equivalently the underlying simple...
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Right-Angled Spaces and Groups
The notion of a “polyhedral product” of a collection of spaces (or pairs of spaces) indexed by the vertex set I of a simplicial complex L is used to...
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Virtual homological eigenvalues and the Weil-Petersson translation length
For any pseudo-Anosov automorphism on an orientable closed surface, an inequality is established by bounding certain growth of virtual homological...
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Algebraic characterisation of relatively hyperbolic special groups
This article is dedicated to the characterisation of the relative hyperbolicity of Haglund and Wise’s special groups. More precisely, we introduce a...