Related papers: A strong Schottky Lemma for nonpositively curved s…

We study structural properties and the harmonic analysis of discrete subgroups of the Euclidean group. In particular, we 1. obtain an efficient description of their dual space, 2. develop Fourier analysis methods for periodic mappings on…

We prove a Morse Lemma for coarsely regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings X. The main application is a simpler coarse geometric characterization of Morse subgroups of the isometry groups…

We construct a Cartan-Hadamard manifold with pinched negative curvature whose group of isometries possesses divergent discrete free subgroups with parabolic elements who do not satisfy the so-called "parabolic gap condition" . This…

We extend the structure theory of Burago--Gromov--Perelman for Alexandrov spaces with curvature bounded below, to the setting of Busemann spaces with non-negative curvature. We prove that any finite-dimensional Busemann space with…

We construct irreducible unitary representations of a finitely generated free group which are weakly contained in the left regular representation and in which a given linear combination of the generators has an eigenvalue. When the…

Let $M$ be a compact nonnegatively curved Riemannian manifold admitting an isometric action by a compact Lie group $\mathsf G$ in a way that the quotient space $M/\mathsf G$ has nonempty boundary. Let $\pi : M \to M/\mathsf G$ denote the…

In this paper, we study representations from the four-punctured sphere group into isometry groups of Gromov-hyperbolic spaces. We prove that the set of simple-stable representations (in analogy with Minsky's notion of primitive-stability)…

A coarse group is a group endowed with a coarse structure so that the group multiplication and inversion are coarse mappings. Let $(X, \mathcal{E})$ be a coarse space and let $\mathfrak{M}$ be a variety of groups different from the variety…

In this article we give necessary and sufficient conditions for an irreducible K\"ahler C-space with $b_2=1$ to have nonnegative or positive quadratic bisectional curvature, assuming the space is not Hermitian symmetric. In the cases of the…

Pure de Sitter, anti de Sitter, and orthogonal gauge theories in four-dimensional Euclidean spacetime are studied. It is shown that, if the theory is asymptotically free and a dynamical mass is generated, then an effective geometry may be…

We prove that certain involutions defined by Vogell and Burghelea-Fiedorowicz on the rational algebraic $K$-theory of spaces coincide. This gives a way to compute the positive and negative eigenspaces of the involution on rational homotopy…

Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles…

We establish relations between both the classical and the dual Garside structures of the braid group and the Burau representation. Using the classical structure, we formulate a non-vanishing criterion for the Burau representation of the…

We construct coarse moduli spaces for `Brill-Noether pairs'. Such a pair consists of a torsion-free sheaf $E$ over an algebraic curve $X$ and a vector subspace $\Lambda$ of its space of sections $H^0(E)$. The construction works for an…

We show that $\mathbb{C}^2$ contains pairs of properly embedded, smooth complex curves that are isotopic through homeomorphisms but not diffeomorphisms of $\mathbb{C}^2$. The construction is based on realizing corks as branched covers of…

A theorem of Furuta and Fintushel-Stern provides a criterion for a collection of Seifert fibred homology spheres to be independent in the homology cobordism group of oriented homology 3-spheres. In this article we use these results and some…

We investigate spacetimes with their singular boundaries as noncommutative spaces. Such a space is defined by a noncommutative algebra on a transformation groupoid $\Gamma = E \times G$, where $E$ is the total space of the frame bundle over…

Given any countable group $G$, we construct uncountably many quasi-isometry classes of proper geodesic metric spaces with quasi-isometry group isomorphic to $G$. Moreover, if the group $G$ is a hyperbolic group, the spaces we construct are…

We study nonnegatively curved metrics on S^2xR^4. First, we prove rigidity theorems for connection metrics; for example, the holonomy group of the normal bundle of the soul must lie in a maximal torus of SO(4). Next, we prove that Wilking's…

We examine the hypothesis that space-time is a product of a continuous four-dimensional manifold times a finite space. A new tensorial notation is developed to present the various constructs of noncommutative geometry. In particular, this…