On submanifolds in locally symmetric spaces of noncompact type

A characterization of submanifolds by a homogeneity condition

A very short proof of the following smooth homogeneity theorem of D. Repovs, E. V. Scepin and the author is presented. Let N be a locally compact subset of a smooth manifold M. Assume that for each two points x,y in N there exist their neighborhoods Ux and Uy in M and a diffeomorphism h : Ux \to Uy such that h(x)=y and h (Ux \cap N) = Uy \cap N. Then N is a smooth submanifold of M.

Pontryagin classes of locally symmetric manifolds

In this note we compute low degree rational Pontryagin classes for every closed locally symmetric manifold of noncompact type. In particular, we answer the question: Which locally symmetric M have at least one nonzero Pontryagin class?

Riemannian manifolds with local symmetry

We give a classification of many closed Riemannian manifolds M whose universal cover possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds $M$ such that Isom$(\widetilde{M})$ has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bundle over a locally homogeneous space. This is inspired by work of Eberlein (for nonpositively curved manifolds) and Farb-Weinberger (for aspherical manifolds), and generalizes work of Frankel (for a semisimple group action). As an application, we characterize simply-connected Riem…

Non compact Euclidean cone 3-manifolds with cone angles less than 2pi

We describe some properties of noncompact Euclidean cone manifolds with cone angles less than c less than 2pi and singular locus a submanifold. More precisely, we describe its structure outside a compact set. As a corollary we classify those with cone angles less than 3pi/2 and those with all cone angles equal to 3pi/2.

On the number of ends of rank one locally symmetric spaces

Let Y be a noncompact rank one locally symmetric space of finite volume. Then Y has a finite number e(Y) > 0 of topological ends. In this paper, we show that for any natural number n, the Y with e(Y) \leq n that are arithmetic fall into finitely many commensurability classes. In particular, there is a constant c_n such that n-cusped arithmetic orbifolds do not exist in dimension greater than c_n. We make this explicit for one-cusped arithmetic hyperbolic n-orbifolds and prove that none exist for n \geq 30.

Gluing locally symmetric manifolds: asphericity and rigidity

We use the reflection group trick to glue manifolds with corners that are Borel-Serre compactifications of locally symmetric spaces of noncompact type and obtain aspherical manifolds. We call these \emph{piecewise locally symmetric} manifolds. This class of spaces provide new examples of aspherical manifolds whose fundamental groups have the structure of a complex of groups. These manifolds typically do not admit a locally $\CAT(0)$ metric. We prove that any self homotopy equivalence of such manifolds is homotopic to a homeomorphism. We compute the group of self homotopy equivalences of such a…

仿射不变子流形上大维数度量的有效等分布

Effective equidistribution of large dimensional measures on affine invariant submanifolds

本文证明了平移表面上不稳定叶层的大维数度量在仿射不变子流形中等分布并给出有效速率，为齐性动力学中的密度与等分布结果提供了类似支撑。

On the quasi-isometric classification of locally compact groups

This (quasi-)survey addresses the quasi-isometry classification of locally compact groups, with an emphasis on amenable hyperbolic locally compact groups. This encompasses the problem of quasi-isometry classification of homogeneous negatively curved manifolds. A main conjecture provides a general description; an extended discussion reduces this conjecture to more specific statements. In the course of the paper, we provide statements of quasi-isometric rigidity for general symmetric spaces of noncompact type and also discuss accessibility issues in the realm of compactly generated locally compa…