Hyperbolic elements in negatively curved groups

We show that in a negatively curved groupG the conjugacy class of any infinite cyclic subgroup contains a straight element, an elementg with |g ⁿ |=n|g|, and thus the translation number of an element in a negatively curved group is rational with uniformly bounded denominator. We also find an upper bound on the cardinality of a finite normal subgroup.     

Boundary dimension in negatively curved spaces

LetX be a negatively curved (Gromov hyperbolic) space. We construct a bound on dim X when a group of isometries acts cocompactly onX. We construct an example of a negatively curved space with infinite-dimensional boundary.     

When is the graph product of hyperbolic groups hyperbolic?

Given a finite simplicial graph , and an assignment of groups to the verticles of , the graph product is the free product of the vertex groups modulo relations implying that adjacent vertex groups commute. We use Gromov's link criteria for cubical complexes and techniques of Davis and Moussang to study the curvature of graph products of groups. By constructing a CAT(–1) cubical complex, it is shown that the graph product of word hyperbolic groups is itself word hyperbolic if and only if the full subgraph in , generated by vertices whose associated groups are finite, satisfies three specific criteria. The construction shows that arbitrary graph products of finite groups are Bridson groups.     

Curvature tensors of singular spaces

A sequence of tensor-valued measures of certain singular spaces (e.g., subanalytic or convex sets) is constructed. The first three terms can be interpreted as scalar curvature, Einstein tensor and (modified) Riemann tensor. It is shown that these measures are independent of the ambient space, i.e., they are intrinsic. In contrast to this, there exists no intrinsic tensor-valued measure corresponding to the Ricci tensor.     

Rigidity in CR geometry: The Schoen-Webster theorem

The Schoen-Webster theorem asserts that a strictly pseudoconvex CR manifold whose automorphism group acts nonproperly is either the standard sphere or the Heisenberg group. The purpose of this paper is to survey successive works around this result and then provide a short geometric proof in the compact case.     

Countably determined sets and a conjecture of C.W. Henson

Partially supported by NSF grants DMS-9208071 and DMS-9100383     

Natural representations of some tilde and petersen type geometries

We determine the universal natural representations (embeddings) of certain sporadic geometries with three points per line.The final revision of this paper was done while the author held a temporary position at Michigan State University.     

The Poincaré-Bertrand Formula for the Bochner-Martinelli Integral

The Poincaré-Bertrand formula and the composition formula for the Bochner-Martinelli integral on piecewise smooth manifolds are obtained. As an application, the regularization problem for linear singular integral equation with Bochner-Martinelli kernel and variable coefficients is discussed.     

Coxeter groups are virtually special

In this paper we prove that every finitely generated Coxeter group has a finite index subgroup that is the fundamental group of a special cube complex. Some consequences include: Every f.g. Coxeter group is virtually a subgroup of a right-angled Coxeter group. Every word-hyperbolic Coxeter group has separable quasiconvex subgroups.     

The non-orientable genus of some metacyclic groups

We describe non-orientable, octagonal embeddings for certain 4-valent, bipartite Cayley graphs of finite metacyclic groups, and give a class of examples for which this embedding realizes the non-orientable genus of the group. This yields a construction of Cayley graphs for which is arbitrarily large, where and are the orientable genus and the non-orientable genus of the Cayley graph.Work supported in part by the Research Council of Slovenia, Yugoslavia and NSF Contract DMS-8717441.Supported by NSF Contract DMS-8601760.     

Fisher information metric and Poisson kernels

A complete Riemannian manifold X with negative curvature satisfying −b²?KX?−a²<0 for some constants a,b, is naturally mapped in the space of probability measures on the ideal boundary ∂X by assigning the Poisson kernels. We show that this map is embedding and the pull-back metric of the Fisher information metric by this embedding coincides with the original metric of X up to constant provided X is a rank one symmetric space of non-compact type. Furthermore, we give a geometric meaning of the embedding.     

Kähler–Einstein metrics on Fano surfaces

We give an exposition of a result of G. Tian, which says that a Fano surface admits a Kähler–Einstein metric precisely when the Lie algebra of holomorphic vector fields is reductive.     

Almost convex groups and the eight geometries

IfM is a closed Nil geometry 3-manifold then ₁(M) is almost convex with respect to a fairly simple geometric generating set. IfG is a central extension or a extension of a word hyperbolic group, thenG is also almost convex with respect to some generating set. Combining these with previously known results shows that ifM is a closed 3-manifold with one of Thurston's eight geometries, ₁(M) is almost convex with respect to some generating set if and only if the geometry in question is not Sol.     

Problème du bord dans les variétés $q$-convexes et phénomène de Hartogs-Bochner

Résumé.   Soient E un fibré vectoriel holomorphe au dessus d'une variétéq-complète X et M une sous-variété réelle de dimension 2p-1 () de E. Nous montrons que le problème du bord pour M est résoluble si et seulement si M est maximalement complexe. Dans le cas où nous retrouvons le théorème de Harvey et Lawson [17]. Comme conséquence nous obtenons une généralisation du théorème de Hartogs-Bochner pour les applications CR-méromorphesà valeurs dans une variétéq-convexe et une généralisation au cas CR-méromorphe du théorème d'extension de Chazal [5].

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Let E be a holomorphic vector bundle over a -complete manifold X and M be a real submanifold of dimension 2p-1 () of E. We prove that M has a solution to the boundary problem in X if and only if M is maximally complex. In the case , this is a result of Harvey and Lawson [17]. As a consequence, we obtain a generalization of the Hartogs-Bochner theorem for CR-meromorphic maps taking their values in -convex manifolds and a generalization to the CR-meromorphic case of a theorem of Chazal [5].

Received: 27 April 2000 / Published online: 23 July 2001