Igor Belegradek

Obstructions to nonnegative curvature and rational homotopy theory

We establish a link between rational homotopy theory and the problem which vector bundles admit a complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if C lies in the class and T is a torus of positive dimension, then most vector bundles over C x T admit no complete nonnegatively curved metrics.

Topological obstructions to nonnegative curvature

Abstract. We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metrics.

ENDOMORPHISMS OF RELATIVELY HYPERBOLIC GROUPS

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a nonelementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group. • If H is a nonparabolic subgroup of a relatively hyperbolic group, and if any isometric H-action on an ℝ-tree is trivial, then H is Hopfian. • If G is a nonelementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a nonelementary relatively hyperbolic quotient that is Hopfian. • Any finitely presented group is isomorphic to a finite index subgroup of Out(H) for some group H with Kazhdan property (T). (This sharpens a result of Ollivier–Wise).

ON CO-HOPFIAN NILPOTENT GROUPS

Co-Hopfian finitely generated torsion-free nilpotent groups are characterized in this paper in terms of their Lie algebra automorphisms. Many examples of such groups are also constructed.

Metrics of positive Ricci curvature on bundles

We construct new examples of manifolds of positive Ricci curvature which, topologically, are vector bundles over compact manifolds of almost nonnegative Ricci curvature. In particular, we prove that if E is the total space of a vector bundle over a compact manifold of nonnegative Ricci curvature, then E × ℝ p admits a complete metric of positive Ricci curvature for all large p.

Rips construction and Kazhdan property (T)

We show that for any non-elementary hyperbolic group H and any finitely presented group Q, there exists a short exact sequence 1 ! N ! G ! Q ! 1, where G is a hyperbolic group and N is a quotient group of H. As an application we construct a hyperbolic group that has the same n-dimensional complex representations as a given finitely generated group, show that adding relations of the form x n D 1 to a presentation of a hyperbolic group may drastically change the group even in case n � 1, and prove that some properties (e.g. properties (T) and FA) are not recursively recognizable in the class of hyperbolic groups. A relatively hyperbolic version of this theorem is also used to generalize results of Ollivier-Wise on outer automorphism groups of Kazhdan groups. Mathematics Subject Classification (2000). 20F67, 20F65, 20F28.

On Mostow rigidity for variable negative curvature

Abstract We prove a finiteness theorem for the class of complete finite-volume Riemannian manifolds with pinched negative sectional curvature, fixed fundamental group, and of dimension ⩾3. One of the key ingredients is that the fundamental group of such a manifold does not admit a small nontrival action on an R -tree.

Intersections in hyperbolic manifolds

We obtain some restrictions on the topology of innite volume hyperbolic manifolds. In particular, for any n and any closed negatively curved manifold M of dimension 3, only nitely many hyperbolic n{manifolds are total spaces of orientable vector bundles over M .

Vector bundles with infinitely many souls

We construct the first examples of manifolds, the simplest one being S^3 x S^4 x R^5, which admit infinitely many complete nonnegatively curved metrics with pairwise nonhomeomorphic souls.

Aspherical manifolds, relative hyperbolicity, simplicial volume and assembly maps

This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. n(I) Any closed aspherical triangulated n-manifold M^n with hyperbolic fundamental group is a retract of a closed aspherical triangulated (n+1)-manifold N^(n+1) with hyperbolic fundamental group. n(II) If B_1,...,B_m are closed aspherical triangulated n-manifolds, then there is a closed aspherical triangulated manifold N of dimension n+1 such that N has nonzero simplicial volume, N retracts to each B_k, and pi_1(N) is hyperbolic relative to pi_1(B_k)s. n(III) Any finite aspherical simplicial complex is a retract of a closed aspherical triangulated manifold with positive simplicial volume and non-elementary relatively hyperbolic fundamental group.