Nina Lebedeva

Dimensions of locally and asymptotically self-similar spaces

We obtain two in a sense dual to each other results: First, that the capacity dimension of every compact, locally self-similar metric space coincides with the topological dimension, and second, that the asymptotic dimension of a metric space, which is asymptotically similar to its compact subspace coincides with the topological dimension of the subspace. As an application of the first result, we prove the Gromov conjecture that the asymptotic dimension of every hyperbolic group G equals the topological dimension of its boundary at infinity plus 1, asdimG = dim@1G + 1. As an application of the second result, we construct Pontryagin surfaces for the asymptotic dimension, in particular, those are first examples of metric spaces X, Y with asdim(X ×Y ) < asdimX+asdimY . Other applications are also given.

On spaces of polynomial growth with no conjugate points

The following generalization of the Hopf conjecture is proved: if the fundamental group of an n-dimensional compact polyhedral space M without boundary and with no conjugate points has polynomial growth, then there exists a finite covering of M by a flat torus. §

Alexandrov spaces with maximal number of extremal points

We show that any n-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of Euclidean n -space by an action of a crystallographic group. We describe all such actions.

Smoothing 3-dimensional polyhedral spaces

We show that 3-dimensional polyhedral manifolds with nonnegative curvature in the sense of Alexandrov can be approximated by nonnegatively curved 3-dimensional Riemannian manifolds.

Dimensions of products of hyperbolic spaces

We give estimates on asymptotic dimensions of products of general hyperbolic spaces with following applications to the hyperbolic groups. We give examples of strict inequality in the product theorem for the asymptotic dimension in the class of the hyperbolic groups; and examples of strict inequality in the product theorem for the hyperbolic dimension. We prove that R is dimensionally full for the asymptotic dimension in the class of the hyperbolic groups.

Local characterization of polyhedral spaces

We show that a compact length space is polyhedral if a small spherical neighborhood of any point is conic.

Exponential growth of spaces without conjugate points

An n-dimensional polyhedral space is a length space M (with intrinsic metric) triangulated into n-simplexes with smooth Riemannian metrics. In the definitions below, we assume that the triangulation is fixed. The boundary of M is the union of the (n− 1)-simplexes of the triangulation that are adjacent to only one (n− 1)-simplex. As usual, a geodesic in M is a naturally parametrized locally shortest curve defined on an interval. We say that M has no conjugate points if any two points in the universal covering space M̃ of M are joined by a unique geodesic. We say that the volume entropy of M̃ is positive if the volume of metric balls in M̃ has at least exponential growth. Now, we state the main result of this paper.

ON THE TOTAL CURVATURE OF MINIMIZING GEODESICS ON CONVEX SURFACES

We give a universal upper bound for the total curvature of minimizing geodesic on a convex surface in the Euclidean space.