Sergei Buyalo

Elements of Asymptotic Geometry

Asymptotic geometry is the study of metric spaces from a large scale point of view, where the local geometry does not come into play. An important class of model spaces are the hyperbolic spaces (in the sense of Gromov), for which the asymptotic geometry is nicely encoded in the boundary at infinity.   In the first part of this book, in analogy with the concepts of classical hyperbolic geometry, the authors provide a systematic account of the basic theory of Gromov hyperbolic spaces. These spaces have been studied extensively in the last twenty years, and have found applications in group theory, geometric topology, Kleinian groups, as well as dynamics and rigidity theory. In the second part of the book, various aspects of the asymptotic geometry of arbitrary metric spaces are considered. It turns out that the boundary at infinity approach is not appropriate in the general case, but dimension theory proves useful for finding interesting results and applications.   The text leads concisely to some central aspects of the theory. Each chapter concludes with a separate section containing supplementary results and bibliographical notes. Here the theory is also illustrated with numerous examples as well as relations to the neighboring fields of comparison geometry and geometric group theory.   The book is based on lectures the authors presented at the Steklov Institute in St. Petersburg and the University of Zurich. It addressed to graduate students and researchers working in geometry, topology, and geometric group theory.

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.

ASYMPTOTIC DIMENSION OF A HYPERBOLIC SPACE AND CAPACITY DIMENSION OF ITS BOUNDARY AT INFINITY

We introduce a quasi-symmetry invariant of a metric space Z called the capacity dimension, cdimZ. Our main result says that for a visual Gromov hyperbolic space X the asymptotic dimension of X is at most the capacity dimension of its boundary at infinity plus 1, asdimX ≤ cdim@1X + 1.

Topological and geometric properties of graph-manifolds

This is a unified exposition of results (obtained by different authors) on the existence of π1-injective immersed and embedded surfaces in graph-manifolds, and also of nonpositively curved metrics on graph-manifolds. The basis for unification is provided by the notion of compatible cohomology classes and by a certain difference equation on the graph of a graph-manifold (the BKN-equation). Criteria for seven different properties of graph-manifolds are given at three levels: at the level of compatible cohomology classes; at the level of solutions of the BKN-equation; and in terms of spectral properties of operator invariants of a graph-manifold.

Capacity dimension and embedding of hyperbolic spaces into a product of trees

We prove that every visual Gromov hyperbolic space X whose boundary at infinity has the finite capacity dimension n admits a quasi-isometric embedding into (n+1)-fold product of metric trees.

Hyperbolic dimension of metric spaces

We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromovs asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however, unlike the asymptotic dimension, the hyperbolic dimension of any Euclidean space R^n is zero (while asdim R^n=n.) This invariant possesses usual properties of dimension like monotonicity and product theorems. Our main result says that the hyperbolic dimension of any Gromov hyperbolic space X (with mild restrictions) is at least the topological dimension of the boundary at infinity plus 1. As an application we obtain that there is no quasi-isometric embedding of the real hyperbolic space H^n into the (n-1)-fold metric product of metric trees stabilized by any Euclidean factor.

On the asymptotic geometry of nonpositively curved graphmanifolds

In this paper we study the Tits geometry of a 3-dimensional graphmanifold of nonpositive curvature. In particular we give an optimal upper bound for the length of nonstandard components of the Tits metric. In the special case of a π/2-metric we determine the whole length spectrum of the nonstandard components.

Geodesics avoiding open subsets in surfaces of negative curvature

We prove existence and non-existence results for geodesics avoiding

Volume entropy of hyperbolic graph surfaces

a

An extremal theorem of Riemannian geometry

-separated sets on a surface of negative curvature.