arXiv subjects
Publications and source records attributed to Viktor Schroeder.
A pair of points in a riemannian manifold makes a secure configuration if the totality of geodesics connecting them can be blocked by a finite set. The manifold is secure if every configuration is secure. We investigate the security of compact, locally symmetric spaces.
We prove that a finitely generated, right-angled, hyperbolic Coxeter group can be quasiisometrically embedded into the product of n binary trees, where n is the chromatic number of the group. As application we obtain certain strongly aperiodic tilings of the Davis complex of these groups.
We study CAT(kappa) spaces X admitting affine functions. We show that there exists a canonical isometric embedding X -> Y x H where H is a Hilbert space, such that every affine function f: X -> R factors as f'o p, where p is the projection to H and f' is an affine function on H.
We prove that a right angled Coxeter group with chromatic number n can be embedded in a bilipschitz way into the product of n locally finite trees. We give applications of this result to various embedding problems and determine the hyperbolic rank of products of exponentially branching trees.
In arXiv math.MG/0207296 we introduced a product construction for locally compact, complete, geodesic hyperbolic metric spaces. In the present paper we define the hyperbolic product for general Gromov-hyperbolic spaces. In the case of roughly geodesic spaces we also analyse the boundary at infinity.
We consider a fairly general class of natural non standard metric products and classify those amongst them, which yield a product of certain type (for instance an inner metric space) for all possible choices of factors of this type (inner metric spaces). We further prove the additivity of the Minkowski rank for a large class of metric products.
Generalizing results due to Brady and Farb we prove the existence of a bilipschitz embedded manifold of pinched negative curvature and dimension m_1+m_2-1 in the product X:=X_1^{m_1} times X_2^{m_2} of two Hadamard manifolds X_i^{m_i} of dimension m_i with pinched negative curvature. Combining this result with a Theorem by Buyalo and Schroeder we prove the additivity of the hyperbolic rank for products of manifolds with pinched negative curvature.
Let (X_i,d_i), i=1,2, be proper geodesic hyperbolic metric spaces. We give a general construction for a ``hyperbolic product'' X_1{times}_h X_2 which is itself a proper geodesic hyperbolic metric space and examine its boundary at infinity.
We introduce a notion of the Euclidean- and the Minkowski rank for arbitrary metric spaces and we study their behaviour with respect to products. We show that the Minkowski rank is additive with respect to metric products, while additivity of the Euclidean rank only holds under additional assumptions, e.g. for Riemannian manifolds. We also study products with nonstandard product metrics.
We introduce a new quasi-isometry invariant $\subcorank X$ of a metric space $X$ called {\it subexponential corank}. A metric space $X$ has subexponential corank $k$ if roughly speaking there exists a continuous map $g:X\to T$ such that for each $t\in T$ the set $g^{-1}(t)$ has subexponential growth rate in $X$ and the topological dimension $\dim T=k$ is minimal among all such maps. Our main result is the inequality $\hyprank X\le\subcorank X$ for a large class of metric spaces $X$ including all locally compact Hadamard spaces, where $\hyprank X$ is maximal topological dimension of $\di Y$ among all $\CAT(-1)$ spaces $Y$ quasi-isometrically embedded into $X$ (the notion introduced by M. Gromov in a slightly stronger form). This proves several properties of $\hyprank$ conjectured by M. Gromov, in particular, that any Riemannian symmetric space $X$ of noncompact type possesses no quasi-isometric embedding $\hyp^n\to X$ of the standard hyperbolic space $\hyp^n$ with $n-1>\dim X-\rank X$.