Hyperbolic rank and subexponential corank of metric spaces

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.Comment: 12 page

Incidence axioms for the boundary at infinity of complex hyperbolic spaces

We characterize the boundary at infinity of a complex hyperbolic space as a compact Ptolemy space that satisfies four incidence axioms.Comment: 48 pages, 3 figure