Hyperbolic rank and subexponential corank of metric spaces

Abstract

We introduce a new quasi-isometry invariant corank X of a metric space X called subexponential corank. A metric space X has subexponential corank k if roughly speaking there exists a continuous map g : X → T such that for each t ∈ 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 rank h X ≤ corank X for a large class of metric spaces X including all locally compact Hadamard spaces, where rank h X is maximal topological dimension of ∂ ∞ 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 rank h conjectured by M. Gromov, in particular, that any Riemannian symmetric space X of noncompact type possesses no quasi-isometric embedding H n → X of the standard hyperbolic space H n with n − 1 > dim X − rank X.

Embedding of Hyperbolic Spaces in the Product of Trees

“…To make this question nontrivial, one should stabilize the product by an additional factor which has arbitrarily large dimension and small growth rate, e.g., by R m . It easily follows from results of our previous paper [BS1] that there is no quasi-isometric embedding H n → X, X = T 1 × • • • × T p × R m , for any p ≤ n − 2 and m ≥ 0. For the projection X → T 1 × • • • × T p defines a subexponential foliation of X of rank p = dim(T 1 × • • • × T p ), therefore, the subexponential corank of X is ≤ p, and by the main result of [BS1], the existence of H n → X implies n − 1 ≤ p. In fact, Theorem 1.1 is optimal w.r.t.…”

“…It easily follows from results of our previous paper [BS1] that there is no quasi-isometric embedding H n → X, X = T 1 × • • • × T p × R m , for any p ≤ n − 2 and m ≥ 0. For the projection X → T 1 × • • • × T p defines a subexponential foliation of X of rank p = dim(T 1 × • • • × T p ), therefore, the subexponential corank of X is ≤ p, and by the main result of [BS1], the existence of H n → X implies n − 1 ≤ p. In fact, Theorem 1.1 is optimal w.r.t. the number of trees in the product stabilized by R m .…”

Hyperbolic dimension of metric spaces

“…In our earlier paper [BS1] we introduced another quasi-isometry invariant of metric spaces, called the subexponential corank. This invariant gives an upper bound for the topological dimension of a Gromov hyperbolic space that can be quasi-isometrically embedded into a given metric space X, rank h (X) ≤ corank(X).…”

“…This invariant gives an upper bound for the topological dimension of a Gromov hyperbolic space that can be quasi-isometrically embedded into a given metric space X, rank h (X) ≤ corank(X). Thus, the corank is a useful tool for finding obstructions to such embeddings, and it works perfectly well in many cases; see [BS1] for the details. However, e.g., for quasi-isometric embeddings H n → T 1 × · · · × T k × R m it gives only k ≥ n−2, while hypdim gives the optimal inequality k ≥ n−1 by Corollary 1.2.…”

Dimension of asymptotic cones of Lie groups

“…However, the dimension of G/K, where K is a maximal compact subgroup of G, is a quasi-isometry invariant of G by a theorem of J. Roe [33,Proposition 3.33 and Corollary 3.35]. Other kinds of dimensions that provide quasi-isometry invariants are, among others, asymptotic dimension [18,14], subexponential corank [9], and hyperbolic dimension [10]. Here we focus on the covering dimension of the asymptotic cone, which was considered in Gromov's book [18,Chapter 2], in Burillo's paper [8] and in [24] in the context of solvable groups, in [3] in the case of the mapping class group, in [15] in connection with asymptotic Assouad-Nagata dimension, and in [21] in the context of nonpositively curved manifolds.…”