Hyperbolic groups with boundary an n-dimensional Sierpinski space

For [Formula: see text], we show that if [Formula: see text] is a torsion-free hyperbolic group whose visual boundary [Formula: see text] is an [Formula: see text]-dimensional Sierpinski space, then [Formula: see text] for some aspherical [Formula: see text]-manifold [Formula: see text] with non-empty boundary. Concerning the converse, we construct, for each [Formula: see text], examples of aspherical manifolds with boundary, whose fundamental group [Formula: see text] is hyperbolic, but with visual boundary [Formula: see text] not homeomorphic to [Formula: see text]. Our examples even support (metric) negative curvature, and have totally geodesic boundary.

A GEOMETRIC STUDY OF WASSERSTEIN SPACES: HADAMARD SPACES

Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space [Formula: see text]. In this paper we investigate the geometry of [Formula: see text] when X is a Hadamard space, by which we mean that X has globally non-positive sectional curvature and is locally compact. Although it is known that, except in the case of the line, [Formula: see text] is not non-positively curved, our results show that [Formula: see text] have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for [Formula: see text] that enables us to prove a non-embeddablity result: if X has the visibility property, then the Euclidean plane does not admit any isometric embedding in [Formula: see text].