The purpose of this book is to describe the global properties of complete simply connected
spaces that are non-positively curved in the sense of AD Alexandrov and to examine the …

Singular spaces with upper curvature bounds and, in particular, spaces of nonpositive
curvature, have been of interest in many fields, including geometric (and combinatorial) …

In this note, we characterize the graphs (1-skeletons) of some piecewise Euclidean
simplicial and cubical complexes having nonpositive curvature in the sense of Gromov's …

We begin by recalling some well-known facts. The natural action of the symmetric group Sn
on Rncan be viewed as a group generated by reflections. The reflections in Sn are the …

Harmonic maps between smooth Riemannian manifolds play a ubiquitous role in differential
geometry. Examples include geodesics viewed as maps, minimal surfaces, holomorphic …

We prove that numerous negatively curved simply connected locally compact polyhedral
complexes, admitting a discrete cocompact group of automorphisms, have automorphism …

We introduce a family of conditions on a simplicial complex that we call local k-largeness
(k≥ 6 is an integer). They are simply stated, combinatorial and easily checkable. One of our …

Given a finitely generated Coxeter group W, I described, in [D, Section 14], a certain
contractible simplicial complex, here denoted| W|, on which W acts properly with compact …

A conjecture of H. Hopf states that if M 2 n is a closed, Riemannian manifold of nonpositive
sectional curvature, then its Euler characteristic, χ (M 2 n), should satisfy (− 1) n χ (M 2 n)≥ …

This short monograph arose as an offshoot of the book on Alexandrov geometry we have
been writing for a number of years. The notes were shaped in a number of lectures given by …