Half dimensional collapse of ends of manifolds of nonpositive curvature

TL;DR

This paper constructs a geometric analog of the rational Tits building for noncompact, finite volume manifolds with nonpositive curvature and proves it has dimension less than half the manifold's dimension.

Contribution

It introduces a new geometric analog of the rational Tits building for nonpositively curved manifolds and establishes a dimension bound.

Findings

Constructed a geometric analog of the rational Tits building.

Proved the analog has dimension less than half the manifold's dimension.

Extended understanding of the structure of nonpositively curved manifolds.

Abstract

This paper accomplishes two things. First, we construct a geometric analog of the rational Tits building for general noncompact, complete, finite volume $n$-manifolds $M$ of bounded nonpositive curvature. Second, we prove that this analog has dimension less than $\lfloor n/2\rfloor$.