Half dimensional collapse of ends of manifolds of nonpositive curvature
  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 $\lfloor n/2\rfloor$. 
Grigori Avramidi, T. Tam Nguyen Phan
math.GT math.DG math.MG