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On ends of finite-volume noncompact manifolds of nonpositive curvature

Ran Ji, Yunhui Wu

Abstract

In this paper we confirm a folklore conjecture which suggests that for a complete noncompact manifold $M$ of finite volume with sectional curvature $-1 \leq K \leq 0$, if the universal cover of $M$ is a visibility manifold, then the fundamental group of each end of $M$ is almost nilpotent.

On ends of finite-volume noncompact manifolds of nonpositive curvature

Abstract

In this paper we confirm a folklore conjecture which suggests that for a complete noncompact manifold of finite volume with sectional curvature , if the universal cover of is a visibility manifold, then the fundamental group of each end of is almost nilpotent.

Paper Structure

This paper contains 8 sections, 25 theorems, 175 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Boundedness implies nilpotency
  4. Nilpotency implies boundedness
  5. Zero algebraic entropy
  6. Proof of Theorem \ref{['mt-1']}
  7. Proofs of Corollary \ref{['c-rank']}, \ref{['c-nov']} and \ref{['c-zero']}
  8. Appendix: a visibility manifold with a finite volume quotient is not Gromov hyperbolic

Key Result

Theorem \oldthetheorem

Conjecture Eb-C is true. Moreover, the index can be bounded from above by a uniform constant depending only on the dimension.

Figures (2)

  • Figure 1:
  • Figure 2:

Theorems & Definitions (55)

  • Conjecture \oldthetheorem
  • Theorem \oldthetheorem
  • Conjecture \oldthetheorem: Modified Milnor Problem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem: Margulis Lemma at Infinity
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • Corollary \oldthetheorem
  • ...and 45 more