#### Volume 24, issue 4 (2020)

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Graph manifolds as ends of negatively curved Riemannian manifolds

### Koji Fujiwara and Takashi Shioya

Geometry & Topology 24 (2020) 2035–2074
##### Abstract

Let $M$ be a graph manifold such that each piece of its JSJ decomposition has the ${ℍ}^{2}×ℝ$ geometry. Assume that the pieces are glued by isometries. Then there exists a complete Riemannian metric on $ℝ×M$ which is an “eventually warped cusp metric” with the sectional curvature $K$ satisfying $-1\le K<0$.

A theorem by Ontaneda then implies that $M$ appears as an end of a $4$–dimensional, complete, noncompact Riemannian manifold of finite volume with sectional curvature $K$ satisfying $-1\le K<0$.

 Dedicated to Professor Kenji Fukaya on his 60th birthday
##### Keywords
ends of manifolds, negative curvature, graph manifold, cusp
##### Mathematical Subject Classification 2010
Primary: 53C20
Secondary: 57M50, 57N10