Topological and geometric properties of graph-manifolds

Buyalo, S.; Svetlov, P.

doi:10.1090/S1061-0022-05-00852-6

Topological and geometric properties of graph-manifolds
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by S. Buyalo and P. Svetlov
Translated by: the authors

St. Petersburg Math. J. 16 (2005), 297-340

DOI: https://doi.org/10.1090/S1061-0022-05-00852-6

Published electronically: March 9, 2005

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Abstract:

This is a unified exposition of results (obtained by different authors) on the existence of $\pi _1$-injective immersed and embedded surfaces in graph-manifolds, and also of nonpositively curved metrics on graph-manifolds. The basis for unification is provided by the notion of compatible cohomology classes and by a certain difference equation on the graph of a graph-manifold (the BKN-equation). Criteria for seven different properties of graph-manifolds are given at three levels: at the level of compatible cohomology classes; at the level of solutions of the BKN-equation; and in terms of spectral properties of operator invariants of a graph-manifold.

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