Finiteness of mapping degrees and PSL(2,R)–volume on graph manifolds

Derbez, Pierre; Wang, Shicheng

doi:10.2140/agt.2009.9.1727

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Finiteness of mapping degrees and ${\rm PSL}(2,\mathbf{R})$–volume on graph manifolds

Pierre Derbez and Shicheng Wang

Algebraic & Geometric Topology 9 (2009) 1727–1749

DOI: 10.2140/agt.2009.9.1727

Abstract

For given closed orientable $3$ –manifolds $M$ and $N$ let $D (M, N)$ be the set of mapping degrees from $M$ to $N$ . We address the problem: For which $N$ is $D (M, N)$ finite for all $M$ ? The answer is known for prime $3$ –manifolds unless the target is a nontrivial graph manifold. We prove that for each closed nontrivial graph manifold $N$ , $D (M, N)$ is finite for any graph manifold $M$ .

The proof uses a recently developed standard form of maps between graph manifolds and the estimation of the $\tilde{PSL} (2, R)$ –volume for a certain class of graph manifolds.

Keywords

graph manifold, nonzero degree maps, volume of a representation

Mathematical Subject Classification 2000

Primary: 57M50

Secondary: 51H20

References

Publication

Received: 23 March 2009

Revised: 17 July 2009

Accepted: 27 July 2009

Published: 2 September 2009

Authors

Pierre Derbez
CMI Technopôle Château-Gombert 39 rue Joliot Curie 13453 Marseille Cedex 13 France

Shicheng Wang
Department of Mathematics Peking University Beijing, 100871 China

Volume 9, issue 3 (2009)