Volume 9, issue 3 (2009)

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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

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 PSL˜(2,R)–volume for a certain class of graph manifolds.

graph manifold, nonzero degree maps, volume of a representation
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 51H20
Received: 23 March 2009
Revised: 17 July 2009
Accepted: 27 July 2009
Published: 2 September 2009
Pierre Derbez
Technopôle Château-Gombert
39 rue Joliot Curie
13453 Marseille Cedex 13
Shicheng Wang
Department of Mathematics
Peking University
Beijing, 100871