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

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

2008-01-13

arxiv.org/abs/0801.1946v3

Mathematics

Geometric Topology

15 pages 4 figures

Scientific paper

For given closed orientable 3-manifolds $M$ and $N$ let $\c{D}(M,N)$ be the set of mapping degrees from $M$ to $N$. We address the problem: For which $N$, $\c{D}(M,N)$ is finite for all $M$? The answer is known in Thurston's picture of closed orientable irreducible 3-manifolds unless the target is a non-trivial graph manifold. We prove that for each closed non-trivial graph manifold $N$, $\c{D}(M,N)$ is finite for all graph manifold $M$. The proof uses a recently developed standard forms of maps between graph manifolds and the estimation of the $\widetilde{\rm PSL}(2,{\R})$-volume for certain class of graph manifolds.

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