#### 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
##### Abstract

For given closed orientable $3$–manifolds $M$ and $N$ let $\mathsc{D}\left(M,N\right)$ be the set of mapping degrees from $M$ to $N$. We address the problem: For which $N$ is $\mathsc{D}\left(M,N\right)$ 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 $N$, $\mathsc{D}\left(M,N\right)$ 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 $\stackrel{˜}{PSL}\left(2,R\right)$–volume for a certain class of graph manifolds.

##### Keywords
graph manifold, nonzero degree maps, volume of a representation
Primary: 57M50
Secondary: 51H20