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

Abstract

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