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

Abstract

For given closed orientable 3-manifolds $M$ and $N$ let $cD(M,N)$ be the set of mapping degrees from $M$ to $N$. We address the problem: For which $N$, $cD(M,N)$ 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 $N$, $cD(M,N)$ 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 $\widetilde{\mathrm{PSL}}(2,\mathbb{R})$-volume for certain class of graph manifolds.

15 pages 4 figures

References in corpus (1)

• Non-zero degree maps between $2n$-manifolds