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

TL;DR

This paper proves that for any closed non-trivial graph manifold N, the set of mapping degrees from any graph manifold M to N is finite, using standard forms of maps and PSL(2,R)-volume estimates.

Contribution

It establishes the finiteness of mapping degrees for all graph manifolds N and M, extending known results to non-trivial graph manifolds.

Findings

Finiteness of mapping degrees for non-trivial graph manifolds N.

Use of standard forms of maps between graph manifolds.

Estimation of PSL(2,R)-volume for certain graph manifolds.

Abstract

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.