## Algebraic & Geometric Topology

### 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 $D(M,N)$ be the set of mapping degrees from $M$ to $N$. We address the problem: For which $N$ is $D(M,N)$ 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 $N$, $D(M,N)$ 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 $PSL˜(2,R)$–volume for a certain class of graph manifolds.

#### Article information

Source
Algebr. Geom. Topol., Volume 9, Number 3 (2009), 1727-1749.

Dates
Received: 23 March 2009
Revised: 17 July 2009
Accepted: 27 July 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513797042

Digital Object Identifier
doi:10.2140/agt.2009.9.1727

Mathematical Reviews number (MathSciNet)
MR2539193

Zentralblatt MATH identifier
1187.57021

#### Citation

Derbez, Pierre; Wang, Shicheng. Finiteness of mapping degrees and ${\rm PSL}(2,\mathbf{R})$–volume on graph manifolds. Algebr. Geom. Topol. 9 (2009), no. 3, 1727--1749. doi:10.2140/agt.2009.9.1727. https://projecteuclid.org/euclid.agt/1513797042

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