Algebraic & Geometric Topology

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

Pierre Derbez and Shicheng Wang

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds
Secondary: 51H20: Topological geometries on manifolds [See also 57-XX]

graph manifold nonzero degree maps volume of a representation


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.

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