We study the set vol(M, G) of volumes of all representations ρ : π1M → G, where M is a closed oriented 3-manifold and G is either Iso+H 3 or Isoe SL2(R).By various methods, including relations between the volume of representations and the Chern-Simons invariants of flat connections, and recent results of surfaces in 3-manifolds, we prove that any 3-manifold M with positive Gromov simplicial volume has a finite cover M with vol( M, Iso+H 3 ) = {0}, and that any non-geometric 3-manifold M containing at least one Seifert piece has a finite cover M with vol( M, Isoe SL2(R)) = {0}.We also find 3-manifolds M with positive simplicial volume but vol(M, Iso+H 3 ) = {0}, and non-trivial graph manifolds M with vol(M, Isoe SL2(R)) = {0}, proving that it is in general necessary to pass to some finite covering to guarantee that vol(M, G) = {0}.Besides we determine vol(M, G) when M supports the Seifert geometry. ContentsDefinition 4.9. Let M be an orientable closed irreducible mixed 3-manifold containing no essential Klein bottles. Let J 0 be a JSJ piece, T 0 be a JSJ torus adjacent to J 0 and ζ 0 be a slope on T 0 . A partial PW subsurface j : R M is said to be virtually bounded by ζ 0 outside J 0 if the boundary ∂R of R is non-empty, covering ζ 0 under j, and if the interiorR of R misses J 0 under j. In this case, the carrier chunk X(R) ⊂ M of R is the unique minimal chunk that contains R, and the carrier boundary of X(R) is the component T 0 ⊂ ∂X(R).Definition 4.10. We say that a partial PW subsurface j : R M is parallel cutting if, for every JSJ torus T ⊂ M , all components of j −1 (T ) in R cover the same slope of T . Virtual existence of partial PW subsurfacesTheorem 4.11. Let M be an orientable closed irreducible mixed 3-manifold containing no essential Klein bottles. Let ζ 0 be a slope on a JSJ torus T 0 adjacent to a JSJ piece J 0 . Then, for some finite coverM of M together with an elevation (J 0 ,T 0 ,ζ 0 ) of the triple (J 0 , T 0 , ζ 0 ), Pierre Derbez I2M UMR 7373
Given a connected real Lie group and a contractible homogeneous proper Gspace X furnished with a G-invariant volume form, a real valued volume can be assigned to any representation ρ : π 1 (M ) → G for any oriented closed smooth manifold M of the same dimension as X. Suppose that G contains a closed and cocompact semisimple subgroup, it is shown in this paper that the set of volumes is finite for any given M . From a perspective of model geometries, examples are investigated and applications with mapping degrees are discussed.
In this paper, it is shown that for any closed orientable 3-manifold with positive simplicial volume, the growth of the Seifert volume of its finite covers is faster than the linear rate. In particular, each closed orientable 3manifold with positive simplicial volume has virtually positive Seifert volume. The result reveals certain fundamental difference between the representation volume of the hyperbolic type and the Seifert type. The proof is based on developments and reactions of recent results on virtual domination and on virtual representation volumes of 3-manifolds.2010 Mathematics Subject Classification. 57M50, 51H20.
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 , D(M, N ) is finite for all M ? The answer is known for prime 3-manifolds unless the target is a non-trivial graph manifold. We prove that for each closed non-trivial graph manifold N , 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 g PSL(2, R)-volume for certain class of graph manifolds.
By constructing certain maps, this note completes the answer of the Question: For which closed orientable 3-manifold N , the set of mapping degrees D(M, N ) is finite for any closed orientable 3-manifold M ? Date: June 9, 2018. 1991 Mathematics Subject Classification. 57M99, 55M25.
Abstract. This paper adresses the following problem: Given a closed orientable three-manifold M , are there at most finitely many closed orientable three-manifolds 1-dominated by M ? We solve this question for the class of closed orientable graph manifolds. More precisely the main result of this paper asserts that any closed orientable graph manifold 1-dominates at most finitely many orientable closed three-manifolds satisfying the Poincaré-Thurston Geometrization Conjecture. To prove this result we state a more general theorem for Haken manifolds which says that any closed orientable three-manifold M 1-dominates at most finitely many Haken manifolds whose Gromov simplicial volume is sufficiently close to that of M .
ABSTRACT. This paper shows that the Seifert volume of each closed non-trivial graph manifold is virtually positive. As a consequence, for each closed orientable prime 3-manifold N , the set of mapping degrees D(M, N ) is finite for any 3-manifold M , unless N is finitely covered by either a torus bundle, or a trivial circle bundle, or the 3-sphere.
Abstract.A natural problem in the theory of 3-manifolds is the question of whether two 3-manifolds are homeomorphic or not. The aim of this paper is to study this problem for the class of closed Haken manifolds using degree one maps.To this purpose we introduce an invariant .N / D .Vol.N /; kN k/, where kN k denotes the Gromov simplicial volume of N and Vol.N / is a 2-dimensional simplicial volume which measures the volume of the base 2-orbifolds of the Seifert pieces of N .After studying the behavior of .N / under the action of non-zero degree maps, we prove that if M and N are closed Haken manifolds such that kM k D jdeg.f /jkN k and Vol.M / D Vol.N / then any non-zero degree map f W M ! N is homotopic to a covering map. As a corollary we prove that if M and N are closed Haken manifolds such that .N / is sufficiently close to .M / then any degree one map f W M ! N is homotopic to a homeomorphism. Mathematics Subject Classification (2000). 57M50, 51H20.
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