Chern--Simons theory, surface separability, and volumes of 3-manifolds

Abstract

We study the set $\mathrm{vol}\left(M,G\right)$ of volumes of all representations $ρ\coπ_1M\to G$, where $M$ is a closed oriented $3$-manifold and $G$ is either ${\rm Iso}_+{\Hi}^3$ or ${\rm Iso}_e\t{\rm SL_2(\R)}$. 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 $\t M$ with ${\rm vol}(\t M,{\rm Iso}_+{\Hi}^3)\ne \{0\}$, and that any non-geometric 3-manifold $M$ containing at least one Seifert piece has a finite cover $\t M$ with ${\rm vol}(\t M,{\rm Iso}_e\t{\rm SL_2(\R)}) \ne \{0\}$. We also find 3-manifolds $M$ with positive simplicial volume but ${\rm vol}(M,{\rm Iso}_+{\Hi}^3)=\{0\}$, and non-trivial graph manifolds $M$ with ${\rm vol}(M,{\rm Iso}_e\t{\rm SL_2(\R)})=\{0\}$, proving that it is in general necessary to pass to some finite covering to guarantee that $\mathrm{vol}(M,G)\ne \{0\}$. Besides we determine $\mathrm{vol}\left(M,G\right)$ when $M$ supports the Seifert geometry.

43 pages. arXiv admin note: substantial text overlap with arXiv:1111.6153

Cited by in corpus (7)

• Virtual Domination of $3$-manifolds
• NonLERFness of arithmetic hyperbolic manifold groups and mixed 3-manifold groups
• Positive simplicial volume implies virtually positive Seifert volume for $3$-manifolds
• Cohomological invariants of representations of 3-manifold groups
• Immersing quasi-Fuchsian surfaces of odd Euler characteristic in closed hyperbolic 3-manifolds
• Virtual domination of 3-manifolds II
• Volume of Seifert representations for graph manifolds and their finite covers