Chern--Simons theory, surface separability, and volumes of 3-manifolds
| Autor: | Derbez, Pierre, Liu, Yi, Wang, Shicheng |
|---|---|
| Rok vydání: | 2013 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1112/jtopol/jtv023 |
| Popis: | We study the set ${\rm vol}\left(M,G\right)$ of volumes of all representations $\rho\co\pi_1M\to G$, 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 ${\rm vol}(M,G)\not=\{0\}$. Besides we determine ${\rm vol}\left(M, G \right)$ when $M$ supports the Seifert geometry. Comment: 43 pages. arXiv admin note: substantial text overlap with arXiv:1111.6153 |
| Databáze: | arXiv |
| Externí odkaz: |
|