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.
Categories: math.GT math.DG