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

We study the set $\mathrm{v}\mathrm{o}\mathrm{l}(M,G)$ of volumes of all representations $\rho \text{co}{\pi }_{1}M\to G$, where $M$ is a closed oriented $3$-manifold and $G$ is either ${\mathrm{I}\mathrm{s}\mathrm{o}}_{+}{\text{Hi}}^3$ 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 $\text{t}M$ with ${\rm vol}(\t M,{\rm Iso}+{\Hi}^3)\ne {0}$,andthatanynon-geometric3-manifold$M$ containing at least one Seifert piece has a finite cover $\text{t}M$ 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 ${\rm vol}(M,{\rm Iso}+{\Hi}^3)={0}$,andnon-trivialgraphmanifolds$M$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 $\mathrm{v}\mathrm{o}\mathrm{l}(M,G)$ when $M$ supports the Seifert geometry.