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

We study the set vol(M,G) vol(M,G) of volumes of all representations ρ:π_1 M→G, where M is a closed oriented 3-manifold and G is either Iso + H^3 or Iso_eSL_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 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, Iso_eSL_2(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, Iso_eSL_2(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.