Singular surfaces, mod 2 homology, and hyperbolic volume, II

If g is an integer ⩾2, and M is a closed simple 3-manifold such that π1(M) has a subgroup isomorphic to a genus-g surface group and dimZ2H1(M;Z2)⩾max(3g−1,6), we show that M contains a closed, incompressible surface of genus at most g. As an application we show that if M is a closed orientable hyperbolic 3-manifold such that , then dimZ2H1(M;Z2)⩽5.