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 π 1 ( M ) has a subgroup isomorphic to a genus-g surface group and dim Z 2 H 1 ( M ; Z 2 ) ⩾ max ( 3 g − 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 Vol M ⩽ 3.08 , then dim Z 2 H 1 ( M ; Z 2 ) ⩽ 5 .