  In this paper, we find lower bounds for volumes of hyperbolic 3-manifolds
with various topological conditions. Let V_3 = 1.01494 denote the volume of a
regular ideal simplex in hyperbolic 3-space. As a special case of the main
theorem, if a hyperbolic manifold M contains an acylindrical surface S, then
Vol(M)>= -2 V_3 chi(S). We also show that if beta_1(M)>= 2, then Vol(M)>= 4/5
V_3.
