We show that for every finite-volume hyperbolic 3-manifold M and every prime p we have dim H 1 (M ; F p ) < 168.05 • vol M . This improves on a result proved by Agol, Leininger and Margalit giving the same inequality with the coefficient 168.05 replaced by 334.08, and on the analogous result with a coefficient of about 260 which would be obtained by combining the arguments due to Agol, Leininger and Margalit with a result due to Böröczky. Our inequality involving homology rank is deduced from a result about the rank of the fundamental group: if M is a finite-volume orientable hyperbolic 3-manifold such that π 1 (M ) is 2-semifree, then rank π 1 (M ) < 1 + λ 0 • vol M , where λ 0 is a certain constant less than 167.79.

The results of Culler and Shalen for 2, 3 or 4-free hyperbolic 3-manifolds are contingent on properties specific to and special about rank two subgroups of a free group.Here we determine what construction and algebraic information is required in order to make a geometric statement about M , a closed, orientable hyperbolic 3-manifold with kfree fundamental group, for any value of k greater than four. Main results are both to show what the formulation of the general statement should be, for which Culler and Shalen's result is a special case, and that it is true modulo a group-theoretic conjecture. A major result is in the k = 5 case of the geometric statement. Specifically, we show that the required grouptheoretic conjecture is in fact true in this case, and so the proposed geometric statement when M is 5-free is indeed a theorem. One can then use the existence of a point P and knowledge about π 1 (M, P ) resulting from this theorem to attempt to improve the known lower bound on the volume of M , which is currently 3.44 [8, Theorem 1.5].

No abstract

We investigate the geometry of closed, orientable, hyperbolic 3-manifolds whose fundamental groups are k-free for a given integer k ≥ 3. We show that any such manifold M contains a point P with the following property: If S is the set of elements of π 1 (M, P ) represented by loops of length < log(2k−1), then for every subset T ⊂ S, we have rank T ≤ k − 3. This generalizes to all k ≥ 3 results proved in [6] and [10], which have been used to relate the volume of a hyperbolic manifold to its topological properties, and it strictly improves on the result obtained in [11] for k = 5. The proof avoids the use of results about ranks of joins and intersections in free groups that were used in [10] and [11].

Under mild topological restrictions, we obtain new linear upper bounds for the dimension of the mod p homology (for any prime p) of a finite-volume orientable hyperbolic 3-manifold M in terms of its volume. A surprising feature of the arguments in the paper is that they require an application of the Four Color Theorem.If M is closed, and either (a) π 1 (M ) has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus 2, 3 or 4, or (b) p = 2, and M contains no (embedded, two-sided) incompressible surface of genus 2, 3 or 4, then dim H 1 (M ; F p ) < 157.763•vol(M ). If M has one or more cusps, we get a very similar bound assuming that π 1 (M ) has no subgroup isomorphic to the fundamental group of a closed, orientable surface of genus g for g = 2, . . . , 8. These results should be compared with those of our previous paper "The ratio of homology rank to hyperbolic volume, I," in which we obtained a bound with a coefficient in the range of 168 instead of 158, without a restriction on surface subgroups or incompressible surfaces. In a future paper, using a much more involved argument, we expect to obtain bounds close to those given by the present paper without such a restriction.The arguments also give new linear upper bounds (with constant terms) for the rank of π 1 (M ) in terms of vol M , assuming that either π 1 (M ) is 9-free, or M is closed and π 1 (M ) is 5-free.