Free Kleinian groups and volumes of hyperbolic {3}-manifolds

Dehn surgery, homology and hyperbolic volume

“…As in [4], we shall say that a group is semifree if it is a free product of free abelian groups; and we shall say that a group is k -semifree if every subgroup of whose rank is at most k is semifree. Note that is 2-semifree if and only if every rank-2 subgroup of is either free or free abelian.…”

The visual core of a hyperbolic 3-manifold

“…On the one hand, one may think of it as a geometric analogue of the main result from [3], which asserts that under the same hypotheses, there is a compact core for the algebraic limit manifold which embeds in the geometric limit manifold. On the other hand, Theorem 4.2 can be thought of as a generalization of the result, proven in [4], that when the algebraic limit is a maximal cusp, the convex core of the algebraic limit manifold embeds in the geometric limit manifold. In fact, when Γ is a maximal cusp, the visual and convex cores of H 3 /Γ coincide.…”

“…In fact, one may view our Theorem 4.2 and the result from [3] that asserts that, under the same assumptions, a compact core for the algebraic limit manifold embeds in the geometric limit manifold, as two different generalizations of Proposition 3.2 from [4]. (2) In general, even if the algebraic limit is a generalized web group, the convex core of the algebraic limit manifold need not embed in the geometric limit manifold.…”

“…Remarks: (1) One may think of Theorem 4.2 as one way to generalize Proposition 3.2 from [4], which shows that if the algebraic limit is a maximal cusp, then the convex core of the algebraic limit manifold embeds in the geometric limit manifold under the covering map. In fact, one may view our Theorem 4.2 and the result from [3] that asserts that, under the same assumptions, a compact core for the algebraic limit manifold embeds in the geometric limit manifold, as two different generalizations of Proposition 3.2 from [4].…”

Incompressible surfaces, hyperbolic volume, Heegaard genus and homology

“…We recall that a group Γ is said to be k-free for a given positive integer k if every subgroup of Γ having rank at most k is free. According to Corollary 9.3 of [1], which was deduced from results in [3], if M is a closed, orientable hyperbolic 3-manifold such that π 1 (M ) is 3-free then vol M > 3.08. Now suppose that M satisfies the hypotheses of Theorem 6.3, but that π 1 (M ) is not 3-free.…”