Incompressible surfaces, hyperbolic volume, Heegaard genus and homology

Abstract

We show that if M is a complete, finite-volume, hyperbolic 3-manifold having exactly one cusp, and if dim Z2 H 1 (M ; Z 2 ) ≥ 6, then M has volume greater than 5.06. We also show that if M is a closed, orientable hyperbolic 3-manifold with dim Z2 H 1 (M ; Z 2 ) ≥ 4, and if the image of the cup product mapZ 2 ) has dimension at most 1, then M has volume greater than 3.08. The proofs of these geometric results involve new topological results relating the Heegaard genus of a closed Haken manifold M to the Euler characteristic of the kishkes of the complement of an incompressible surface in M .

Volume and homology of one-cusped hyperbolic 3–manifolds

“…1 .N / were injective then˛n 1 would be a boundary slope, in contradiction to our choice of n 1 . On the other hand, if As we mentioned in the Introduction, in [10] it will be shown that the hypothesis of Proposition 10.1 implies conclusion (a). The proof of this stronger result uses Proposition 10.1.…”

“…In work with DeBlois [10], the following strictly stronger version of Proposition 10.1 will be proved: If M is a complete, finite-volume, orientable hyperbolic manifold having exactly one cusp, such that dim Z 2 H 1 .M I Z 2 / 7, then vol M > 5:06. The proof of the stronger result takes Proposition 10.1 as a starting point, and involves a careful analysis of the case where M contains an incompressible surface of genus 2.…”

Volume and topology of bounded and closed hyperbolic $3$-manifolds

“…Then M has volume greater than 6.89. Theorem 1.3 is analogous to [6,Theorem 6.5], and follows in a similar way: we apply Theorem 1.1 and the results of Miyamoto and KojimaMiyamoto discussed above, using work of Agol et al [2], to the output of the topological theorem below. The notation in the statement is taken from [6].…”

“…Theorem 1.3 is analogous to [6,Theorem 6.5], and follows in a similar way: we apply Theorem 1.1 and the results of Miyamoto and KojimaMiyamoto discussed above, using work of Agol et al [2], to the output of the topological theorem below. The notation in the statement is taken from [6]. In particular, below and in the remainder of this paper, we will use the term "simple" as it is defined in [6, Definitions 1.1], which differs from its usage in [17] mentioned above.…”

“…Nonetheless it seems to be the only result of its kind in the literature. Theorem 1.2 will be deduced by combining Theorem 1.1 with results from [6,8]. The transition from these results to Theorem 1.2 involves two other results, Theorems 1.3 and 1.4 below, which are of independent interest.…”