Volume and homology of one-cusped hyperbolic 3-manifolds (Q2426837)

Let \(M\) be a complete, finite-volume, orientable hyperbolic 3-manifold having exactly one cusp, such that \(\pi_1(M)\) has no subgroup isomorphic to a genus 2 surface group. Suppose that either (a) \(\dim_{\mathbb{Z}_p}H_1(M;\mathbb{Z}_p)\geq 5\) for some prime \(p\), or (b) \(\dim_{\mathbb{Z}_2}H_1(M;\mathbb{Z}_2)\geq 4\), and the subspace of \(H^2(M;\mathbb{Z}_2)\) spanned by the image of the cup product \(H_1(M;\mathbb{Z}_2)\times H^1(M;\mathbb{Z}_2)\to H^2(M;\mathbb{Z}_2)\) has dimension at most one. Then the main result of the paper states that the volume of \(M\) is greater than 5.06. Conversely, one can think of this result as saying that if the one-cusped hyperbolic 3-manifold \(M\) has volume at most 5.06, then \(\dim_{\mathbb{Z}_p}H^1(M;\mathbb{Z}_p)\leq 4\) for every prime \(p\); and that if in addition one assumes that the span of the image of the cup product has dimension at most one, then \(\dim_{\mathbb{Z}_2}H^1(M; \mathbb{Z}_2)\leq 3\). The authors do not know whether these bounds on homology are sharp. The Weeks-Hodgson census contains many examples of one-cusped manifolds with volume less than 5.06 for which \(\dim_{\mathbb{Z}_2}H^1(M;\mathbb{Z}_2)= 3\). The authors have not calculated the cup product for these examples. Another results says that if one assumes furthermore that \(\dim_{\mathbb{Z}_2}H_1(M; \mathbb{Z}_2)\geq 7\), then the volume of \(M\) is greater than 3.66.