On the complexity and volume of hyperbolic 3-manifolds

Abstract

We compare the volume of a hyperbolic 3-manifold M of finite volume and the complexity of its fundamental group. 1

Volume vs. Complexity of Hyperbolic Groups

“…The conjecture holds for non-uniform arithmetic lattices [13], and a slightly weaker bound was recently obtained in [15]. Other measures of complexity that are bounded from above by the co-volume include: Betti numbers of non-positively curved manifolds [2]; rank(G) for lattices in various symmetric spaces [3,12,14]; log( T or(H k (G; Z) ) for manifolds with normalized bounded negative curvature [1]; a relative T-invariant for hyperbolic 3-manifolds [10].…”