Lower bounds on volumes of hyperbolic Haken 3-manifolds

TL;DR

This paper establishes new lower bounds on the volumes of hyperbolic Haken 3-manifolds based on their topological features, such as the presence of acylindrical surfaces and Betti number conditions.

Contribution

It provides explicit volume lower bounds for hyperbolic 3-manifolds with specific topological properties, advancing understanding of geometric constraints.

Findings

Volume bounds for manifolds with acylindrical surfaces

Volume bounds for manifolds with Betti number ≥ 2

Quantitative relations between topology and hyperbolic volume

Abstract

In this paper, we find lower bounds for volumes of hyperbolic 3-manifolds with various topological conditions. Let V_3 = 1.01494 denote the volume of a regular ideal simplex in hyperbolic 3-space. As a special case of the main theorem, if a hyperbolic manifold M contains an acylindrical surface S, then Vol(M)>= -2 V_3 chi(S). We also show that if beta_1(M)>= 2, then Vol(M)>= 4/5 V_3.