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Theorem 0.1. Suppose M is a closed hyperbolic 3-manifold. Given a presentation of π1M let L be the sum of the word-lengths of the relations and n the number of relations of length at least 3. Then volume(M) < π(L− 2n). Proof. A pleated disc is a map covered by a map of a disc into H such that there is a triangulation of the disc with vertices only on the boundary of the disc and with the property that the image of each 2-simplex is a geodesic 2-simplex in H. A presentation of M gives a set of generators and relations. For simplicity, we will assume every relation has word length at least 3. This may be realized geometrically by a map f : S −→M of a 2-complex S which induces an isomorphism of π1S onto π1M. The map f may be homotoped so that edges map to geodesics and f restricted to each 2-cell is a pleated disc. The area of a pleated disc is at most π times the number of 2-simplices. The boundary of a 2-cell, D, in S represents a relation, and the number of 2-simplices in D is the number of edges in ∂D minus 2. The number of edges in ∂D is the word length of the relation represented by ∂D. Thus the total surface area of f(S) is at most π(L− 2n). Let X be the closure of a component of M − f(S); then X lifts to H. For otherwise, there is a loop γ in X which is not contractible in M. Since S is mapped into M − γ, the isomorphism f∗ : π1S −→ π1M factors through π1(M − γ). Thus the composite π1M ∼= π1S −→ π1(M − γ) −→ π1M is the identity, where the second map is induced by inclusion. Now M is aspherical, hence π2(M − γ) = 0 because otherwise, by the sphere theorem, γ would be contained inside a ball and thus contractible in M. Hence M − γ is a K(π, 1) and thus the first homomorphism is induced by a continuous map M −→ (M − γ). Thus the composite M −→ (M − γ) −→M is a π1-isomorphism, hence a homotopy equivalence. Consideration of the induced map on H3 gives a contradiction: H3(M) −→ H3(M − γ) −→ H3(M) since the composite is an isomorphism and M is closed.

The volume of a closed hyperbolic 3-manifold is bounded by 𝜋 times the length of any presentation of its fundamental group