Geometry and rank of fibered hyperbolic 3–manifolds

Geometry and Rank of Fibered Hyperbolic 3-manifolds

Recall that the rank of a finitely generated group is the minimal number of elements needed to generate it. In [Whi02], M. White proved that the injectivity radius of a closed hyperbolic 3-manifold M is bounded above by some function of rank(π1(M)). Building on a technique that he introduced, we determine the ranks of the fundamental groups of a large class of hyperbolic 3-manifolds fibering ov...

Geometry and Rank of Fibered

Recall that the rank of a finitely generated group is the minimal number of elements needed to generate it. M. White [Whi02] has proven that the injectivity radius of M is bounded above by some function of rank(π1(M)). Building on a technique that he introduced, we show that if M is a hyperbolic 3-manifold fibering over the circle with fiber Σg and rank(π1(M)) 6= 2g + 1, then the diameter of M ...

The Geometry of Rank 2 Hyperbolic Root Systems

Let ∆ be a rank 2 hyperbolic root system. Then ∆ has generalized Cartan matrix H(a, b) = ( 2 −b −a 2 ) indexed by a, b ∈ Z with ab ≥ 5. If a 6= b, then ∆ is non-symmetric and is generated by one long simple root and one short simple root, whereas if a = b, ∆ is symmetric and is generated by two long simple roots. We prove that if a 6= b, then ∆ contains an infinite family of symmetric rank 2 hy...

From Hyperbolic Polynomials to Real Fibered Morphisms

On Hyperbolic 4-manifolds Fibered over Surfaces

In this paper we prove that for any hyperbolic 4-manifold M , which is R 2 bundle over surface , the absolute value of the Euler number e() of the bration : M ! is not greater than exp(exp(10 8 j(()j)). This result partially corroborates a conjecture of Gromov, Lawson and Thurston that je()j j(()j.