On the Complexity and Volume of Hyperbolic Manifolds
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ar X iv :0 81 1. 42 74 v1 [ ma th. GT ] 26 N ov 20 08 On the Complexity and Volume of Hyperbolic 3-Manifolds. Thomas Delzant and Leonid Potyagailo Abstract We compare the volume of a hyperbolic 3-manifold M of finite volume and the complexity of its fundamental group. 1 1 Introduction. Complexity of 3-manifolds and groups. One of the most striking corollaries of the recent solution of the geometrization conjecture for 3-manifolds is the fact that every aspherical 3- manifold is uniquely determined by its fundamental group. It seems to be natural to think that a topological/geometrical description of a 3-manifold M produces the simplest way to describe its fundamental group π1(M); on the other hand, the simplest way to define the group π1(M) gives rise to the most efficient way to describe M. More precisely, we want to compare the complexity of 3-manifolds and their fundamental groups. The study of the complexity of 3-manifolds goes back to the classical work of H. Kneser [K]. Recall that the Kneser complexity invariant k(M) is defined to be the minimal number of simplices of a triangulation of the manifold M . The main result of Kneser is that this complexity serves as a bound of the number of embedded incompressible 2-spheres in M , and bounds the numbers of factors in a decomposition ofM as a connected sum.
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- torsion any
- group
- since every
- every finitely generated
- finite discrete
- all parabolic
- manifold
- kneser complexity
- hyperbolic
