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ALGEBRAIC AND GEOMETRIC TOPOLOGY, vol.14, no.6, pp.3141-3184, 2014 (SCI-Expanded)
The log 3 theorem, proved by Culler and Shalen, states that every point in the hyperbolic 3-space H-3 is moved a distance at least log 3 by one of the noncommuting isometries xi or eta of H-3 provided that xi and eta generate a torsion-free, discrete group which is not cocompact and contains no parabolic. This theorem lies in the foundations of many techniques that provide lower estimates for the volumes of orientable, closed hyperbolic 3-manifolds whose fundamental groups have no 2-generator subgroup of finite index and, as a consequence, gives insights into the topological properties of these manifolds.