Abstract: The entropy of a quasifuchsian group agrees with the Hausdorff dimension of its limit set, and the entropy gives rise to an analytic function on the space of marked quasifuchsian groups. We find an unbounded open neighborhood of the Fuchsian locus in quasifuchsian space so that the only critical points of the entropy function lie on the Fuchsian locus. We also find an open neighborhood of the Fuchsian locus so that (the adjoint of) any quasifuchsian group in the neighborhood arises as the linear part of a proper affine action of the surface group on the Lie algebra of SL(2,C).
Both of these results are obtained by studying the infinitesimal behavior of bending deformations of quasifuchsian groups. This is joint work with Martin Bridgeman and Andres Sambarino.
