G. Brock Williams

Earthquakes and circle packings

We prove that earthquakes on hyperbolic surfaces can be approximated by discrete earthquakes constructed using circle packings. Consequently, we obtain a combinatorial version of Thurston’s Earthquake Theorem. Any surface can be approximated by combinatorial earthquakes of a packable surface. This provides a controlled combinatorial method for deforming hyperbolic surfaces.

Approximation of Quasisymmetries Using Circle Packings

The Universal Teichmüller Space has recently gained the attention of physicists as a setting for developing string theory. Two common models for this space are the set of quasicircles and the set of quasisymmetries. The link between these models lies in the fact that the Riemann maps from

The sharp bound for the deformation of a disc under a hyperbolically convex map

\mathbb{D}

Noncompact surfaces are packable

and

Combinatorial excursions in moduli space

\mathbb{D}^*

Sharp bounds for the Schwarzian derivative for hyperbolically convex functions

to the complementary domains of quasicircles induce quasisymmetric automorphisms of