Combination Theorems for Discrete Convergence Groups

The goal of this thesis is to give an overview on combination theorems for Kleinian groups in 3-dimensional real hyperbolic space, and then generalize these to discrete convergence groups. A combination theorem, for us, will be a criterion under which two subgroups can be combined to form a new subgroup with a prescribed presentation while preserving geometrical properties of the initial subgroups. The inspiration for the generalizations are the two combination theorems by Maskit, which deal with amalgamated free products and HNN extensions.

The majority of the arguments Maskit uses involve the dynamics of a Kleinian group acting on the boundary of hyperbolic 3-space. In particular, he leverages a property of Kleinian groups called convergence dynamics. The more general setting we will introduce involves forgetting about the geometry inside hyperbolic space and just retaining these convergence dynamics on the boundary. This gives rise to discrete convergence groups, which includes Kleinian groups, along with discrete subgroups of the isometries of any rank 1 symmetric space, and many more examples.

These new combination theorems constitute joint work with Theodore Weisman.