The classical Klein Combination Theorem offers sufficient conditions to construct new Kleinian groups. Building on this foundation, Maskit introduced profound generalizations of this theorem, expanding its applicability. A special feature of Maskit Combination Theorems is that they furnish sufficient conditions that ensure the resulting combined group retains desirable geometric properties, such as convex cocompactness or geometric finiteness.
More recently, Anosov groups have emerged as a natural higher-rank extension of the convex cocompact Kleinian groups, exhibiting their robust geometric and dynamical characteristics. In this talk, I will discuss my joint work with Michael Kapovich on Combination Theorems tailored specifically for Anosov groups.