UC Santa Barbara

This thesis presents three projects whose common thread is the study of representations of orbifold and surface groups. These projects move from studying orbifold groups in PSL(2,R) using tools from hyperbolic geometry to exploring surface subgroups in PSL(n,R) with n > 2 using algebraic, dynamic, and geometric group theoretic approaches. In the first project, we build a new infinite family of non-commensurable pseudomodular groups obtained via the jigsaw method. The second project is concerned with obtaining families of Zariski dense rational surface group representations into SL(n,R) for odd n > 2 by bending orbifold representations. The final project uses the composition of Hitchin representations into PSL(3,R) with a generalization of the irreducible representation from PSL(2,R) to PSL(n,R) to construct families of Anosov representations outside the Hitchin component.