Arithmetic Fuchsian groups of genus two

A hyperbolic 2- or 3-orbifold M is called arithmetic if M = H^2/G or H^3/G where G is an arithmetic Fuchsian or Kleinian group, respectively. Considerable work has been done in the last two decades classifying these groups. I will explain how arithmetic Fuchsian and Kleinian groups can be described in terms of quaternion algebras over number fields and discuss some results and open questions regarding these groups. Time permitting, I will outline the classification of derived arithmetic Fuchsian groups of genus 2.