  In these lectures we consider how algebraic properties of discrete subgroups
of Lie groups restrict the possible actions of those groups on surfaces. The
results show a strong parallel between the possible actions of such a group on
the circle $S^1$ and the measure preserving actions on surfaces.
  Our aim is the study of the (non)-existence of actions of lattices in a large
class of non-compact Lie groups on surfaces. A definitive analysis of the
analogous question for actions on $S^1$ was carried out by \'E. Ghys. Our
approach is topological and insofar as possible we try to isolate properties of
a group which provide the tools necessary for our analysis. The two key
properties we consider are almost simplicity and the existence of a distortion
element. Both will be defined and described in the lectures.
  Our techniques are almost all from low dimensional dynamics. But we are
interested in how algebraic properties of a group -- commutativity, nilpotence,
etc. affect the possible kinds of dynamics which can occur. For most of the
results we will consider groups of diffeomorphisms which preserve a Borel
probability measure.
