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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.

We consider groups of orientation-preserving real analytic diffeomorphisms of the circle which have a finite image under the rotation number function. We show that if such a group is nondiscrete with respect to the $C^1$-topology then it has a finite orbit. As a corollary, we show that if such a group has no finite orbit then each of its subgroups contains either a cyclic group of finite index or a nonabelian free subgroup.

We study groups of circle diffeomorphisms whose action on the cylinder $\mathcal C=\mathbb S^1\times \mathbb S^1\setminus \Delta$ preserves a volume form. We first show that such a group is topologically conjugate to a subgroup of $\rm{PSL}(2,\mathbb R)$, then discuss the existence of a differentiable conjugacy.

We prove two theorems of reduction of cocycles taking values in the group of diffeomorphisms of the circle. They generalise previous results obtained by the author concerning rigidity for smooth actions on the circle of Kazhdan's groups and higher rank lattices.

This book covers many of the recent results on group actions on the circle, with an emphasis in the differentiable case.