# Decomposition and inertia subgroups of a group of birational automorphisms of an algebraic surface

I will discuss a problem of whether an automorphism of an algebraic surface of positive entropy can fix a smooth rational curve on the surface pointwisely. This problem was raised by A. Coble almost a hundred years ago and the first such example has been found only recently by John Lesieutre. I will discuss an example of a group of automorphisms of a rational or a K3 surface that is isomorphic to a non-elementary discrete group of isometries of a hyperbolic space that leaves a smooth rational curve invariant and hence admits a non-trivial homomorphism to the group of Moebius transformations. The problem of injectivity of this map is related to the problem of freeness of some rotation group associated with a regular tetrahedron.