In 1979 D.Mumford constructed a celebrated example of a fake projective plane,
but the same paper contains an outline of a much more general construction
of "fake" algebraic surfaces using groups acting on buildings.
We'll discuss explicit constructions of such groups,
old and new, and their connections with the algebraic surfaces.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/5312