'Strange' rational maps and spectra of graphs and groups

I will present a few results concerning spectral theory of
graphs and groups that are based on the use of multidimentional rational
maps and self-similar groups. It will be explained how these maps
appear, how they help to compute the spectrum, and why they are
``strange'' (or better to say, non typical). As part of the story there
will be a short survey on spectra of infinite graphs and groups with
some old and new results belonging to the speaker and his collaborators
(L.Bartholdi, A.Dudko, D.Lenz, T.Nagnibeda, B.Simanek, A.Zuk). The
relation to the random Schrodinger operator will be mentioned at the end
of the talk if time permits.