On a group associated to $z^2-1$
  We construct a group acting on a binary rooted tree; this discrete group mimics the monodromy action of iterates of $f(z)=z^2-1$ on associated coverings of the Riemann sphere.   We then derive some algebraic properties of the group, and describe for that specific example the connection between group theory, geometry and dynamics.   The most striking is probably that the quotient Cayley graphs of the group (aka ``Schreier graphs'') converge to the Julia set of $f$. 
Laurent Bartholdi and Rostislav I. Grigorchuk