On a group associated to $z^2-1$

Details On a group associated to $z^2-1$ On a group associated to $z^2-1$

2002-03-23

arxiv.org/abs/math/0203244v1

Mathematics

Group Theory

10 pages, 3 figures

Scientific paper

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 $f$.

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