On a group associated to z²-1

2002/03/23 by Laurent Bartholdi, Bartholdi, Laurent, Rostislav I. Grigorchuk +1
Mathematics · #20E08 #20F32 #37F10 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20E08 #msc:20F32 #msc:37F10

paper · pdf · doi:10.48550/arxiv.math/0203244

10 pages, 3 figures

arxiv created 2002/03/23 · arxiv updated 2009/11/30

Abstract

We construct a group acting on a binary rooted tree; this discrete group mimics the monodromy action of iterates of f(z)=z²-1 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.

Related

• Iterated Monodromy Groups
• The Spectral Problem, Substitutions and Iterated Monodromy
• Notes on the Schreier graphs of the Grigorchuk group
• Amenability of groups acting on trees
• A Contracting Fractal Group, Schreier Graphs and The Spectra