Iterated monodromy group of the map $z \to z^2+i$ as an example of L-presented group

Geometry & Topology
Event time: 
Tuesday, March 22, 2011 - 12:30pm to Monday, March 21, 2011 - 8:00pm
431 DL
Dmytro Savchuk
Speaker affiliation: 
Binghamton University
Event description: 

A relatively recent discovery (mainly due Nekrashevych) of the beautiful connection between holomorphic dynamics and groups generated by automata is realized by iterated monodromy groups. Here is a rough idea: with every (nice) d-fold self-covering of a (nice) topological space X one can associate the group acting on the tree of the preimages of any point in X by automorphisms. It can be shown that under certain labeling of the tree this group can be generated by states of an automaton, which, in turn, may be used to understand the structure of the group.

In particular, it is known that many iterated monodromy groups admit L-presentation. L-presented groups are finitely generated groups that admit a presentation involving finitely many relators and their iterations by substitution. Such presentations are at the first level of complexity after the finite presentations and quite often provide the simplest way to describe a group that is not finitely presented. Further, such presentations can be used to embed a group into a finitely presented group in a way that preserves many properties of the original group. In this talk I will concentrate on an iterated monodromy group of the map $z \to z^2+i$. After an introduction to the basic concepts, I will show how to compute the automaton generating this group and give an idea how to obtain an L-presentation.