Texas A&M University, Department of
Mathematics, 216 Milner Hall, 26th of January 2005, 3:00-3:50
Groups and Dynamics Seminar
Square complexes
and automaton groups
Zoran Sunik of Texas A&M University
Despite of the fact that two randomly chosen automorphisms of the
regular rooted tree almost surely generate a free group of rank 2,
explicit constructions of faithful actions of free non-abelian groups
on rooted trees are not as common. Examples of free groups generated by
finite automata were constructed by Brunner and Sidki, as well as by
Oliinyk in the late 1990's. However, these constructions do not provide
self-similar actions. An interesting candidate for a free group
generated by a finite automaton providing self-similar action on the
binary rooted tree was produced by Aleshin in the early 1980's, but to
this date the question of freeness of this group has not been resolved.
We will describe a recent construction by Glasner and Mozes involving
VH-T square complexes and arithmetic lattices in PGL(Q_p) x PGL(Q_p)
which provides the first known examples of finite automata leading to
self-similar actions of free non-abelian groups on rooted trees.