Texas A&M University, Department of
Mathematics, 216 Milner Hall, 14th of February 2007, 3:00-3:50
Groups and Dynamics Seminar
Reidemeister
numbers and the twisted Burnside-Frobenius Theorem
Evgeny Troitskiy of Moscow State
University, Russia
Let G be a (countable discrete) group and f be an automorphism of G. A
twisted conjugacy, or f-conjugacy, class in G is the equivalence class
defined by "x is equivalent to g x f(g^-1), for g in G". If f=Id this
is exactly the definition of usual conjugacy classes. For a number of
applications it is very important to identify the number R(f) of such
classes (Reidemeister number) and the number of fixed points of the
associated homeomorphism of some (appropriate) dual object (provided
R(f) is finite).
We present results concerning polycyclic-by-finite groups. We also plan
to discuss some important examples, to give a complete answer to a
"weak question" (based on a noncommutative Riesz representation
theorem), and to give some relations to the twisted Dehn problem. This
general presentation in some parts needs advanced Functional Analysis.
(parts of the talk are based on joint works with A. Felshtyn and A.
Vershik)