[Binghamton Geometry/Topology Seminar] Mar 1

[Dmytro Savchuk]

                 BINGHAMTON GEOMETRY/TOPOLOGY SEMINAR

*Date:*  Thursday, Mar 1, 2012

*Time:*  2:50-3:50pm
*Place:*  Library North 2205 followed by coffee/tea in the Anderson Reading
Room.

*Speaker: *Phu Chung (SUNY Buffalo)
*Title: **Homoclinic groups, von Neumann algebras and algebraic actions *
*
Abstract:*   Homoclinic points describe the asymptotic behavior of group
actions on spaces and play an important role in general theory of dynamical
systems. In 1999, Doug Lind and Klaus Schmidt established relations between
homoclinic points and entropy properties for expansive algebraic actions of
Z^d. Their proof depends heavily on the commutative factorial Noetherian
ring structure of the integral group ring of Z^d.
In a joint work with Hanfeng Li, we extend their results to expansive
algebraic actions of polycyclic-by-finite groups. We use three ingredients
to do this: character- izations of expansive algebraic actions, local
entropy theory for actions of countable amenable groups on compact groups,
and comparison between entropies of dual algebraic actions.
Applying our results to the field of von Neumann algebras, we get a
positive answer to a question of Deninger about the Fuglede-Kadison
determinant to the case group is amenable. We also prove that for an
amenable group, an element in the integral group ring is a non-zero divisor
if and only if the entropy of corresponding principal algebraic action is
finite.
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