Property (T), fixed point properties and
Masato Mimura (Tohoku U., EPFL Lausanne)
Kazhdan's property (T) is one of the most
fundamental properties in the analytic counterpart of geometric group
theory. By Delorme--Guichardet theorem, this property is equivalent to
the fixed point property (FH) for locally compact sigma-compact groups:
A group is said to have property (FH) if every (continuous) action of it
on every Hilbert space by affine isometries admits a global fixed point.
We overview these two notions (the original definition of property (T)
has totally different nature to (FH)), and proceed to their strengthening
defined by replacing Hilbert spaces with other Banach spaces. They have
applications to coarse geometry via "(ordinary or Banach) expander graphs".
We discuss theory on these topics, including recent developments.
No serious backgrounds are assumed in this lecture series. Prerequisites
are similarities to examples of infinite groups (definitions of SL(n,Z) and
SL(n,R), etc.) and to basic functional analysis (definitions of Hilbert
spaces, orthogonal complement, unitary representations; Banach spaces,
continuous duals, reflexivity, quotient Banach spaces, Lebesgue and sequence
L_p spaces, etc.). Some definitions will be recalled during the lectures,
This will be a two-week course consisting of 6 x 2 hours of lectures and additional problem sessions led by Biswarup Das. The course is aimed at Master and PhD students (It is worth 3 ECTS points).
The lectures will take place on:
Tuesday, 3.10, 14:00-16:00, room 605
Wednesday, 4.10, 14:00-16:00, room 605
Friday, 6.10, 12:00-15:00, room 605
Tuesday, 10.10, 14:00-17:00, room 605
Wednesday, 11.10, 14:00-16:00, room 607
Thursday, 12.10, 12:00-14:00, room 605
Friday, 13.10, 12:00-16:00, room 605
The problem sessions will take place on
Tuesdays, 12:00--14:00, room 606. There will be at least six problem sessions during six weeks, starting Tuesday, 3.10.