Michelle Bucher (University of Geneva)

Title:
Continuous cocycles on the Furstenberg boundary and applications to bounded cohomology

Abstract:

Group cohomology comes in many variations. The standard Eilenberg-MacLane group cohomology is the cohomology of the cocomplex { f:G^q+1→ R | f is G-invariant} endowed with its natural homogeneous coboundary operator. Now whenever a property P of such cochains is preserved under the coboundary one can obtain the corresponding P-group cohomology. P could be: continuous, measurable, L⁰, bounded, alternating, etc. Sometimes these various cohomology groups are known to differ (eg P=empty and P=continuous for most topological groups), in other cases they are isomorphic (eg P=empty and P=alternating (easy), P=continuous and P=L⁰ (a highly nontrivial result by Austin and Moore valid for locally compact second countable groups)).

In 2006, Monod conjectured that for semisimple connected, finite center, Lie groups of noncompact type, the natural forgetful functor induces an isomorphism between continuous bounded cohomology and continuous cohomology (which is typically very wrong for discrete groups). I will focus here on the injectivity and show its validity in several new cases including isometry groups of hyperbolic n-spaces in degree 4, known previously only for n=2 by a tour de force due to Hartnick and Ott. 

This is joint work with Alessio Savini.

Schedule:
09:30-11:00 RTG Lecture 1 (Roman Sauer), SR. 2.058
11:00-11:30 Get-Together with Speaker, SR. 1.058
11:45-12:45 Colloquium, SR. 1.067
13:15 Lunch at Max Rubner Institute | Daily Menu
14:30 Coffee in math building, SR. 1.058
14:45-15:15 Informal meeting of PhD students, Room 1.062
15:15-16:45 RTG Lecture 1 (Roman Sauer), SR. 3.069

KA