6 feb 2025 -- 14:30
Aula Riunioni - Dipartimento di Matematica, Pisa
Abstract.
Group cohomology comes in many variations. The standard Eilenberg-MacLane group cohomology is the cohomology of the cocomplex $\{ f:G^{q+1}\rightarrow \mathbb{R}\mid f \mathrm{\ is \ } G\mathrm{-invariant}\}$ 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, $L^0$, 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=$L^0$ (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. Monod recently proved that all such continuous (bounded) cohomology classes can be represented by measurable (bounded) cocycles on the Furstenberg boundary. Our main result is that these cocycles can be chosen to be continuous on a subset of full measure. In the real hyperbolic case, this subset of full measure is the set of distinct tuples of points, easily leading to the injectivity in degree 4.
This is joint work with Alessio Savini.