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Convegno
“HYPERBOLIC MANIFOLDS, THEIR SUBMANIFOLDS AND FUNDAMENTAL GROUPS”
IMPA, Rio de Janeiro, January 06–10. The conference will be followed by a workshop aimed at graduate students on similar topics on January 13 –17, 2025.
The main subject of the conference is about recent developments in hyperbolic geometry and hyperbolic group theory also covering some other closely related areas. The subsequent workshop will provide an in depth introduction to several key topics of the conference.
organizzato da: Alan Reid (Rice University) Matthew Stover (Temple University) Mikhail Belolipetsky (IMPA) Plinio Murillo (Universidade Federal Fluminense) Stefano Francaviglia (University of Bologna)
Elenco seminari
Gennaio
dal giorno
06/01/2025
al giorno
10/01/2025
06/01/2025
al giorno
10/01/2025
Roberto Frigerio
Relazione all'interno del convegno: Hyperbolic Manifolds, Their Submanifolds and Fundamental Groups
Seminario di algebra e geometria
Constructing higher degree non-trivial bounded coho-
mology classes is a very challenging task. For surface groups and
free groups, bounded cohomology is very rich in degree 2 and 3,
and a natural question is whether one can build non-trivial classes
in higher degrees by taking the cup product of lower-dimensional
classes. For hyperbolic manifolds, there exists a well defined map
Ψ• associating to every closed differential form a bounded coho-
mology class via integration over straight simplices. Classes in the
image of this map are usually called De Rham classes, and, in de-
gree 2, they span an infinite-dimensional subspace of the bounded
cohomology space of the manifold.
We prove that, in suitable degrees, Ψ• is a homomorphism of al-
gebras, i.e. it sends the wedge product of closed differential forms to
the cup product of the associated bounded cohomology classes. As a
corollary, the cup product of two De Rham classes vanishes, provided
that its degree exceeds the dimension of the manifold. This result
complements several recent vanishing results for the cup product of
De Rham classes.