Numerical invariants for measurable cocycles

Sarti, Filippo (2022) Numerical invariants for measurable cocycles, [Dissertation thesis], Alma Mater Studiorum Università di Bologna. Dottorato di ricerca in Matematica, 34 Ciclo. DOI 10.48676/unibo/amsdottorato/10160.
Documenti full-text disponibili:
[img] Documento PDF (English) - Richiede un lettore di PDF come Xpdf o Adobe Acrobat Reader
Disponibile con Licenza: Salvo eventuali più ampie autorizzazioni dell'autore, la tesi può essere liberamente consultata e può essere effettuato il salvataggio e la stampa di una copia per fini strettamente personali di studio, di ricerca e di insegnamento, con espresso divieto di qualunque utilizzo direttamente o indirettamente commerciale. Ogni altro diritto sul materiale è riservato.
Download (810kB)

Abstract

The theory of numerical invariants for representations can be generalized to measurable cocycles. This provides a natural notion of maximality for cocycles associated to complex hyperbolic lattices with values in groups of Hermitian type. Among maximal cocycles, the class of Zariski dense ones turns out to have a rigid behavior. An alternative implementation of numerical invariants can be given by using equivariant maps at the level of boundaries and by exploiting the Burger-Monod approach to bounded cohomology. Due to their crucial role in this theory, we prove existence results in two different contexts. Precisely, we construct boundary maps for non-elementary cocycles into the isometry group of CAT(0)-spaces of finite telescopic dimension and for Zariski dense cocycles into simple Lie groups. Then we approach numerical invariants. Our first goal is to study cocycles from complex hyperbolic lattices into the Hermitian group SU(p,q). Following the theory recently developed by Moraschini and Savini, we define the Toledo invariant by using the pullback along cocycles, also by involving boundary maps. For cocycles Γ × X → SU(p,q) with 1<p<q<+∞, we prove that maximality and Zariski density imply superrigidity in the sense of Zimmer, namely such cocycles come from representations PU(1,n) → SU(p,q) of the ambient group. As a consequence, there is no Zariski dense such cocycle when 1<p<q. Then we move to cocycles Γ × X → PU(p,∞) where PU(p,∞) is the infinite dimensional version of PU(p,q). We show that maximal cocycles are reducible, namely that, modulo cohomology, their image is contained in a finite dimensional algebraic subgroup of PU(p,∞). Finally, we classify Zariski dense measurable cocycles Γ × X → G from finitely generated groups into Hermitian groups not of tube-type. Precisely, we show that the pullback of the Kahler class completely determines the cohomology class of such cocycles.

Abstract
Tipologia del documento
Tesi di dottorato
Autore
Sarti, Filippo
Supervisore
Co-supervisore
Dottorato di ricerca
Ciclo
34
Coordinatore
Settore disciplinare
Settore concorsuale
Parole chiave
measurable cocycle, rigidity, complex hyperbolic lattice, boundary, Hermitian space, CAT(0)-space, bounded cohomology, ergodic
URN:NBN
DOI
10.48676/unibo/amsdottorato/10160
Data di discussione
20 Giugno 2022
URI

Altri metadati

Statistica sui download

Gestione del documento: Visualizza la tesi

^