12 research outputs found
Maximal representations of complex hyperbolic lattices in SU(m,n)
Let denote a lattice in , with greater than 1. We show
that there exists no Zariski dense maximal representation with target
if . The proof is geometric and is based on the study of the rigidity
properties of the geometry whose points are isotropic -subspaces of a
complex vector space endowed with a Hermitian metric of signature
and whose lines correspond to the dimensional subspaces of on
which the restriction of has signature .Comment: 41 pages, 2 figures, accepted for pubblication in GAF
Boundary maps and maximal representations on infinite dimensional Hermitian symmetric spaces
We define a Toledo number for actions of surface groups and complex
hyperbolic lattices on infinite dimensional Hermitian symmetric spaces, which
allows us to define maximal representations. When the target is not of tube
type we show that there cannot be Zariski-dense maximal representations, and
whenever the existence of a boundary map can be guaranteed, the representation
preserves a finite dimensional totally geodesic subspace on which the action is
maximal. In the opposite direction we construct examples of geometrically dense
maximal representation in the infinite dimensional Hermitian symmetric space of
tube type and finite rank. Our approach is based on the study of boundary maps,
that we are able to construct in low ranks or under some suitable
Zariski-density assumption, circumventing the lack of local compactness in the
infinite dimensional setting.Comment: Comments are welcome! The maximality assumption was unfortunately
missing in Theorem 1.1 and 1.4 of the first versio
Bounded cohomology and the simplicial volume of the product of two surfaces
Il volume simpliciale è un invariante omotopico definito a partire dall'omologia singolare il cui calcolo esplicito è molto difficile. Scopo della tesi è calcolare il volume simpliciale del prodotto di due superfici seguendo un lavoro di Bucher-Karlsson del 2008 dopo aver introdotto tutti gli strumenti utili, in particolare la coomologia continua (e limitata) di gruppi topologici e le medie per gruppi topologici. Sarà necessario, per calcolare il volume simpliciale del prodotto di due superfici, dimostrare il principio di proporzionalità (tra volume simpliciale e volume Riemanniano) per spazi localmente simmetrici
Isometric Embeddings in Bounded Cohomology
This paper is devoted to the construction of norm-preserving maps between
bounded cohomology groups. For a graph of groups with amenable edge groups we
construct an isometric embedding of the direct sum of the bounded cohomology of
the vertex groups in the bounded cohomology of the fundamental group of the
graph of groups. With a similar technique we prove that if (X,Y) is a pair of
CW-complexes and the fundamental group of each connected component of Y is
amenable, the isomorphism between the relative bounded cohomology of (X,Y) and
the bounded cohomology of X in degree at least 2 is isometric. As an
application we provide easy and self-contained proofs of Gromov Equivalence
Theorem and of the additivity of the simplicial volume with respect to gluings
along \pi_1-injective boundary components with amenable fundamental group.Comment: The text overlaps with the submission http://arxiv.org/abs/1205.1022
by the same author
Basmajian-type inequalities for maximal representations
International audienceFor suitable metrics on the locally symmetric space associated to a maximal representation, we prove inequalities between the length of the boundary and the lengths of orthogeodesics that generalize the classical Basmajian's identity from Teichmueller theory. Any equality characterizes diagonal embeddings
Positive crossratios, barycenters, trees and applications to maximal representations
We study metric properties of maximal framed representations of fundamental groups of surfaces in symplectic groups over real closed fields, interpreted as actions on Bruhat-Tits buildings endowed with adapted Finsler norms. We prove that the translation length can be computed as intersection with a geodesic current, give sufficient conditions guaranteeing that such a current is a multicurve, and, if the current is a measured lamination, construct an isometric embedding of the associated tree in the building. These results are obtained as application of more general results of independent interest on positive crossratios and actions with compatible barycenters
The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups
We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional. In an appendix we use the same techniques to give a cohomological proof of a lower bound, originally due to Brock, on the volume of the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed surface in terms of its Teichmiiller translation distance