arXiv subjects

Alessandra Iozzi

Publications and source records attributed to Alessandra Iozzi.

24 records · Page 2Linked to original sources

arXiv2008

Tight homomorphisms and Hermitian symmetric spaces

We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetric spaces. We show that tight maps behave in a functorial way with respect to the Shilov boundary and use this to prove a general structure theorem for tight homomorphisms. Furthermore we classify all tight embeddings of the Poincare' disk.

math.DG↗

arXiv2008

Surface group representations with maximal Toledo invariant

We develop the theory of maximal representations of the fundamental group of a compact connected oriented surface with boundary, into a group of Hermitian type. For any such representation we define the Toledo invariant, for which we establish properties such as uniform boundedness on the representation variety, additivity under connected sum of surfaces and congruence relations. We thus obtain geometric properties of the maximalrepresentations, that is representations whose Toledo invariant achieves the maximum value: we show that maximal representations have discrete image, are faithful and completely reducible and they always preserve a maximal tube type subdomain. This extends to the case of a general Hermitian group some of the properties of the representations in Teichmuller space, as well as results due to Goldman, Toledo, Hernandez, Bradlow--Garcia-Prada--Gothen. An announcement of these results in the case of surfaces without boundary -- where the role of tube type domains had already been emphasized -- appeared in 2003 by the same authors. The congruence relations for the Toledo invariant involve a rotation number function related to a continuous homogeneous quasimorphism which gives an explicit way to compute the Toledo invariant. This rotation number generalizes constructions due to Ghys, Barge--Ghys, and Clerc--Koufany. We establish moreover properties of boundary maps associated to maximal representations which generalize naturally, for the causal structure of the Shilov boundary, monotonicity properties of quasiconjugations of the circle. This, together with the congruence relations leads to the result that the subset of maximal representations is always real semialgebraic.

math.DG↗

arXiv2005

Maximal Representations of Surface Groups: Symplectic Anosov Structures

Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Gamma be the fundamental group of a compact orientable surface of genus at least 2. We survey the study of maximal representations, that is the subset of Hom(Gamma,G) which is a union of components characterized by the maximality of the Toledo invariant. Then we concentrate on the particular case G=SP(2n,R), and we show that the image of Gamma under any maximal representation is a discrete faithful realization of Gamma as a Kleinian group of complex motions in X with an associated Anosov system, and whose limit set in an appropriate compactification of X is a rectifiable circle.

math.DG↗

arXiv2005

Bounded Cohomology and Deformation Rigidity in Complex Hyperbolic Geometry

We develop further basic tools in the theory of continuous bounded cohomology of locally compact groups. We apply this tools to establish a Milnor-Wood type inequality in a very general context and to prove a global rigidity result which was originally announced by the authors with a sketch of a proof using bounded cohomology techniques and then proven by Koziarz and Maubon using harmonic map techniques. As a corollary one obtains that a lattice in SU(p,1) cannot be deformed nontrivially in SU(q,1), if either p is at least 2 or the lattice is cocompact. This generalizes to noncocompact lattices a theorem of Goldman and Millson.

math.MG↗

arXiv2001

Tessellations of homogeneous spaces of classical groups of real rank two

Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a "tessellation" if there is a discrete subgroup D of G, such that D acts properly discontinuously on G/H, and the double-coset space D\G/H is compact. Note that if either H or G/H is compact, then G/H has a tessellation; these are the obvious examples. It is not difficult to see that if G has real rank one, then only the obvious homogeneous spaces have tessellations. Thus, the first interesting case is when G has real rank two. In particular, R.Kulkarni and T.Kobayashi constructed examples that are not obvious when G = SO(2,2n) or SU(2,2n). H.Oh and D.Witte constructed additional examples in both of these cases, and obtained a complete classification when G = SO(2,2n). We simplify the work of Oh-Witte, and extend it to obtain a complete classification when G = SU(2,2n). This includes the construction of another family of examples. The main results are obtained from methods of Y.Benoist and T.Kobayashi: we fix a Cartan decomposition G = KAK, and study the intersection of KHK with A. Our exposition generally assumes only the standard theory of connected Lie groups, although basic properties of real algebraic groups are sometimes also employed; the specialized techniques that we use are developed from a fairly elementary level.

math.RT↗

arXiv2000

Cartan-decomposition subgroups of SU(2,n)

We give explicit, practical conditions that determine whether or not a closed, connected subgroup H of G = SU(2,n) has the property that there exists a compact subset C of G with CHC = G. To do this, we fix a Cartan decomposition G = K A K of G, and then carry out an approximate calculation of the intersection of KHK with A, for each closed, connected subgroup H of G. This generalizes the work of Hee Oh and Dave Witte for G = SO(2,n).

math.RT↗