Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

Abstract. 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 Poincaré disk.

show abstract

“…For tight embeddings of CH n into classical Hermitian symmetric spaces of rank 2 this can be deduced from [17].…”

Section: Corollary 4 Let G a Connected Algebraic Group Defined Over mentioning

confidence: 99%

Tight Homomorphisms and Hermitian Symmetric Spaces

Burger¹,

Iozzi²,

Wienhard

2009

Geom. Funct. Anal.

View full text Add to dashboard Cite

show abstract

“…The picture for rank one targets was completed independently by Koziarz and Maubon [KM08a] and Burger and Iozzi [BI08]: any maximal representation of a lattice in SU(1, p) with values in SU(1, q) admits an equivariant totally geodesic holomorphic embedding H p C → H q C . Koziarz and Maubon generalized this result to the situation in which the target group is classical of rank 2 and the lattice is cocompact [KM08b]. 1 It is conjectured that every maximal representation of a complex hyperbolic lattice with target a Hermitian Lie group is superrigid, namely it extends, up to a representation of Γ in the compact centralizer of the image, to a representation of the ambient group SU(1, p).…”

Section: Introductionmentioning

confidence: 99%

Maximal representations of complex hyperbolic lattices into SU(m,n)

Pozzetti

2015

Geom. Funct. Anal.

View full text Add to dashboard Cite

Let Γ denote a lattice in SU(1, p), with p greater than 1. We show that there exists no Zariski dense maximal representation with target SU(m, n) if n > m > 1. The proof is geometric and is based on the study of the rigidity properties of the geometry whose points are isotropic m-subspaces of a complex vector space V endowed with a Hermitian metric h of signature (m, n) and whose lines correspond to the 2m dimensional subspaces of V on which the restriction of h has signature (m, m).

show abstract

“…For more general Kähler manifolds, not even a Milnor-Wood inequality is available. The most relevant work here is again due to Koziarz and Maubon [KM10], who considered a slight variation of the Toledo invariant for complex varieties of general type, and were able to replicate the results in [KM08] in this broader setting. In this paper we prove that both the inequality and the study of the case of equality can be proved if one knows beforehand that the representation admits a ρ-equivariant holomorphic (or anti-holomorphic) map: Theorem 1.2.…”

Section: Introductionmentioning

confidence: 86%

Rigidity of maximal holomorphic representations of Kähler groups

Spinaci

2015

Int. J. Math.

View full text Add to dashboard Cite

Abstract. We investigate representations of Kähler groups Γ " π 1 pXq to a semisimple non-compact Hermitian Lie group G that are deformable to a representation admitting an (anti)-holomorphic equivariant map. Such representations obey a Milnor-Wood inequality similar to those found by Burger-Iozzi and Koziarz-Maubon. Thanks to the study of the case of equality in Royden's version of the Ahlfors-Schwarz Lemma, we can completely describe the case of maximal holomorphic representations. If dim C X ě 2, these appear if and only if X is a ball quotient, and essentially reduce to the diagonal embedding Γ ă SUpn, 1q Ñ SUpnq, qq ãÑ SUpp, qq. If X is a Riemann surface, most representations are deformable to a holomorphic one. In that case, we give a complete classification of the maximal holomorphic representations, that thus appear as preferred elements of the respective maximal connected components.

show abstract

Representations of complex hyperbolic lattices into rank 2 classical Lie groups of Hermitian type

Cited by 17 publications

References 28 publications

Tight Homomorphisms and Hermitian Symmetric Spaces

Tight Homomorphisms and Hermitian Symmetric Spaces

Maximal representations of complex hyperbolic lattices into SU(m,n)

Rigidity of maximal holomorphic representations of Kähler groups

Contact Info

Product

Resources

About