Tight homomorphisms and Hermitian symmetric spaces

2007/10/30 by Marc Burger, Burger, Marc, Alessandra Iozzi +3
Mathematics · #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG

paper · pdf · doi:10.48550/arxiv.0710.5641

51 pages, no figure. The exposition has been improved, especially in the Introduction

openalex publication_date 2007/10/30 · arxiv created 2008/09/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Poincare' disk.

Citations

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

Related

• Classification of tight homomorphisms
• Tight maps and holomorphicity
• Bounded Cohomology and Geometry
• Tight holomorphic maps, a classification
• On rigidity of locally symmetric spaces