Tight homomorphisms and Hermitian symmetric spaces

TL;DR

This paper introduces and studies tight homomorphisms into groups with nonvanishing bounded cohomology, focusing on Lie groups of Hermitian type, and classifies tight embeddings of the Poincaré disk.

Contribution

It develops a structure theory for tight homomorphisms into Hermitian Lie groups and classifies all tight embeddings of the Poincaré disk.

Findings

Tight homomorphisms induce tight totally geodesic maps.

Tight maps are functorial with respect to the Shilov boundary.

Complete classification of tight embeddings of the Poincaré disk.

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.