On splitting rank of non-compact type symmetric spaces and bounded cohomology

TL;DR

This paper introduces the concept of splitting rank for higher rank symmetric spaces of non-compact type, computes it explicitly for irreducible cases, and studies the surjectivity of the comparison map in bounded cohomology for spaces without small factors.

Contribution

It defines the splitting rank for symmetric spaces, computes it explicitly for all irreducible cases, and establishes surjectivity results for the comparison map in bounded cohomology.

Findings

Explicit splitting rank for each irreducible symmetric space.

Surjectivity of the comparison map in degrees above splitting rank + 2.

Applicable to symmetric spaces without small direct factors.

Abstract

Let $X=G/K$ be a higher rank symmetric space of non-compact type, where $G$ is the connected component of the isometry group of $X$. We define the splitting rank of $X$, denoted by $\text{srk}(X)$, to be the maximal dimension of a totally geodesic submanifold $Y\subset X$ which splits off an isometric $\mathbb R$-factor. We compute explicitly the splitting rank for each irreducible symmetric space. For an arbitrary (not necessarily irreducible) symmetric space, we show that the comparison map $\eta:H^{*}_{c,b}(G,\mathbb{R})\rightarrow H^{*}_c(G,\mathbb{R})$ is surjective in degrees $*\geq \text{srk}(X)+2$, provided $X$ has no small direct factors.