Isometric Embeddings in Bounded Cohomology

TL;DR

This paper develops isometric embeddings in bounded cohomology for graphs of groups and CW-complex pairs, providing new tools for understanding bounded cohomology and applications to geometric topology.

Contribution

It constructs norm-preserving maps between bounded cohomology groups for graphs of groups and CW-complex pairs, advancing the understanding of their structure.

Findings

Constructs isometric embeddings for bounded cohomology of graphs of groups.

Proves isometric isomorphisms for relative bounded cohomology in CW-complex pairs.

Provides simplified proofs of Gromov's Equivalence Theorem and additivity of simplicial volume.

Abstract

This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X,Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X,Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along \pi_1-injective boundary components with amenable fundamental group.