Relating boundary and interior solutions of the cohomological equation for cocycles by isometries of negatively curved spaces. The Livsic case

TL;DR

This paper investigates the reducibility of cocycles by isometries in negatively curved spaces, establishing that boundary reducibility implies interior reducibility under certain conditions.

Contribution

It demonstrates that boundary reducibility in a H"older class guarantees interior reducibility for cocycles by isometries of Gromov hyperbolic spaces.

Findings

Boundary cocycle reducibility implies interior reducibility.

Reduces the problem of interior reducibility to boundary analysis.

Applicable in the Livsic setting for hyperbolic spaces.

Abstract

We consider the reducibility problem of cocycles by isometries of Gromov hyperbolic metric spaces in the Livsic setting. We show that provided that the boundary cocycle (that acts on a compact space) is reducible in a suitable H\"older class, then the original cocycle by isometries (that acts on an unbounded space) is also reducible.