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

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.