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We consider the reducibility problem of cocycles by isometries of Gromovhyperbolic metric spaces in the Livsic setting. We show that provided that theboundary cocycle (that acts on a compact space) is reducible in a suitableH\"older class, then the original cocycle by isometries (that acts on anunbounded space) is also reducible.

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