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We consider cocycles of isometries on spaces of nonpositive curvature $H$. Weshow that the supremum of the drift over all invariant ergodic probabilitymeasures equals the infimum of the displacements of continuous sections underthe cocycle dynamics. In particular, if a cocycle has uniform sublinear drift,then there are almost invariant sections, that is, sections that movearbitrarily little under the cocycle dynamics. If, in addition, $H$ is asymmetric space, then we show that almost invariant sections can be madeinvariant by perturbing the cocycle.

A geometric path from zero Lyapunov exponents to rotation cocycles