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We consider the wave maps equation for maps from $(1+1)$-dimensional Minkowski space into a compact Lie group. The Gibbs measure of this model corresponds to a Brown- ian motion on the Lie group, which is a natural object from stochastic differential geometry. Our main result is the invariance of the Gibbs measure under the wave maps equation and is the first result of this kind for any geometric wave equation. The proof combines techniques from differential geometry, partial differential equations, and probability theory.