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The Furstenberg–Poisson boundary and CAT(0) cube complexes



We show under weak hypotheses that $\unicode[STIX]{x2202}X$ , the Roller boundary of a finite-dimensional CAT(0) cube complex $X$ is the Furstenberg–Poisson boundary of a sufficiently nice random walk on an acting group $\unicode[STIX]{x1D6E4}$ . In particular, we show that if $\unicode[STIX]{x1D6E4}$ admits a non-elementary proper action on $X$ , and $\unicode[STIX]{x1D707}$ is a generating probability measure of finite entropy and finite first logarithmic moment, then there is a $\unicode[STIX]{x1D707}$ -stationary measure on $\unicode[STIX]{x2202}X$ making it the Furstenberg–Poisson boundary for the $\unicode[STIX]{x1D707}$ -random walk on $\unicode[STIX]{x1D6E4}$ . We also show that the support is contained in the closure of the regular points. Regular points exhibit strong contracting properties.



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