The Roller boundary of a countable CAT(0) cube complex of dimension n has Borel asymptotic dimension at most n, and its component equivalence relation is smooth.
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Borel asymptotic dimension of the Roller boundary of finite dimensional CAT(0) cube complexes
The Roller boundary of a countable CAT(0) cube complex of dimension n has Borel asymptotic dimension at most n, and its component equivalence relation is smooth.