Every virtually special action of a countable group on a CAT(0) cube complex yields a hyperfinite orbit equivalence relation on its Roller boundary.
Borel asymptotic dimension of the Roller boundary of finite dimensional CAT(0) cube complexes
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abstract
We prove that for any countable finite dimensional CAT(0) cube complex, the Borel median graph on its Roller compactification has the Borel asymptotic dimension bounded from above by its dimension.
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Hyperfiniteness of the boundary action of virtually special groups
Every virtually special action of a countable group on a CAT(0) cube complex yields a hyperfinite orbit equivalence relation on its Roller boundary.