Right-angled hexagon tilings of the hyperbolic plane

We study isometry-invariant probability measures on the space $Ω$ of tilings of the hyperbolic plane with right-angled hexagons of varying shapes. We prove that, for each measure $μ$ in a certain natural family of measures on right-angled hexagons, there is an isometry-invariant measure on $Ω$ whose marginal distribution on tiles is $μ$.

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