We define for every positive Anosov representation of a nonabelian free group into SO(2n,2n-1) a family of R⁴ⁿ⁻¹-valued cocycles which induce proper affine actions on R⁴ⁿ⁻¹. We construct fundamental domains in R⁴ⁿ⁻¹ bounded by generalized crooked planes for these affine actions, and deduce that the quotient manifolds are homeomorphic to handlebodies.
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Burelle et al. (2024) studied this question.
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