Schottky presentations of positive representations

Abstract

We define for every positive Anosov representation of a nonabelian free group into SO(2n, 2n − 1) a family of R 4n−1 -valued cocycles which induce proper affine actions on R 4n−1 . We construct fundamental domains in R 4n−1 bounded by generalized crooked planes for these affine actions, and deduce that the quotient manifolds are homeomorphic to handlebodies.

Domains of discontinuity in oriented flag manifolds

“…We will give a brief overview of generalized Schottky groups and their properties before showing how they fit into our framework. More details and proofs can be found in [6] and [7]. For all odd dimensions, the partial cyclic order on  {1} is given as follows.…”

“…In [7], generalized Schottky groups in $$G=\operatorname{PSL}(n,\mathbb {R})$$ are introduced. The construction relies on the existence of a partial cyclic order on the space $$G/B_0 = \mathcal {F}_{\lbrace 1\rbrace }$$, which is an oriented version of Fock–Goncharov triple positivity [12] and Labourie's 3‐hyperconvexity [29].…”

“…In [7], generalized Schottky groups in 𝐺 = PSL(𝑛, ℝ) are introduced. The construction relies on the existence of a partial cyclic order on the space 𝐺∕𝐵 0 =  {1} , which is an oriented version of Fock-Goncharov triple positivity [12] and Labourie's 3-hyperconvexity [29].…”

“…The following result allows us to apply our theory of domains of discontinuity to purely hyperbolic generalized Schottky representations. Theorem 7.8 [7]. Let 𝜌 ∶ Γ = 𝐹 𝑘 → PSL(2𝑛 + 1, ℝ) be a purely hyperbolic generalized Schottky representation.…”