2022/03/08 by Suhyoung Choi, Choi, Suhyoung
Mathematics · #53C15 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Primary 57M50 #Secondary 53A20
paper · pdf · doi:10.48550/arxiv.2203.03968
openalex publication_date 2022/03/08 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
Let N be a complete affine manifold An/Γ of dimension n where Γ is an affine transformation group and K(Γ, 1) is realized as a finite CW-complex. N has a partially hyperbolic holonomy group if the tangent bundle pulled over the unit tangent bundle over a sufficiently large compact part splits into expanding, neutral, and contracting subbundles along the geodesic flow. We show that if the holonomy group is partially hyperbolic of index k, k < n/2, then cd(Γ) ≤ n-k. Moreover, if a finitely-presented affine group Γ acts on An properly discontinuously and freely with the k-Anosov linear group for k ≤ n/2, then cd(Γ) ≤ n-k. Also, there exists a compact collection of n-k-dimensional affine subspaces where Γ acts on. The techniques here are mostly from coarse geometry.