Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Complete affine manifolds with Anosov holonomy groups

Suhyoung Choi

Abstract

Let $N$ be a complete affine manifold $\mathbb{A}^n/Γ$ of dimension $n$, where $Γ$ is an affine transformation group acting on the complete affine space $\mathbb{A}^n$, and $K(Γ, 1)$ is realized as a finite CW-complex. $N$ has a $k$-partially hyperbolic holonomy group if the tangent bundle pulled back over the unit tangent bundle of a sufficiently large compact subset splits into expanding, neutral, and contracting subbundles along the geodesic flow, where the expanding and contracting subbundles are $k$-dimensional with $k < n/2$. In part 1, we will demonstrate that the complete affine $n$-manifold has a $P$-Anosov linear holonomy group for a parabolic subgroup $P$ of $\mathrm{GL}(n, \mathbb{R})$ if and only if it has a partially hyperbolic linear holonomy group. This had never been done over the full general linear group before this paper. Part 1 will primarily employ representation theory techniques. In part 2, we demonstrate that if the holonomy group is partially hyperbolic of index $k$, where $k < n/2$, then $\mathrm{cd}(Γ) \leq n-k$. Moreover, if a finitely-presented affine group $Γ$ acts properly discontinuously and freely on $\mathbb{A}^n$ with a $k$-Anosov linear subgroup for $k \leq n/2$, then $\mathrm{cd}(Γ) \leq n-k$. Also, there exists a compact collection of $n-k$-dimensional affine subspaces where $Γ$ acts. The techniques employed here mostly stem from the coarse geometry theory. Canary and Tsouvalis previously proved the same result using the powerful method of Bestvina and Mess for word hyperbolic groups; however, our approach differs in that our method projects the holonomy cover to a stable affine subspace, and we plan to generalize to relative Anosov groups.

Complete affine manifolds with Anosov holonomy groups

Abstract

Let be a complete affine manifold of dimension , where is an affine transformation group acting on the complete affine space , and is realized as a finite CW-complex. has a -partially hyperbolic holonomy group if the tangent bundle pulled back over the unit tangent bundle of a sufficiently large compact subset splits into expanding, neutral, and contracting subbundles along the geodesic flow, where the expanding and contracting subbundles are -dimensional with . In part 1, we will demonstrate that the complete affine -manifold has a -Anosov linear holonomy group for a parabolic subgroup of if and only if it has a partially hyperbolic linear holonomy group. This had never been done over the full general linear group before this paper. Part 1 will primarily employ representation theory techniques. In part 2, we demonstrate that if the holonomy group is partially hyperbolic of index , where , then . Moreover, if a finitely-presented affine group acts properly discontinuously and freely on with a -Anosov linear subgroup for , then . Also, there exists a compact collection of -dimensional affine subspaces where acts. The techniques employed here mostly stem from the coarse geometry theory. Canary and Tsouvalis previously proved the same result using the powerful method of Bestvina and Mess for word hyperbolic groups; however, our approach differs in that our method projects the holonomy cover to a stable affine subspace, and we plan to generalize to relative Anosov groups.

Paper Structure

This paper contains 32 sections, 32 theorems, 97 equations, 4 figures.

Table of Contents

  1. Hyperbolic bundles and Anosov representations
  2. Introduction
  3. Notation
  4. Main result
  5. History of the field relevant to this result
  6. Outline of Part 1
  7. Acknowledgments
  8. Preliminary
  9. Convergences of geodesics
  10. $(G,X)$-structures and affine manifolds
  11. Gromov hyperbolic spaces
  12. Flow space $G\hat{M}$ quasi-isometric to $\partial_\infty^{(2)}\hat{M} \times {\mathbb{R}}$
  13. Partial hyperbolicity and $P$-Anosov properties
  14. Some background
  15. $P$-Anosov representations
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $N$ be a complete affine $n$-manifold where $K(\pi_1(N), 1)$ is realized by a finite complex. Let $\rho: \pi_1(N)\rightarrow {\mathsf{GL}}(n, {\mathbb{R}})$ is a linear part of an affine holonomy homomorphism $\rho'$. Then $\rho$ is $P$-Anosov for any parabolic subgroup $P$ of ${\mathsf{GL}}(n,

Figures (4)

  • Figure 1: The proof of Lemma \ref{['lem:geolim-p2']}
  • Figure 2: The proof of Lemma \ref{['lem:Crough-p2']}
  • Figure 3: The proof of Proposition \ref{['prop:Finalpart-p2']}
  • Figure 4: A close-up of the proof of Proposition \ref{['prop:Finalpart-p2']}.

Theorems & Definitions (62)

  • Definition 1.1: Partial hyperbolicity
  • Theorem 1.1
  • Theorem 1.2: Theorem \ref{['thm:main-p2']}
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4: Theorem 8.3.C of Gromov Gromov83, Theorem 60 of Mineyev Mineyev
  • ...and 52 more