论文标题
与Anosov全能组完全仿射歧管
Complete affine manifolds with Anosov holonomy groups
论文作者
论文摘要
令$ n $为一个完整的仿射歧管$ \ mathbb {a}^n/γ$ dimension $ n $，其中$γ$是作用于完整仿射空间$ \ mathbb {a}^n $和$ k（γ，1）$的仿射转换组。 $ n $具有$ k $的双曲线固体组，如果切线束在单位切线上拉回了足够大的紧凑型子集的单位切线捆绑包中，以扩展，中性和收缩的子划分，沿着大地测量流的扩展和收缩的子捆绑，在该捆绑中，扩展和合同的分支为$ k $ $ k $ - $ k $ dimimections $ k $ k <n/2 $ <n/2 $。在第1部分中，我们将证明完整的Aggine $ n $ -Manifold具有$ p $ -Anosov线性固体组，用于抛物线子组$ p $ $ \ mathrm {gl}（n，\ mathbb {r}）$，并且只有当时只有一个部分超级型线性纯态度组。在本文之前，这从未在整个通用线性群体上完成。第1部分将主要采用代表理论技术。在第2部分中，我们证明，如果整体组是索引$ k $的部分双曲线，其中$ k <n/2 $，则$ \ mathrm {cd}（γ）（γ）\ leq n-k $。此外，如果有限的仿射组$γ$与$ k $ -Anosov线性亚组适用于$ k \ leq n/2 $，然后在$ \ mathbb {a}^n $上正确地表现，然后是$ k $ -Anosov线性子组，然后$ k \ leq n/2 $，然后$ \ \ \ \ \ m athrm {cd}（cd}（γ）\ leq n-k $。此外，还存在$ n-k $二维仿射子空间的紧凑型集合，其中$γ$ ACTS。这里采用的技术主要源于粗糙的几何理论。 Canary和Tsouvalis先前使用强大的BESTVINA和WORD双曲线组的杂物证明了相同的结果；但是，我们的方法有所不同，因为我们的方法将固体覆盖范围投影到稳定的仿射子空间，我们计划将其推广到相对的Anosov群体。
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.
