论文标题
正确凸结构的空间
The space of properly-convex structures
论文作者
论文摘要
假设$ g $是有限生成的组，$ \ nathcal {c}（g）$由所有$ρ：g \ to \ operatorName {pgl}（n+1，\ mathbb {r}）$组成，在$ \ mathbb {r} r} $ mathb n $ n $ c} $（p）中，将其适当地设置为$ \ \ mathbb {p）。然后，$ \ MATHCAL {C}（G）$的图像在字符品种中关闭。 假设$ g $不包含无限，普通，阿贝尔子组和$ \ Mathcal {d}（g）\ subset \ mathcal \ Mathcal {c}（g）$是适当的convex $ n $ n $ manifolds的子集的子集。然后，图像$ \ MATHCAL {D}（G）$在字符品种中关闭。 如果$ m $是紧凑型$ n $ -manifold的内部，而$ g =π_1M$如上所述，并且$ m $要么关闭，或者$π_1M$包含一个子组的无限索引同构为$ \ $ \ m athbb {z}}^{n-1}^{n-1} $，然后是$ \ m mathcal chint $ \ m atecal {d iS g）如果此外，$ m $是紧凑型歧管$ n $的内部，使得$ \ partial n $的每个组成部分都是$π_1$ - 注射剂，并有限地覆盖了圆环，则$ \ nathcal {d}（g）$的每个元素都是$ m $ a $ m $ is a IS a Union an Is an Compont $ m Math的整体，g）半代数集。
Suppose $G$ is finitely generated group and $\mathcal{C}(G)$ consists of all $ρ:G\to\operatorname{PGL}(n+1,\mathbb{R})$ for which there exists a properly convex set in $\mathbb{R}\mathbb{P}^n$ that is preserved by $ρ(G)$. Then the image of $\mathcal{C}(G)$ is closed in the character variety. Suppose $G$ does not contain an infinite, normal, abelian subgroup and $\mathcal{D}(G)\subset\mathcal{C}(G)$ is the subset of holonomies of properly-convex $n$-manifolds with fundamental group $G$. Then the image $\mathcal{D}(G)$ is closed in the character variety. If $M$ is the interior of a compact $n$-manifold and $G=π_1M$ is as above, and either $M$ is closed, or $π_1M$ contains a subgroup of infinite index isomorphic to $\mathbb{Z}^{n-1}$, then $\mathcal{D}(G)$ is closed. If, in addition, $M$ is the interior of a compact manifold $N$ such that every component of $\partial N$ is $π_1$-injective, and finitely covered by a torus, then every element of $\mathcal{D}(G)$ is the holonomy of a properly-convex structure on $M$, and $\mathcal{D}(G)$ is a union of connected components of a semi-algebraic set.
