论文标题
在RCD空间的基本组上
On fundamental groups of RCD spaces
论文作者
论文摘要
我们获得有关$ rcd^{\ ast}（k，n）$空间的基本组的结果，以前在其他条件下（例如平滑度或较低的截面曲率边界）所知。对于固定的$ k \ in \ mathbb {r} $，$ n \ in [1，\ infty）$，$ d> 0 $，我们显示以下内容， $ \ bullet $有$ c> 0 $，以便每$ rcd^{\ ast}（k，n）$ space $ x $ x $ diameter $ \ leq d $，其基本组$π_1（x）$都是由最多在$ c $元素生成的。 $ \ bullet $有$ \ tilde {d}> 0 $，以至于对于每个$ rcd^{\ ast}（k，k，n）$ x $ diameter $ \ leq d $带有紧凑型通用封面$ \ tilde {x} $的$ x $ x $ timm $（x} $ $ \ bullet $如果$ rcd^{\ ast}的顺序（0，n）$ spaces $ x_i $ x_i $的$ \ leq d $ \ leq d $ and rectififiable dimension $ n $，以至于它们的通用覆盖$ \ tilde {x} _i $ commisies $ n ins commistion $ nisted gromov-hausdor $ hausdor $ n $ n $ $ c> 0 $这样，对于每个$ i $，基本组$π_1（x_i）$包含一个索引$ \ leq c $的亚巴子亚组。 $ \ bullet $如果$ rcd^{\ ast}（k，n）$ spaces $ x_i $ x_i $的顺序，则$ x_i $ $ \ leq d $和可重新处理的尺寸$ n $使它们的通用覆盖$ \ tilde {x} _i $是压实的，并在点数$ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $有$ c> 0 $，因此对于每个$ i $，基本组$π_1（x_i）$包含一个索引$ \ leq c $的Abelian子组。 $ \ bullet $如果$ rcd^{\ ast}的顺序（k，n）$ spaces $ x_i $带有第一个betti $ $ \ geq r $ \ geq r $和可回座的尺寸$ n $在gromov-hausdorff sissecles complation $ m $ $ m $ y $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x $ x。 主要工具是Gigli的分裂定理，Mondino的爆炸属性 - 纳伯（Naber），$ rcd^{\ ast} $ rcd^{\ ast}（k，n）$ spaces的半局部简化连接性，以及Guijarro的Isometry组结构，是Guijarro和第一作者。
We obtain results about fundamental groups of $RCD^{\ast}(K,N)$ spaces previously known under additional conditions such as smoothness or lower sectional curvature bounds. For fixed $K \in \mathbb{R}$, $N \in [1,\infty )$, $D > 0 $, we show the following, $\bullet$ There is $C>0$ such that for each $RCD^{\ast}(K,N)$ space $X$ of diameter $\leq D$, its fundamental group $π_1(X)$ is generated by at most $C$ elements. $\bullet$ There is $\tilde{D}>0$ such that for each $RCD^{\ast}(K,N)$ space $X$ of diameter $\leq D$ with compact universal cover $\tilde{X}$, one has diam$(\tilde{X})\leq \tilde{D}$. $\bullet$ If a sequence of $RCD^{\ast}(0,N)$ spaces $X_i$ of diameter $\leq D$ and rectifiable dimension $n$ is such that their universal covers $\tilde{X}_i$ converge in the pointed Gromov--Hausdorff sense to a space $X$ of rectifiable dimension $n$, then there is $C>0$ such that for each $i$, the fundamental group $π_1(X_i)$ contains an abelian subgroup of index $\leq C$. $\bullet$ If a sequence of $RCD^{\ast}(K,N)$ spaces $X_i$ of diameter $\leq D$ and rectifiable dimension $n$ is such that their universal covers $\tilde{X}_i$ are compact and converge in the pointed Gromov--Hausdorff sense to a space $X$ of rectifiable dimension $n$, then there is $C>0$ such that for each $i$, the fundamental group $π_1(X_i)$ contains an abelian subgroup of index $\leq C$. $\bullet$ If a sequence of $RCD^{\ast}(K,N)$ spaces $X_i$ with first Betti number $\geq r$ and rectifiable dimension $ n$ converges in the Gromov--Hausdorff sense to a compact space $X$ of rectifiable dimension $m$, then the first Betti number of $X$ is at least $r + m - n$. The main tools are the splitting theorem by Gigli, the splitting blow-up property by Mondino--Naber, the semi-locally-simple-connectedness of $RCD^{\ast}(K,N)$ spaces by Wang, and the isometry group structure by Guijarro and the first author.
