On fundamental groups of RCD spaces

Abstract

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 $\pi_1(X)$ 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 $\pi_1(X_i)$ 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 $\pi_1(X_i)$ 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.