On fundamental groups of RCD spaces

We obtain results about fundamental groups of $RC{D}^{\ast }(K,N)$ spaces previously known under additional conditions such as smoothness or lower sectional curvature bounds. For fixed $K\in \mathbb{R}$, $N\in [1,\mathrm{\infty })$, $D>0$, we show the following, $\bullet$ There is $C>0$ such that for each $RC{D}^{\ast }(K,N)$ space $X$ of diameter $\le D$, its fundamental group ${\pi }_1(X)$ is generated by at most $C$ elements. $\bullet$ There is $\tilde{D}>0$ such that for each $RC{D}^{\ast }(K,N)$ space $X$ of diameter $\le D$ with compact universal cover $\tilde{X}$, one has diam$(\tilde{X})\le \tilde{D}$. $\bullet$ If a sequence of $RC{D}^{\ast }(0,N)$ spaces $X_i$ of diameter $\le D$ 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 $\le C$. $\bullet$ If a sequence of $RC{D}^{\ast }(K,N)$ spaces $X_i$ of diameter $\le D$ 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 $\le C$. $\bullet$ If a sequence of $RC{D}^{\ast }(K,N)$ spaces $X_i$ with first Betti number $\ge r$ 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 $RC{D}^{\ast }(K,N)$ spaces by Wang, and the isometry group structure by Guijarro and the first author.