On fundamental groups of RCD spaces

Abstract

Abstract We obtain results about fundamental groups of RCD ∗ ⁢ ( K , N ) {\mathrm{RCD}^{\ast}(K,N)} spaces previously known under additional conditions such as smoothness or lower sectional curvature bounds. For fixed K ∈ ℝ {K\in\mathbb{R}} , N ∈ [ 1 , ∞ ) {N\in[1,\infty)} , D > 0 {D>0} , we show the following: • There is C > 0 {C>0} such that for each RCD ∗ ⁢ ( K , N ) {\mathrm{RCD}^{\ast}(K,N)} space X of diameter ≤ D {\leq D} , its fundamental group π 1 ⁢ ( X ) {\pi_{1}(X)} is generated by at most C elements. • There is D ~ > 0 {\tilde{D}>0} such that for each RCD ∗ ⁢ ( K , N ) {\mathrm{RCD}^{\ast}(K,N)} space X of diameter ≤ D {\leq D} with compact universal cover X ~ {\tilde{X}} , one has diam ⁡ ( X ~ ) ≤ D ~ {\operatorname{diam}(\tilde{X})\leq\tilde{D}} . • If a sequence of RCD ∗ ⁢ ( 0 , N ) {\mathrm{RCD}^{\ast}(0,N)} spaces X i {X_{i}} of diameter ≤ D {\leq D} and rectifiable dimension n is such that their universal covers X ~ i {\tilde{X}_{i}} converge in the pointed Gromov–Hausdorff sense to a space X of rectifiable dimension n, then there is C > 0 {C>0} such that for each i, the fundamental group π 1 ⁢ ( X i ) {\pi_{1}(X_{i})} contains an abelian subgroup of index ≤ C {\leq C} . • If a sequence of RCD ∗ ⁢ ( K , N ) {\mathrm{RCD}^{\ast}(K,N)} spaces X i {X_{i}} of diameter ≤ D {\leq D} and rectifiable dimension n is such that their universal covers X ~ i {\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 {C>0} such that for each i, the fundamental group π 1 ⁢ ( X i ) {\pi_{1}(X_{i})} contains an abelian subgroup of index ≤ C {\leq C} . • If a sequence of RCD ∗ ⁢ ( K , N ) {\mathrm{RCD}^{\ast}(K,N)} spaces X i {X_{i}} with first Betti number ≥ r {\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 {r+m-n} . The main tools are the splitting theorem by Gigli, the splitting blow-up property by Mondino and Naber, the semi-locally-simple-connectedness of RCD ∗ ⁢ ( K , N ) {\mathrm{RCD}^{\ast}(K,N)} spaces by Wang, the isometry group structure by Guijarro and the first author, and the structure of approximate subgroups by Breuillard, Green and Tao.