Fundamental-group epimorphisms in noncollapsed Ricci limits

Abstract

Let $\varphi:G\twoheadrightarrow Q$ be an epimorphism of finitely presented groups. We characterize when $\varphi$ is induced, under fixed markings, by continuous Gromov--Hausdorff approximation maps for a sequence of metrics on a fixed closed manifold, with a uniform lower Ricci bound, bounded diameter and a positive lower volume bound. A realization with a compact semilocally simply connected limit exists in some dimension if and only if $\ker\varphi$ is finitely normally generated by finite-order elements of $G$. Every such epimorphism has a realization on a fixed closed oriented four-manifold with constant positive volume, two-sided Ricci bounds and uniformly bounded $L^2$ curvature. The limit is an orbifold with one isolated flat singularity for each nonidentity element in a chosen finite list of finite-order normal generators of the kernel. We construct two resolutions of one orbifold whose fundamental groups are Thompson's group $V$ and the trivial group, and whose scalar eigenvalues have the same limit at each fixed index. We also give a five-dimensional nonnegative-Ricci sequence with infinite dihedral fundamental group and simply connected limit. Finally, in every fixed dimension, under uniform lower Ricci, diameter and volume bounds, a heat integral comparing a manifold with its universal cover diverges exactly when the homotopy systole tends to zero.