Fundamental-group epimorphisms in noncollapsed Ricci limits
Abstract
Let φ:G↠ Q be an epimorphism of finitely presented groups. We characterize when φ 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 kerφ 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 L2 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.
Related
- Homological obstructions to local contractibility in noncollapsed Ricci limits
- TOPOLOGICAL SINGULARITIES IN THE CODIMENSION-TWO STRATUM OF NONCOLLAPSED RICCI LIMITS
- Positive bisectional curvature, Ricci mass, and the fundamental group
- Instability of the Fundamental Group for Ricci-Flat Kähler 4-Manifolds
- Topological regularity and stability of noncollapsed spaces with Ricci curvature bounded below
BibTeX
@misc{nifa2026fundamentalgroup,
title = {{Fundamental-group epimorphisms in noncollapsed Ricci limits}},
author = {Anass Nifa},
howpublished = {HAL},
year = {2026},
month = may,
doi = {10.5281/zenodo.22770177},
url = {https://hal.science/hal-05769943v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fundamental-group epimorphisms in noncollapsed Ricci limits
AU - Anass Nifa
PY - 2026
DA - 2026/05/25
PB - HAL
DO - 10.5281/zenodo.22770177
UR - https://hal.science/hal-05769943v1
AB - Let φ:G↠ Q be an epimorphism of finitely presented groups. We characterize when φ 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 kerφ 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.
ER -