Fundamental-group epimorphisms in noncollapsed Ricci limits

Nifa, Anass

doi:10.5281/zenodo.22770177

Preprint · v2

Fundamental-group epimorphisms in noncollapsed Ricci limits

Anass Nifa25 May 2026

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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.

MSC 2020

53C21: Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57M10: Covering spaces and low-dimensional topology
57R65: Surgery and handlebodies
20E06: Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.22770177
All versions: 10.5281/zenodo.22704690
Subjects: Fundamental groups; Ricci curvature
Earlier title: Normal generation of fundamental-group kernel in noncollapsed Ricci limits
Research index: OpenAlex record
Language: English

Cite this paper

Anass Nifa. Fundamental-group epimorphisms in noncollapsed Ricci limits. Preprint, Zenodo, v2 (2026). doi:10.5281/zenodo.22770177

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References

As listed in the manuscript bibliography.

[1]
M. T. Anderson, Short geodesics and gravitational instantons, J. Differential Geom. 31 (1990), no. 1, 265–275. https://doi.org/10.4310/jdg/1214444097.
[2]
C. Bleak and M. Quick, The infinite simple group 𝑉 of Richard J. Thompson: presentations by permutations, Groups Geom. Dyn. 11 (2017), no. 4, 1401–1436. https://doi.org/10.4171/GGD/433.
[3]
C. Brena, Instability of the fundamental group for non-collapsed Ricci-limits, J. Funct. Anal. 290 (2026), no. 7, article 111343. https://doi.org/10.1016/j.jfa.2026.111343. Construction references are to arXiv:2505.17263v1 (2025).
[4]
R. Fintushel and R. J. Stern, Rational blowdowns of smooth 4-manifolds, J. Differential Geom. 46 (1997), no. 2, 181–235. https://doi.org/10.4310/jdg/1214459932. The numbered lemma is cited from arXiv:alg-geom/9505018v1 (1995).
[5]
N. Gigli, A. Mondino and G. Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proc. Lond. Math. Soc. (3) 111 (2015), no. 5, 1071–1129. https://doi.org/10.1112/plms/pdv047. Numbered statements are cited from arXiv:1311.4907v1 (2013).
[6]
R. Jiang, H. Li and H. Zhang, Heat kernel bounds on metric measure spaces and some applications, Potential Anal. 44 (2016), no. 3, 601–627. https://doi.org/10.1007/s11118-015-9521-2. Numbered statements are cited from arXiv:1407.5289v2 (2015).
[7]
K. Kröncke and Á. Szabó, Optimal coordinates for Ricci-flat conifolds, Calc. Var. Partial Differential Equations 63 (2024), article 188. https://doi.org/10.1007/s00526-024-02780-y.
[8]
J. Pan and G. Wei, Semi-local simple connectedness of non-collapsing Ricci limit spaces, J. Eur. Math. Soc. (JEMS) 24 (2022), no. 12, 4027–4062. https://doi.org/10.4171/JEMS/1166.
[9]
D. Panov and A. Petrunin, Telescopic actions, Geom. Funct. Anal. 22 (2012), no. 6, 1814–1831. https://doi.org/10.1007/s00039-012-0194-3. Numbered statements are cited from arXiv:1104.4814v6 (2012).
[10]
C. Sormani and G. Wei, Hausdorff convergence and universal covers, Trans. Amer. Math. Soc. 353 (2001), no. 9, 3585–3602. https://doi.org/10.1090/S0002-9947-01-02802-1.
[11]
I. Şuvaina, ALE Ricci-flat Kähler metrics and deformations of quotient surface singularities, Ann. Global Anal. Geom. 41 (2012), no. 1, 109–123. https://doi.org/10.1007/s10455-011-9273-1. Numbered statements are cited from arXiv:1102.2672v1 (2011).
[12]
Ž. Virk, 1-Dimensional intrinsic persistence of geodesic spaces, J. Topol. Anal. 12 (2020), 169–207. https://doi.org/10.1142/S1793525319500444. Numbered statements are cited from arXiv:1709.05164v4 (2024).
[13]
J. Wang, Ricci limit spaces are semi-locally simply connected, J. Differential Geom. 128 (2024), no. 3, 1301–1314. https://doi.org/10.4310/jdg/1729092461.

[reference-1] [1]
M. T. Anderson, Short geodesics and gravitational instantons, J. Differential Geom. 31 (1990), no. 1, 265–275. https://doi.org/10.4310/jdg/1214444097.

[reference-2] [2]
C. Bleak and M. Quick, The infinite simple group 𝑉 of Richard J. Thompson: presentations by permutations, Groups Geom. Dyn. 11 (2017), no. 4, 1401–1436. https://doi.org/10.4171/GGD/433.

[reference-3] [3]
C. Brena, Instability of the fundamental group for non-collapsed Ricci-limits, J. Funct. Anal. 290 (2026), no. 7, article 111343. https://doi.org/10.1016/j.jfa.2026.111343. Construction references are to arXiv:2505.17263v1 (2025).

[reference-4] [4]
R. Fintushel and R. J. Stern, Rational blowdowns of smooth 4-manifolds, J. Differential Geom. 46 (1997), no. 2, 181–235. https://doi.org/10.4310/jdg/1214459932. The numbered lemma is cited from arXiv:alg-geom/9505018v1 (1995).

[reference-5] [5]
N. Gigli, A. Mondino and G. Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proc. Lond. Math. Soc. (3) 111 (2015), no. 5, 1071–1129. https://doi.org/10.1112/plms/pdv047. Numbered statements are cited from arXiv:1311.4907v1 (2013).

[reference-6] [6]
R. Jiang, H. Li and H. Zhang, Heat kernel bounds on metric measure spaces and some applications, Potential Anal. 44 (2016), no. 3, 601–627. https://doi.org/10.1007/s11118-015-9521-2. Numbered statements are cited from arXiv:1407.5289v2 (2015).

[reference-7] [7]
K. Kröncke and Á. Szabó, Optimal coordinates for Ricci-flat conifolds, Calc. Var. Partial Differential Equations 63 (2024), article 188. https://doi.org/10.1007/s00526-024-02780-y.

[reference-8] [8]
J. Pan and G. Wei, Semi-local simple connectedness of non-collapsing Ricci limit spaces, J. Eur. Math. Soc. (JEMS) 24 (2022), no. 12, 4027–4062. https://doi.org/10.4171/JEMS/1166.

[reference-9] [9]
D. Panov and A. Petrunin, Telescopic actions, Geom. Funct. Anal. 22 (2012), no. 6, 1814–1831. https://doi.org/10.1007/s00039-012-0194-3. Numbered statements are cited from arXiv:1104.4814v6 (2012).

[reference-10] [10]
C. Sormani and G. Wei, Hausdorff convergence and universal covers, Trans. Amer. Math. Soc. 353 (2001), no. 9, 3585–3602. https://doi.org/10.1090/S0002-9947-01-02802-1.

[reference-11] [11]
I. Şuvaina, ALE Ricci-flat Kähler metrics and deformations of quotient surface singularities, Ann. Global Anal. Geom. 41 (2012), no. 1, 109–123. https://doi.org/10.1007/s10455-011-9273-1. Numbered statements are cited from arXiv:1102.2672v1 (2011).

[reference-12] [12]
Ž. Virk, 1-Dimensional intrinsic persistence of geodesic spaces, J. Topol. Anal. 12 (2020), 169–207. https://doi.org/10.1142/S1793525319500444. Numbered statements are cited from arXiv:1709.05164v4 (2024).

[reference-13] [13]
J. Wang, Ricci limit spaces are semi-locally simply connected, J. Differential Geom. 128 (2024), no. 3, 1301–1314. https://doi.org/10.4310/jdg/1729092461.