Almost-cyclic fundamental groups: normality, linear models, and dimension three

Definitions and mathematical statement

Fix an odd integer $n\geq3$ and consider closed connected Riemannian $n$-manifolds $(M,g)$ with strictly positive sectional curvature. Their fundamental groups are finite: compactness gives a positive uniform lower curvature bound for each metric, and the corresponding positive Ricci bound applies to the complete universal cover. For a subgroup $C\leq\pi_1(M)$, its index $[\pi_1(M):C]$ is the number of left cosets. A subgroup is cyclic if it is generated by one element; the trivial subgroup is allowed.

Is there a constant $J(n)<\infty$, depending only on dimension, such that every manifold in this class has a cyclic subgroup $C$ with \[ [\pi_1(M):C]\leq J(n)? \] The metric, manifold, and cyclic subgroup may vary; the index bound must not depend on a curvature pinching ratio, a diameter or volume bound, or the order of the fundamental group. Normality of the cyclic subgroup is not required. The question bounds the noncyclic portion of the group rather than its total cardinality, which can be arbitrarily large even among constant-curvature examples. One may equivalently ask whether every such manifold has a finite covering of uniformly bounded degree whose fundamental group is cyclic. No isometric torus action or other local or global symmetry is assumed. The restriction to odd dimension is part of the source problem and avoids replacing it with the different even-dimensional fundamental-group restrictions.

Short English statement

In each odd dimension, do all positively curved closed manifolds have fundamental groups containing a cyclic subgroup of uniformly bounded index?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20003178 (AIM-TOPOLOGY-0266), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Manifolds with nonnegative sectional curvature, item 53. Original workshop question underlying AIM-TOPOLOGY-0266. https://aimath.org/WWN/nnsectcurvature/nnsectcurvature.pdf.