Fukaya–Yamaguchi conjecture on bounded abelian-subgroup index

Prove the Fukaya–Yamaguchi conjecture that, for complete manifolds with non-negative sectional curvature, the index of an abelian subgroup of the fundamental group is bounded by a constant depending only on the dimension.

Background

The paper places this conjecture in the broader context of rigidity phenomena in which non-negative curvature forces virtually nilpotent structures to become virtually abelian. Fukaya and Yamaguchi proposed a uniform strengthening of the fact that the fundamental group of a compact non-negatively curved manifold is virtually abelian: under non-negative sectional curvature, the index of an abelian subgroup should admit a bound depending solely on the manifold’s dimension.

The authors explicitly note that this conjecture remains unresolved. They contrast it with the analogous assertion for non-negative Ricci curvature, which was recently disproved, emphasizing that the sectional-curvature version is still an open problem.

References

We conjecture that this constant should in fact be universal, depending only on $n$ and not on $\theta$.

— Fundamental Groups in the Five Dimensional Pan-Rong Conjecture  (2609.09080 - Huang, 8 Sep 2026) in Remark following Theorem 1.3, Section 1, Introduction

Fukaya and Yamaguchi conjectured that, under non-negative sectional curvature, the index of the abelian subgroup should be bounded by a constant depending only on the dimension. This conjecture remains open.

— Ollivier--Ricci Curvature on Groups of Polynomial Growth  (2608.14232 - Brena et al., 14 Aug 2026) in Subsection “Nilpotent structures and curvature”

These conjectures are:

The fundamental group of a non-negatively curved m-manifold is C(m)-abelian, i.e. it contains an abelian subgroup of index at most C(m), where C(m)<\infty is a constant which only depends on the dimension m.

— Torus actions, almost non-negative curvature and fundamental groups  (2608.27249 - Wiemeler, 27 Aug 2026) in Conjecture in Section 1, Introduction