A group is called almost cyclic if it contains an element aa such that every element of the group is conjugate to a power of aa. V. S. Guba constructed a non-cyclic finitely generated almost cyclic group in 1986, but it is not finitely presented, and the question posed by D. V. Anosov (Kourovka Notebook Problem 8.8b) whether a non-cyclic finitely presented almost cyclic group exists remains open. Rademacher and Taimanov (2023) identified this as the exact remaining algebraic gap in the program of proving that every compact manifold with infinite fundamental group carries at least two geometrically distinct closed geodesics: if such a group (necessarily torsion-free, with aa of infinite order) were the fundamental group of a closed K(π,1)K(\pi,1)-manifold, no known method would force a second closed geodesic on it. The problem asks both for the resolution of the group-existence question and for the geometric second-geodesic statement in this exceptional case.

Background

A closed geodesic on a Riemannian or Finsler manifold is a periodic solution of the geodesic equation. A Finsler metric is a Riemannian-type metric given by a possibly non-Euclidean norm function FF on each tangent space; it is called reversible if F(v)=F(−v)F(v)=F(-v) for every tangent vector vv, as holds for a Riemannian metric, so that a geodesic traversed backwards is again a geodesic. Two closed geodesics are called geometrically distinct if neither is an iterate of the other. By a theorem of Lyusternik and Fet, every compact Riemannian manifold carries at least one closed geodesic, and a long-standing program asks how many geometrically distinct closed geodesics must exist. When the fundamental group π1(M)\pi_1(M) is infinite, each free homotopy class (i.e., conjugacy class in π1(M)\pi_1(M)) of loops contains a length-minimizing closed geodesic, so the conjugacy structure of π1(M)\pi_1(M) controls the supply of closed geodesics. Bangert and Hingston (1984) proved that if π1(M)\pi_1(M) is infinite abelian then MM carries infinitely many geometrically distinct closed geodesics, and Ballmann, Thorbergsson and Ziller (1981) obtained infinitely many closed geodesics when all closed geodesics of the metric are hyperbolic (the linearized return map along the geodesic has no eigenvalue of modulus 1) and π1(M)\pi_1(M) either contains a torsion element or has two iterates of a nontrivial element with coinciding conjugacy classes.

Rademacher and Taimanov (2023) reduced the remaining case: if a compact manifold with torsion-free nontrivial infinite fundamental group carried a Riemannian metric with only one geometrically distinct closed geodesic, then π1(M)\pi_1(M) must be almost cyclic, meaning there is an element aa with ⟨a⟩≅Z\langle a\rangle\cong\mathbb{Z} (the cyclic subgroup generated by aa is infinite cyclic) such that every g∈π1(M)g\in\pi_1(M) can be written g=hamh−1g=h a^m h^{-1} for some hh and integer mm; moreover the abelianization H1(M;Z)H_1(M;\mathbb{Z}) must be finite cyclic. They then observed, in their Remark 1, that the existence of a non-cyclic finitely presented almost cyclic group is an open problem: Problem 8.8 in the Kourovka Notebook (the long-running Novosibirsk collection of unsolved problems in group theory), posed by D. V. Anosov in 1982, asked in part (a) whether a non-cyclic finitely generated such group exists — answered yes by Guba (1986/87), whose example is a finitely generated divisible group (a group in which every element has nn-th roots for all nn) — while part (b), the finitely presented version, remains open. Guba's group is infinitely related, and a group is finitely presented (admits a presentation with finitely many generators and finitely many defining relations) precisely when it can occur as the fundamental group of a compact manifold; a closed K(π,1)K(\pi,1)-manifold (also called aspherical: its universal cover is contractible, so all homotopy is carried by the fundamental group) additionally requires the group to be torsion-free. Following unpublished work of B. Ikenaga, such groups are called almost cyclic; Ikenaga (2005) proved that solvable almost cyclic groups and one-relator almost cyclic groups (groups admitting a presentation with a single defining relator) are cyclic, so a non-cyclic example must be neither solvable nor a one-relator group. Recent work of Shelukhin and Zhang (2023) shows that if for some m≥2m\ge 2 the mm-th homotopy group πm(M)\pi_m(M) — an abelian group on which π1(M)\pi_1(M) acts by change of base point — has nontrivial invariants or coinvariants under this action, then a C4C^4-generic Riemannian metric carries infinitely many geometrically distinct non-contractible closed geodesics, and that if moreover π1(M)\pi_1(M) has infinitely many conjugacy classes then every Riemannian metric does. Since these conclusions require nontrivial πm(M)\pi_m(M) for some m≥2m\ge 2, they do not reach closed K(π,1)K(\pi,1)-manifolds; combined with the reduction above, the almost-cyclic case, where the entire group is conjugate into a single infinite cyclic subgroup, remains the precise remaining obstruction in the torsion-free setting.

Problem Statement

Two linked open questions arising from Rademacher–Taimanov (2023, Remark 1): the first is Kourovka Notebook Problem 8.8 b) posed by D. V. Anosov, and the second formalizes the limitation of method recorded there:

(1) Does there exist a non-cyclic finitely presented group GG containing an element aa such that every element of GG is conjugate to a power of aa (i.e., for every g∈Gg\in G there exist h∈Gh\in G and m∈Zm\in\mathbb{Z} with g=hamh−1g=h a^m h^{-1})? For the geometric application below the group must moreover be torsion-free with ⟨a⟩≅Z\langle a\rangle\cong\mathbb{Z}.

(2) If such a torsion-free finitely presented almost cyclic group exists and is the fundamental group of a closed K(π,1)K(\pi,1)-manifold MM, does every Riemannian metric (or reversible Finsler metric) on MM nevertheless carry at least two geometrically distinct closed geodesics? An affirmative answer would close, within the closed K(π,1)K(\pi,1) setting, the last remaining case left by the Rademacher–Taimanov reduction for compact manifolds with torsion-free infinite fundamental group, in the program of proving that every compact manifold with infinite fundamental group admits at least two geometrically distinct closed geodesics.

References

  1. Hans-Bert Rademacher and Iskander A. Taimanov, The second closed geodesic, the fundamental group, and generic Finsler metrics, Mathematische Zeitschrift 302 (2023), no. 1, 629–640. DOI: 10.1007/s00209-022-03062-z, https://arxiv.org/abs/2011.01909
  2. E. I. Khukhro and V. D. Mazurov (eds.), Unsolved Problems in Group Theory: The Kourovka Notebook, No. 21, Sobolev Institute of Mathematics, Novosibirsk, 2026 (problem 8.8 b posed by D. V. Anosov, 8th issue, 1982). arXiv: 1401.0300v45, https://arxiv.org/abs/1401.0300
  3. V. S. Guba, A finitely generated complete group, Mathematics of the USSR-Izvestiya 29 (1987), no. 2, 233–277 [Russian original: Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 883–924]. DOI: 10.1070/im1987v029n02abeh000969, https://doi.org/10.1070/im1987v029n02abeh000969
  4. Bruce Ikenaga, Almost cyclic groups, arXiv preprint math/0511400 (2005), https://arxiv.org/abs/math/0511400
  5. W. Ballmann, G. Thorbergsson, and W. Ziller, Closed geodesics and the fundamental group, Duke Mathematical Journal 48 (1981), no. 3, 585–588. DOI: 10.1215/s0012-7094-81-04832-8, https://doi.org/10.1215/s0012-7094-81-04832-8
  6. Victor Bangert and Nancy Hingston, Closed geodesics on manifolds with infinite abelian fundamental group, Journal of Differential Geometry 19 (1984), no. 1, 277–282. DOI: 10.4310/jdg/1214438679, https://doi.org/10.4310/jdg/1214438679
  7. Egor Shelukhin and Jun Zhang, Remark on non-contractible closed geodesics and homotopy groups, arXiv:2307.13877 (2023), https://arxiv.org/abs/2307.13877