The fundamental group of compact manifolds without conjugate points
A complete Riemannian manifold M has no conjugate points if every Jacobi vector field Y along a geodesic \(\gamma\) of M vanishes at most once on \(\gamma\). Equivalently, a manifold M has no conjugate points if the universal Riemannian cover \(\tilde M\) admits a unique geodesic joining any two given distinct points. Manifolds with nonpositive sectional curvature are the most common examples of manifolds without conjugate points. Consequently, it is challenging to extend geometric properties of nonpositively curved manifolds to the larger class of manifolds without conjugate points. These extensions are often difficult since most of the convexity properties of nonpositively curved manifolds are absent in a general manifold without conjugate points. The outstanding problem of this type is the Hopf problem to show that any n-torus without conjugate points is flat. A closely related direction of research is to determine algebraic properties of the fundamental group of a manifold M without conjugate points; such a manifold M is a K(\(\pi\),1)- space and the fundamental group contains the homotopy of M. In particular one may consider properties expressible in terms of the word length metric. If M is compact then its fundamental group \(\Gamma\) is finitely generated. If \(S=\{\gamma_ 1,...,\gamma_ p\}\) is a fixed set of generators for \(\Gamma\), then one may define a word length metric \(| |\) by setting \(| \phi |\), for any element \(\phi\) of \(\Gamma\), to be the minimum length of a word in the generators \(\{\gamma_ i\}\) and their inverses that represents \(\phi\). Different generating sets \(S_ 1,S_ 2\) define equivalent word length metrics \(| |_ 1\) and \(| |_ 2\); that is, 1/c\(\leq | \phi |_ 1/| \phi |_ 2\leq c\) for some \(c>0\) and all \(\phi\in \Gamma\). A finitely generated subgroup \(\Gamma_ 0\) in \(\Gamma\) is called straight if the word length metric on \(\Gamma_ 0\) is equivalent to the restriction of the word length metric on \(\Gamma\) to \(\Gamma_ 0.\) In this paper the authors prove the following main theorem. Let N be a compact manifold that admits an analytic metric without conjugate points. Then every abelian subgroup of the fundamental group of N is straight. As corollaries the authors obtain two results: Theorem A. Let N be a compact manifold that admits a \(C^{\infty}\) metric without conjugate points. Then every nilpotent subgroup of the fundamental group of N is abelian. Theorem B. Let N be as in the main theorem. Then every solvable subgroup \(\Sigma\) of the fundamental group of N is a Bieberbach group. In particular \(\Sigma\) has a finite index abelian subgroup. The real analyticity assumption in the main theorem may be weakened to a certain weak rectifiability condition on the set of points that lie on shortest closed geodesics in a given free homotopy class. The main technical result is an extension to compact manifolds without conjugate points of certain basic properties about Busemann functions in the universal cover \(\tilde M\) that are known for nonpositively curved manifolds.
scientific article; zbMATH DE number 3984055