Each Abelian subgroup of the fundamental group of a compact and locally simply connected $d$-dimensional length space with no conjugate points is isomorphic to $\mathbb{Z}^k$ for some $0 \leq k \leq d$. It follows from this and previously known results that each solvable subgroup of the fundamental group is a Bieberbach group. In the Riemannian setting, this may be proved using a novel property of the asymptotic norm of each Abelian subgroup.

中文翻译：

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无共轭点空间基本群的阿贝尔子群

对于一些$0 \leq k \leq d$，没有共轭点的紧凑且局部简单连接的$d$维长度空间的基本群的每个阿贝尔子群与$\mathbb{Z}^k$同构。从这个和先前已知的结果可以得出，基本群的每个可解子群都是比伯巴赫群。在黎曼设置中，这可以使用每个阿贝尔子群的渐近范数的新性质来证明。