Abelian subgroups of the fundamental group of a space with no conjugate points

TL;DR

The paper addresses the structure of fundamental groups of compact locally simply connected length spaces with no conjugate points, proving that every Abelian subgroup is free abelian of rank $k$ with $0\le k\le d$, and that solvable subgroups are Bieberbach groups. It combines an asymptotic norm $|\cdot|_\infty$ on Abelian subgroups with Lipschitz extension techniques to produce equivariant maps that yield rank bounds and rigid algebraic conclusions. In the Riemannian setting, a Busemann-function-based approach constructs equivariant maps and a bilinear form $B$ that mimics inner-product properties, supporting the identification of $\mathbb{Z}^k$-subgroups and the Bieberbach-type conclusion. Collectively, the results reinforce the viewpoint that spaces with no conjugate points resemble nonpositively curved spaces at the level of fundamental groups, with precise abelian and solvable subgroup structure characterized.

Abstract

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.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Lemma 3
• Lemma 4
• Lemma 5
• proof : Proof of Lemma \ref{['solid cone']}