论文标题
曼哈顿大地测量学和双曲线群的度量结构空间的边界
Manhattan geodesics and the boundary of the space of metric structures on hyperbolic groups
论文作者
论文摘要
对于任何非元素双曲线组$γ$，我们在$γ$上的公制结构空间中找到了外部自动形态的地球材料，配备了thurston Metric在TechiMüllerSpace上的thurston Metric。我们为该空间构建和研究一个边界，并表明它包含了许多众所周知的伪指标，包括来自$ \ text {cat}（0）$ cube complexs，真实树和圆锥形的cayley图的动作。作为推论，我们推断出长度频谱刚度结果，曼哈顿曲线的规律性结果，Anosov表示的最佳生长率以及有关翻译距离函数连续扩展到地理电流空间的连续扩展的结果。利用我们的结果来进行大地电流，我们以负面的猜想解决了bonahon的猜想。
For any non-elementary hyperbolic group $Γ$, we find an outer automorphism invariant geodesic bicombing for the space of metric structures on $Γ$ equipped with a symmetrized version of the Thurston metric on Techimüller space. We construct and study a boundary for this space and show that it contains many well-known pseudo metrics including those coming from actions on $\text{CAT}(0)$ cube complexes, real trees and coned-off Cayley graphs. As corollaries we deduce length spectrum rigidity results, regularity results for Manhattan curves, optimal growth rate results for Anosov representations and results regarding continuous extensions of translation distance functions to the space of geodesic currents. Using our results for geodesic currents we settle a conjecture of Bonahon in the negative.
