论文标题
粗糙的注入性，分层双曲线和半透明度
Coarse injectivity, hierarchical hyperbolicity, and semihyperbolicity
论文作者
论文摘要
我们将三类的非姿态弯曲度量空间联系起来：分层双曲线空间，粗糙的内脏空间和强烈的快捷空间。我们表明，每个分层双曲线空间都允许一个新的注入性的新指标。新的度量标准是原始度量标准的准图表，并保留在层次上双曲空间的自动形态下。我们表明，每个均匀界限几何形状的粗略注入度度量空间都是强烈的捷径。因此，层次上的双曲线组（包括映射的表面映射组）是粗略的，并且是粗略的注射式组是强烈的捷径。 使用这些结果，我们推断出层次上的双曲线组的几个重要特性，包括它们是半活纤维，存在可解决的共轭问题，具有有限的有限亚组的共轭类别，并且它们没有有限生成的阿贝尔亚组。在途中，我们表明层次双曲线组的层次上的准六子群具有限制的包装。
We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely injective spaces, and strongly shortcut spaces. We show that every hierarchically hyperbolic space admits a new metric that is coarsely injective. The new metric is quasi-isometric to the original metric and is preserved under automorphisms of the hierarchically hyperbolic space. We show that every coarsely injective metric space of uniformly bounded geometry is strongly shortcut. Consequently, hierarchically hyperbolic groups -- including mapping class groups of surfaces -- are coarsely injective and coarsely injective groups are strongly shortcut. Using these results, we deduce several important properties of hierarchically hyperbolic groups, including that they are semihyperbolic, have solvable conjugacy problem, have finitely many conjugacy classes of finite subgroups, and that their finitely generated abelian subgroups are undistorted. Along the way we show that hierarchically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing.
