关于$Out(F_n)$的曲线复模拟的几何性质
On the geometry of a proposed curve complex analogue for $Out(F_n)$
摘要：
自由群的外自同构群$\Out$多年来一直是一个活跃的研究对象，但它的几何还没有很好地理解。最近，努力已经集中在寻找一个双曲复杂的$\Out$的行为，在类似的曲线复杂的映射类群。在这里，我们专注于这些建议的类似物之一：边缘分裂复杂$\ESC$，相当于称为分离球复杂。我们的1-骨架代数特征的测地线路径，并使用我们的表征，找到这个图中的点之间的距离的下界。我们的距离计算使我们能够在$\ESC$中找到任意维数的拟平坦。这表明，$\ESC$：不是双曲的，有无限的渐近维数，是这样的，每个渐近锥面是无限维的。这些拟平坦包含$\Out$的一个可约元的无界轨道。因此，在$\ESC$和其他提出的曲线复形类似物之间不存在粗略的$\Out$ -等变拟等距，包括正则自由分裂复形$\FSC$、（非平凡相交）自由因子分解复形$\FFZC$和自由因子复形$\FFC$，这使得希望这些复形中的一些是双曲的。
英文摘要：
The group $\Out$ of outer automorphisms of the free group has been an object of active study for many years, yet its geometry is not well understood. Recently, effort has been focused on finding a hyperbolic complex on which $\Out$ acts, in analogy with the curve complex for the mapping class group. Here, we focus on one of these proposed analogues: the edge splitting complex $\ESC$, equivalently known as the separating sphere complex. We characterize geodesic paths in its 1-skeleton algebraically, and use our characterization to find lower bounds on distances between points in this graph. Our distance calculations allow us to find quasiflats of arbitrary dimension in $\ESC$. This shows that $\ESC$: is not hyperbolic, has infinite asymptotic dimension, and is such that every asymptotic cone is infinite dimensional. These quasiflats contain an unbounded orbit of a reducible element of $\Out$. As a consequence, there is no coarsely $\Out$-equivariant quasiisometry between $\ESC$ and other proposed curve complex analogues, including the regular free splitting complex $\FSC$, the (nontrivial intersection) free factorization complex $\FFZC$, and the free factor complex $\FFC$, leaving hope that some of these complexes are hyperbolic.
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