On the geometry of a proposed curve complex analogue for $Out(F_n)$
On the geometry of a proposed curve complex analogue for $Out(F_n)$
复制标题
关于 $Out(F_n)$ 的拟议曲线复数模拟的几何形状
DOI：
--
复制
发表时间：
2010
期刊：
影响因子：
--
通讯作者：
D. Savchuk
中科院分区：
文献类型：
--
作者：
Lucas Sabalka;D. Savchuk
自由群的外自同构中的群$Out$多年来一直是一个活跃的研究对象，但它的几何结构却没有被很好地理解。最近，人们一直致力于寻找$out$作用于其上的双曲复形，类似于映射类群的曲线复形。在这里，我们集中在这些建议的类似物之一：边分裂复合体$Esc$，相当于被称为分离球复合体。我们对其1-骨架中的测地线路径进行了代数刻画，并利用我们的刻画找到了图中各点之间距离的下界。
我们的距离计算允许我们在$Esc$中找到任意维度的准平坦。这表明$Esc$：不是双曲的，有无穷个渐近维，并且每个渐近锥都是无穷维的。这些拟平坦包含一个由可约元素$out$组成的无界轨道。因此，在$ESC$和其他被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$FSC$，(非平凡交)自由因式分解复形$FFZC$和自由因子复形$FFC$，这使得人们希望这些复形中的一些是双曲的。
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.