论文标题
关于同源性增长和$ \ ell^2 $ -betti $ \ mathrm {out}（w_n）$的数字
On the homology growth and the $\ell^2$-Betti numbers of $\mathrm{Out}(W_n)$
论文作者
论文摘要
令$ n \ ge 3 $，然后让$ \ mathrm {out}（w_n）$为免费的coxeter组的外部自动形态组$ w_n $等级$ n $。我们研究了同源组的维度（在任何字段中具有系数$ \ mathbb {k} $）的增长，并按照$ \ mathrm {out}（w_n）$的有限索引子组的偏僻序列。我们表明，在所有学位上，最多$ \ lfloor \ frac {n} {2} \ rfloor-1 $，这些betti数字在子组的索引中生长。当$ \ mathbb {k} = \ mathbb {q} $，通过lück的近似定理，这意味着所有$ \ ell^2 $ -betti $ \ mathrm {out}（out}（w_n）$的数字$ \ mathrm {out}相比之下，与$ n-2 $相当的顶级，Gaboriau和Noûs的论点意味着$ \ ell^2 $ -betti号码不会消失。我们还证明，积分同源性的扭转增长是肌关系。我们对这些结果的证明取决于Abért，Bergeron，Frączyk和Gaboriau的最新方法。一个关键要素是证明$ w_n $的部分基础的复合体具有尺寸$ \ lfloor \ lfloor \ frac {n} {2} {2} \ rfloor-1 $的同型类型。
Let $n\ge 3$, and let $\mathrm{Out}(W_n)$ be the outer automorphism group of a free Coxeter group $W_n$ of rank $n$. We study the growth of the dimension of the homology groups (with coefficients in any field $\mathbb{K}$) along Farber sequences of finite-index subgroups of $\mathrm{Out}(W_n)$. We show that, in all degrees up to $\lfloor\frac{n}{2}\rfloor-1$, these Betti numbers grow sublinearly in the index of the subgroup. When $\mathbb{K}=\mathbb{Q}$, through Lück's approximation theorem, this implies that all $\ell^2$-Betti numbers of $\mathrm{Out}(W_n)$ vanish up to degree $\lfloor\frac{n}{2}\rfloor-1$. In contrast, in top dimension equal to $n-2$, an argument of Gaboriau and Noûs implies that the $\ell^2$-Betti number does not vanish. We also prove that the torsion growth of the integral homology is sublinear. Our proof of these results relies on a recent method introduced by Abért, Bergeron, Frączyk and Gaboriau. A key ingredient is to show that a version of the complex of partial bases of $W_n$ has the homotopy type of a bouquet of spheres of dimension $\lfloor\frac{n}{2}\rfloor-1$.
