Rational Growth and Almost Convexity of Higher-Dimensional Torus Bundles-猫眼课题宝

高维环面丛的合理增长和近似凸性

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2015

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作者：

Corey Bregman

给定SL（N，）$，形成半直积$G=^N 时间_A $在哪里$$ factor作用于$^$A$。这样的$G$自然产生的基本组的$N$维环面丛纤维的圆。本文证明了：若A有不同的特征值不落在单位圆上，则对于某个生成元集，存在一个具有有理增长级数的有限指标子群HleqG。与此相反，我们表明，如果$A$有至少一个特征值不躺在单位圆，那么$G$是不是几乎凸的任何生成集。

Given a matrix $Ain SL(N,)$, form the semidirect product $G=^N times_A $ where the $$ factor acts on $^N$ by $A$. Such a $G$ arises naturally as the fundamental group of an $N$-dimensional torus bundle which fibers over the circle. In this paper we prove that if $A$ has distinct eigenvalues not lying on the unit circle, then there exists a finite index subgroup $Hleq G$ possessing rational growth series for some generating set. In contrast, we show that if $A$ has at least one eigenvalue not lying on the unit circle, then $G$ is not almost convex for any generating set.