论文标题
可解决群体的行动的增长
Growth of actions of solvable groups
论文作者
论文摘要
鉴于有限生成的组$ g $，我们对$ g $的所有忠实动作图的常见几何属性感兴趣。在本文中，我们专注于它们的成长。我们说，如果每个忠实的$ g $ -s $ x $满足$ \ mathrm {vol} _ {g，x}（n）\ succcurlyeq f（n）$，则一组$ g $具有Schreier增长差距$ f（n）$。在这里，我们研究了有限生成的可解决基团的Schreier生长差距。 我们证明，如果Metabelian $ g $有限呈现或不含扭转，那么$ G $具有Schreier增长差距$ n^2 $，前提是$ G $几乎不是Abelian。我们还证明，如果$ g $是一个krull dimension $ k $的一组，那么$ g $具有Schreier增长差距$ n^k $。例如，花圈产品$ C_P \ WR \ MATHBB {Z}^D $具有Schreier增长差距$ n^d $，而$ \ Mathbb {Z} \ wr \ Mathbb {Z}^d $具有Schreier Growth Gap $ n^{D+1} $。这些下限很锋利。对于可解决的有限Prüfer等级的组，我们建立了一个Schreier增长差距$ \ exp（n）$，前提是$ G $几乎不是nilpotent。这涵盖了在$ \ mathbb {q} $上线性的所有可解决的组。最终，对于包括线性的可解决的群体，我们建立了Schreier增长差距$ n^2 $。
Given a finitely generated group $G$, we are interested in common geometric properties of all graphs of faithful actions of $G$. In this article we focus on their growth. We say that a group $G$ has a Schreier growth gap $f(n)$ if every faithful $G$-set $X$ satisfies $\mathrm{vol}_{G, X}(n)\succcurlyeq f(n)$, where $\mathrm{vol}_{G, X}(n)$ is the growth of the action of $G$ on $X$. Here we study Schreier growth gaps for finitely generated solvable groups. We prove that if a metabelian group $G$ is either finitely presented or torsion-free, then $G$ has a Schreier growth gap $n^2$, provided $G$ is not virtually abelian. We also prove that if $G$ is a metabelian group of Krull dimension $k$, then $G$ has a Schreier growth gap $n^k$. For instance the wreath product $C_p \wr \mathbb{Z}^d$ has a Schreier growth gap $n^d$, and $\mathbb{Z} \wr \mathbb{Z}^d$ has a Schreier growth gap $n^{d+1}$. These lower bounds are sharp. For solvable groups of finite Prüfer rank, we establish a Schreier growth gap $\exp(n)$, provided $G$ is not virtually nilpotent. This covers all solvable groups that are linear over $\mathbb{Q}$. Finally for a vast class of torsion-free solvable groups, which includes solvable groups that are linear, we establish a Schreier growth gap $n^2$.
