论文标题
控制某些组扩展的LEF增长
Controlling LEF growth in some group extensions
论文作者
论文摘要
我们研究了有限生成的LEF组$γ$的LEF增长函数，该$γ$衡量了有限组的命令，该命令在$γ$上的单词度量中允许球的局部嵌入。我们证明，$ n！$和$ \ exp（\ exp（n））$之间的任何足够平滑的增加功能都接近某些有限生成的组的LEF增长函数。这是通过估计$ fsym（ω）\ rtimesγ$的某些半领产品的LEF增长来实现的，其中$γ\ curvearrowrowrowrowrowrowrowrowchrightω$是一种适当的传递动作，而$ fsym（ω）$是$ $ω$的最有限支持的排列。证明中的一个关键工具是确定有限呈现的子组的序列，这些序列具有简短的“相对”呈现。同样，我们还获得了$e_Ω（r）\ rtimesγ$的某些组的LEF增长的估计，对于$ r $，适当的Unital环和$e_Ω（r）$ $ aut_r（r [ω]）$的子组与所有转移产生的均与基础$ω$相关。
We study the LEF growth function of a finitely generated LEF group $Γ$, which measures the orders of finite groups admitting local embeddings of balls in a word metric on $Γ$. We prove that any sufficiently smooth increasing function between $n!$ and $\exp(\exp(n))$ is close to the LEF growth function of some finitely generated group. This is achieved by estimating the LEF growth of some semidirect products of the form $FSym (Ω) \rtimes Γ$, where $Γ\curvearrowright Ω$ is an appropriate transitive action, and $FSym (Ω)$ is the group of finitely supported permutations of $Ω$. A key tool in the proof is to identify sequences of finitely presented subgroups with short "relative" presentations. In a similar vein we also obtain estimates on the LEF growth of some groups of the form $E_Ω (R) \rtimes Γ$, for $R$ an appropriate unital ring and $E_Ω (R)$ the subgroup of $Aut_R (R[Ω])$ generated by all transvections with respect to basis $Ω$.
