论文标题
自由metabelian群体的IA-Automormormthism的商组
Quotient groups of IA-automorphisms of free metabelian groups
论文作者
论文摘要
对于一个正整数$ n $，带有$ n \ geq 2 $，让$ m_n $是一个免费的Metabelian等级$ n $。对于$ c \ in \ mathbb {n} $，令$γ_c（m_n）$为$ m_n $的下中央系列的$ c $ - 任期。对于$ c \ geq 2 $，令$ {\ rm i} _ {c} {\ rm a}（m_n）$为$ {\ rm aut}的子组（m_ {n}）$，由所有自动化构图构成$ m_n/γ_c（m_n）$。在本文中，我们研究商组$ {\ cal l}^{c}（{\ rm ia}（m_ {n}））= {\ rm i} _ {c} {c} {\ rm a} {\ rm a} $ c $。对于$ c \ geq 2 $，我们显示$γ_{c}（{\ rm ia}（m_ {2}））= {\ rm i} _ {c+1} {c+1} {\ rm a}（m_ {2}}））$。 For $n = 3$, we show $γ_{3}({\rm IA}(M_{3})) \neq {\rm I}_{4}{\rm A}(M_{3})$ and so, the Andreadakis' conjecture (for a free metabelian group) is not valid for $n = 3$ and $c = 3$.对于$ n \ geq 4 $和$ c \ geq 3 $，我们证明$ {\ cal l}^{c} {c}（{\ rm ia}（m_ {n}））=γ_{c-1}（c-1}（{\ rm ia} a}（m_ {n}）/{\ rm i} _ {c+1} {\ rm a}（m_ {n}）$。
For a positive integer $n$, with $n \geq 2$, let $M_n$ be a free metabelian group of rank $n$. For $c \in \mathbb{N}$, let $γ_c(M_n)$ be the $c$-th term of the lower central series of $M_n$. For $c \geq 2$, let ${\rm I}_{c}{\rm A}(M_n)$ be the subgroup of ${\rm Aut}(M_{n})$ consisting of all automorphisms inducing the identity mapping on $M_n/γ_c(M_n)$. In this paper, we study the quotient groups ${\cal L}^{c}({\rm IA}(M_{n})) = {\rm I}_{c}{\rm A}(M_n)/{\rm I}_{c+1}{\rm A}(M_n)$ for all $n$ and $c$. For $c \geq 2$, we show $γ_{c}({\rm IA}(M_{2})) = {\rm I}_{c+1}{\rm A}(M_{2}))$. For $n = 3$, we show $γ_{3}({\rm IA}(M_{3})) \neq {\rm I}_{4}{\rm A}(M_{3})$ and so, the Andreadakis' conjecture (for a free metabelian group) is not valid for $n = 3$ and $c = 3$. For $n \geq 4$ and $c \geq 3$, we prove that ${\cal L}^{c}({\rm IA}(M_{n})) = γ_{c-1}({\rm IA}(M_{n})){\rm I}_{c+1}{\rm A}(M_{n})/{\rm I}_{c+1}{\rm A}(M_{n})$.
