Remarks on IA--automorphisms of groups
Bludov V.V.
An automorphism $\varphi$ of a group $G$ is called IA-automorphism if
$\varphi$ induces the identical automorphism on $G/G'$.
The set of all IA-automorphisms of a group $G$ forms the subgroup
$\mbox{IA }G$ and $\mbox{Inn }G<\mbox{IA }G<\mbox{Aut } G$.
V. Shpilrain \cite{s} studied endomorphisms of free
metabelian groups and posed two problems:
{\sc Problem 1.} Suppose $F$ is a free group and $\varphi\in \mbox{Aut } F$
is a non-inner IA-automorphism with a non-trivial fixed point. Is it true
that $\varphi$ has a fixed point inside $[F,f]$?
{\sc Problem 2.} Is it true that every IA-automorphism of a free
metabelian group of finite rank has a non-trivial fixed point?
Another problem concerning IA-automorphisms was posed in the list
``OPEN PROBLEMS in combinatorial group theory'' of
``MAGNUS PROJECT''
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http://zebra.sci.ccny.cuny.edu/web/problems
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(M3) If $G$ is free metabelian, is $\mbox{IA }G/\mbox{Inn }G$ torsion-free?
In the talk we give answers to Problems 1 and 2 of V. Shpilrain, and to
Problem M3.
{\sc Example 1.} Let G be a free group with free generators
$a_1,\dots,a_n,b$ and let $\varphi$ be an automorphism
of $G$ extending the mapping: $$a_i\rightarrow a_i^{a_1},\quad i=1,\dots,n,
\quad b\rightarrow b^{a_2}.$$
Fixed points of $\varphi$ form a cyclic subgroup $$.
{\sc Example 2.} Let G be a free solvable group with free
generators $a_1,\dots, a_n$, $b$, $n>1$ and let $\varphi$ be an automorphism
of $G$ extending the mapping: $$a_i\rightarrow a_i^c,\quad i=1,\dots,n,
\quad b\rightarrow [a_1,a_2]b.$$
Direct calculations in a suitable basis of $G$ show that $\varphi$ is
IA-automorphism without non-trivial fixed points.
{\sc Theorem 1.} If $G$ is a free polynilpotent group then
$\mbox{IA }G/\mbox{Inn }G$ is torsion-free.
\begin{thebibliography}{1}
\bibitem{s} V. Shpilrain. Fixed pointsof endomorphisms of a free metabelian
groups. {\it Math. Proc. Cambridge Philos. Soc.} (1998), {\bf 123},
75--83.
\end{thebibliography}