Let $G$ be a finitely generated group and $\varphi:G\to \operatorname{Aut}(\mathbb C)$ a homomorphism, where $\operatorname{Aut}(\mathbb C)$ is the group of complex affine transfromations $a z+b$.


Can we find a torsion free-subgroup $H$ of $G$ with finite index? And can we find a normal subgroup $H$ which is torsion-free with finite index?



Torsion-free subgroup of affine group
Yes. More generally, for any field $K$ we have an embedding of $\operatorname{Aff}(K^n)$ in $\operatorname{GL}\_{n+1}(K)$, and so if $K$ has characteristic zero we can apply Selberg's lemma to conclude that a finitely generated group of affine transformations of $K^n$ is virtually torsion-free. The normal core of any finite index torsion-free subgroup will be a normal finite index torsion-free subgroup.


