Following this [question](https://mathoverflow.net/questions/214893/infinite-groups-of-finite-exponent-inside-of-sl2-c) I wonder about the following. Examples of infinite torsion groups which are linear in zero characteristic are infinite groups of roots of unit. 


1. Are there other examples which do not contain examples as above as subgroups?
2. Are there non-abelian examples? (How far can they be from abelian?)


(Note that the groups have to be infinitely generated and not of finite exponent.)



How bad can an infinite linear torsion group be?
If $K$ is a field of characteristic zero and $G\subset\mathrm{GL}\_d(K)$ is torsion, then $G$ is virtually abelian, and more precisely the Zariski closure of $G$ is a virtual torus (i.e. its unit component is a torus).


Indeed, $G$ is locally finite, so is a directed union of its finite subgroups; since by Jordan-Zassenhaus its finite subgroups have abelian subgroups of bounded index, $G$ is virtually abelian. So if $H$ is the unit component of its Zariski closure, then $H$ is abelian. So $H=H\_u\times H\_s$ with $H\_u$ unipotent, but the projection of a locally finite group on $H\_u$ has to be trivial, thus $H\_u=1$. So $H=H\_s$ is a torus.


