Following this question 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.)

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    $\begingroup$ Certainly there are non-Abelian examples: the semidirect product of the group of all torsion elements in a fixed maximal torus and the Weyl group of that maximal torus. You could ask whether every example is virtually Abelian. $\endgroup$ – Jason Starr Aug 17 '15 at 21:15
  • $\begingroup$ Thanks. I suspected something like that exists that is why I put the question in brackets. $\endgroup$ – Yiftach Barnea Aug 17 '15 at 21:47

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.

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  • $\begingroup$ That is a beautiful answer. Are you using the Tits alternative to conclude local finiteness? $\endgroup$ – Jason Starr Aug 18 '15 at 11:36
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    $\begingroup$ No need for the Tits alternative it follows from Burnside's original result. $\endgroup$ – Yiftach Barnea Aug 18 '15 at 14:11
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    $\begingroup$ Yes, no need. The Burnside result is a 1-generator version of Tits' alternative and could be called the Burnside alternative (any linear group is either locally finite or contains an infinite cyclic subgroup). $\endgroup$ – YCor Aug 18 '15 at 15:54

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