> <b> Question:</b> For which "interesting" classes of finitely generated groups is
> it known whether every infinite group in the class has an element of infinite order?

Some examples:

  1. For finitely generated abelian groups the question trivially has a positive answer.

  2. For recursively presented groups the answer is negative.
     An example of a periodic such group is the [Grigorchuk group][1].

  3. For finitely presented groups, if I recall correctly, the question is still open.

The motivation for this question is that I am trying to find out whether the class of
finitely generated subgroups of the group discussed [here][2] has this property.
Extensive systematic searches by computer have been inconclusive so far,
i.e. the results found so far neither point to a reason why there are always elements
of infinite order nor do they reveal a way to construct a periodic group.
Knowing suitable other classes of groups for which the answer is known might help me
in getting further.

  [1]: http://en.wikipedia.org/wiki/Grigorchuk_group
  [2]: http://mathoverflow.net/questions/112469/characterization-of-the-elements-of-an-infinite-simple-group