Question: For which "interesting" classes of finitely generated groups is it known whether every infinite group in the class has an element of infinite order?
For finitely generated abelian groups the question trivially has a positive answer.
For recursively presented groups the answer is negative. An example of a periodic such group is the Grigorchuk group.
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 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.