
> 
>  **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?
> 
> 
> 


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](http://en.wikipedia.org/wiki/Grigorchuk_group).
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](https://mathoverflow.net/questions/112469/characterization-of-the-elements-of-an-infinite-simple-group) 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.



Classes of finitely generated groups for which it is known whether they contain periodic groups
Since the comments are already answering the question I am copying here the answers so far as Community Wiki. Please add new answers here.


* Linear groups
* Groups with a regular language of unique normal forms. This includes hyperbolic groups, automatic groups and groups with a finite complete rewriting system.


(This is very easily proved. By the Pumping Lemma for regular languages, such a language would contain a subset of the form $\{ uv^nw∣n≥0 \}$ for words $u,v,w$, and so the group element represented by $v$ must have infinite order.) 


* Relatively hyperbolic groups with "non-trivial" peripheral structure. See Agol's comment below.
* Elementary amenable groups - the smallest class of groups that contains finite and abelian groups and closed under taking extensions, direct unions, subgroups and factor groups. In particular all solvable groups belong here.


