Classes of finitely generated groups for which it is known whether they contain periodic groups 
 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:


*

*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.
 A: 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.
A: Obviously, all free Burnside groups $B(m,n)$ are recursively presentable, so examples with the uniform torsion are possible here. Of course, the proof is much more complicated than for Grigorchuk's group.
For finitely presented groups the question is opened, even for the case of unbounded exponents. The closest result is by I.Ivanov-Pogodaev and A.Kanel-Belov, that gives an example of finitely presented infinite nil-semigroup $\Pi_0$, i.e. a semigroup where every element is equal in some power to an element $0$, for which identities $0x=0$, $x0=0$ hold for every $x \in \Pi_0$.
