I have looked around in the literature on group theory and geometric group theory and this looks to be an open question as far as I can tell (by torsion group, I mean as usual a group in which every element has finite order).


I was wondering if anyone has recently made any progress on this question or if there is some review article which looks at the possibilities?



Can a torsion-free group be quasi-isometric to a torsion group?
This is one of many open questions in geometric group theory related to quasi-isometries. Proving things about invariance under quasi-isometries is generically quite tricky, as quasi-isometries do not even need to be continuous. Some other open questions:


* Is the Haagerup property invariant under quasi-isometries? (see comment by YCor for recent work on this one)
* Is the rapid decay property invariant under quasi-isometries?
* Is the property of having uniform exponential growth invariant under quasi-isometries?
* Are random finitely presented groups quasi-isometry rigid?
* How can fundamental groups of compact $3$-manifolds be classified up to quasi-isometry?


