15
$\begingroup$

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?

Improve this question
$\endgroup$
6
  • 4
    $\begingroup$ The trivial group is both torsion-free and torsion. But apart from this exception I think it's an open question. Already, one can ask if being torsion is a QI-invariant. $\endgroup$
    – YCor
    Commented Apr 20, 2020 at 20:49
  • $\begingroup$ If torsion is in fact an invariant under quasi-isometries, wouldn't that give a negative answer to my question? $\endgroup$
    – Hollis Williams
    Commented Apr 20, 2020 at 21:15
  • 1
    $\begingroup$ Yes of course. But on the other hand, proving that my question has a negative answer might be easier. Of course which question is harder depends on what's true! $\endgroup$
    – YCor
    Commented Apr 20, 2020 at 21:21
  • 3
    $\begingroup$ One can also ask if every finitely generated group is quasi-isometric to a torsion-free group. $\endgroup$
    – Moishe Kohan
    Commented Apr 21, 2020 at 18:53
  • 2
    $\begingroup$ @MoisheKohan The answer is no. Indeed it follows from Eskin-Fisher-Whyte that every group QI to a lamplighter group (finite)$\wr\mathbf{Z}$ has an infinite locally finite normal subgroup. I also guess that $\mathrm{SL}_d(\mathbf{F}_p[t])$ is a counterexample for $d\ge 3$ but I doubt it's known. $\endgroup$
    – YCor
    Commented Apr 26, 2020 at 19:01

1 Answer 1

Reset to default
2
$\begingroup$

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?
Improve this answer
$\endgroup$
1
  • 3
    $\begingroup$ The first one (on Haagerup) has been settled by Carette by combining work of Whyte [on QI between generalized Baumslag-Solitar groups) and myself–Valette (on the Haagerup property for generalized Baumslag-Solitar groups). But I'm not sure Whyte result on which it relies is confirmed. It roughly says "if 2 generalized BS groups (+ assumptions) have their associated acting groups at finite Hausdorff distance, then they are QI", so in principle it's not the hardest part of Whyte's preprint (the hardest part, not needed here, goes the reverse direction). $\endgroup$
    – YCor
    Commented Apr 26, 2020 at 19:05

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .