9
$\begingroup$

I know of Tarski monsters and the Burnside Problem. I would like to know if there is an infinite finitely generated group $G$ such that for any $g$ and $h$ in $G$, the subgroup generated by $\{g,h\}$ is finite.

I am also interested in related questions:

  • For which $(m,n)$ does there exist an infinite $m$-generated group every $n$-generated subgroup of which is finite?

  • For which $(m,n,k)$ does there exist an infinite $m$-generated group every $n$-generated subgroup of which has cardinality at most $k$?

Thank you.

$\endgroup$
  • 2
    $\begingroup$ M. Ershov, Golod-Shafarevich groups: a survey. Internat. J. Algebra and Computation, vol. 22 (2012) $\endgroup$ – Moishe Kohan Feb 1 '17 at 1:12
  • 2
    $\begingroup$ Did you see theorem 3.3 in that paper? I think it answers at least your 2nd question. $\endgroup$ – Moishe Kohan Feb 1 '17 at 6:49
  • 2
    $\begingroup$ And your first question, of course. A look at details of the proof can give you some estimates for the last question. I doubt there is a complete answer for that one. $\endgroup$ – Moishe Kohan Feb 1 '17 at 7:35
  • 1
    $\begingroup$ I think it would be helpful if somebody answers the question at this stage! $\endgroup$ – Derek Holt Feb 1 '17 at 21:08
  • 1
    $\begingroup$ @IanAgol Theorem 3.3 solves the (d,d-1)-case, and solving (d,d-1) implies solving (d,n) for n<d-1 (just taking the same group), so "we don't even have to look at the proof". $\endgroup$ – user56097 Feb 1 '17 at 21:59
8
$\begingroup$

Questions 1 and 2 are answered by Golod's theorem, see Theorem 3.3 in the survey:

M. Ershov, Golod-Shafarevich groups: a survey. Internat. J. Algebra and Computation, vol. 22 (2012).

Namely, for every $d\ge 2$ there exists an infinite $d$-generated group such that every $d-1$-generated subgroup is finite.

(He even gets a finite $p$-group for the given prime number $p$ but this is irrelevant.)

As for the last question, I am not sure, most likely it is unknown. I suggest to ask Ershov directly.

| cite | improve this answer | |
$\endgroup$
  • 2
    $\begingroup$ This is also true for $d = 1$. $\endgroup$ – Geoffrey Irving Feb 2 '17 at 0:59
  • 2
    $\begingroup$ Aren't all 0-generated groups finite? $\endgroup$ – Geoffrey Irving Feb 2 '17 at 4:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.