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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.

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    $\begingroup$ M. Ershov, Golod-Shafarevich groups: a survey. Internat. J. Algebra and Computation, vol. 22 (2012) $\endgroup$ Feb 1, 2017 at 1:12
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    $\begingroup$ Did you see theorem 3.3 in that paper? I think it answers at least your 2nd question. $\endgroup$ Feb 1, 2017 at 6:49
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    $\begingroup$ 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. $\endgroup$ Feb 1, 2017 at 7:35
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    $\begingroup$ I think it would be helpful if somebody answers the question at this stage! $\endgroup$
    – Derek Holt
    Feb 1, 2017 at 21:08
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    $\begingroup$ @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, 2017 at 21:59

1 Answer 1

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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.

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    $\begingroup$ This is also true for $d = 1$. $\endgroup$ Feb 2, 2017 at 0:59
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    $\begingroup$ Aren't all 0-generated groups finite? $\endgroup$ Feb 2, 2017 at 4:03

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