# Finitely generated group every 2-generated subgroup of which is finite

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.

• M. Ershov, Golod-Shafarevich groups: a survey. Internat. J. Algebra and Computation, vol. 22 (2012) – Moishe Kohan Feb 1 '17 at 1:12
• Did you see theorem 3.3 in that paper? I think it answers at least your 2nd question. – Moishe Kohan Feb 1 '17 at 6:49
• 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. – Moishe Kohan Feb 1 '17 at 7:35
• I think it would be helpful if somebody answers the question at this stage! – Derek Holt Feb 1 '17 at 21:08
• @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". – user56097 Feb 1 '17 at 21:59

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.)
• This is also true for $d = 1$. – Geoffrey Irving Feb 2 '17 at 0:59