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How many groups there are (up to isomorphism) with a presentation with at most $n$ generators and with relators of length at most $3$? I don't expect there exist a sharp solution, since I know that the group isomorphism problem is undecidable (even though I wonder if it's easier when we give an upper bound on both the number of generators and the length of the relators). On the other hand there is an obvious combinatoric estimate $$N\leq 2^n\cdot 2^{n^3},$$ choosing among all possible generators and relations. This estimate is very rough though, as for example we may pick a lot of relations among generators that don't even appear in $S$.

So, what is the best estimate we can make?

I would be happy either to have nice combinatoric estimates (for example it can go down by a factor of $n!$ simply by looking at permutations, can we do better?), but expecially if there were group theory arguments, even if they apply to specific classes of groups. For example I would be satisfied to have an estimate for the number of hyperbolic groups with such a presentation.

Dinisaur
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    Any group presentation can be transformed by Tietze transformations to a presentation of this form, so you cannot expect the isomorphism problem to be any easier. – Derek Holt Nov 02 '23 at 19:13
  • @DerekHolt thats interesting! Can it be done without increasing the number of generators? – Dinisaur Nov 02 '23 at 23:49
  • No, you usually have to increase the number of generators. The result is easy to prove. – Derek Holt Nov 03 '23 at 00:16
  • Well then, shouldn't the isomorphism problem be decidable if we restrict it to the class of $n$-bounded short presentations? After all we can enumerate them... I guess it would be harder, and maybe still undecidable, if we restrict it to the calss of groups that admit a $n$-bounded short presentation... – Dinisaur Nov 03 '23 at 10:34
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    Are you interested in the better "obvious combinatoric estimates" like $n^{n^2 + 1}$, or are you looking for improvements that involve some (non-trivial) group theory? – Mees de Vries Nov 07 '23 at 16:23
  • I was hoping for some group theory to bring down the estimate as much as possible, but even some combinatoric improvement would be acceptable in absence of any other ideas – Dinisaur Nov 07 '23 at 16:59

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