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What are some tools -- either theoretical/by hand or algorithmic/by computer -- that are useful for doing computations in finitely presented groups?

In my particular case, I'm working with a finitely presented group $G=Mod(S_{0,n})$, the mapping class group of an $n$-punctured sphere. Inside there I have a subgroup $H \approx \mathbb{Z}^2$. After adding some relations to $G$ -- namely, killing high powers of the half twist Artin generators -- I expect that the image of $H$ in the quotient should now have rank less than 2. It would be great to actually have in my hand an element of $H$ which is killed by these new relations, and I doubt that guessing words in the generators of $H$ and fiddling with relations for hours by hand would be very productive.

If folks have general suggestions about tools they use for these sorts of things (not just for my question in particular), that would also be useful.

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  • $\begingroup$ This question is very broad. There are some possible answers — indeed. I’m working on new ones right now! — but it would help to have some sense of what kind of group the quotient of $G$ is. $\endgroup$
    – HJRW
    Feb 16, 2021 at 6:19
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    $\begingroup$ Edited to be a bit more specific $\endgroup$
    – Ethan Dlugie
    Feb 16, 2021 at 6:36
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    $\begingroup$ Could you give a specific example, including the presentation of $G$, the subgroup $H$, and some possible extra relations. $\endgroup$
    – Derek Holt
    Feb 16, 2021 at 9:02
  • $\begingroup$ It could be that the quotient is hyperbolic. Killing high powers of Dehn twists is done in papers by Rémi Coulon. $\endgroup$
    – markvs
    Feb 20, 2021 at 9:37
  • $\begingroup$ Yes, this is why I expect the $\mathbb{Z}^2$ subgroup to be killed. But I'd like to see more directly which elements of this subgroup are in the kernel. $\endgroup$
    – Ethan Dlugie
    Feb 21, 2021 at 15:46

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