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I'm looking to compute normalizers of finite subgroups of $\mathrm{GL}(n, \mathbb{Z})$ and its possible that they are infinite but they are always finitely presented. For $\mathrm{GL}(n, \mathbb{Z})$ the solution is actually described here:

https://www.ams.org/journals/mcom/1973-27-121/S0025-5718-1973-0333025-7/S0025-5718-1973-0333025-7.pdf

My question is whether there's a general method for computing finitely presented normalizers when the group and subgroup presentations are known? The subgroup is finite.

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    $\begingroup$ Could you clarify a couple of things? The title says “finitely generated” but the question says “finitely presented”. Also, do you want a procedure that, given a subgroup, outputs a generating set for the normaliser? Or a presentation for the normaliser? $\endgroup$
    – HJRW
    Mar 19 at 7:14
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    $\begingroup$ If your input consists just of finite presentations of the group and subgroup then the problem is almost certainly theoretically undecidable. $\endgroup$
    – Derek Holt
    Mar 19 at 8:15
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    $\begingroup$ @DerekHolt I'm pretty sure that this is decidable (even the stronger version, i.e., outputting presentations). These are lattices in "computable" Zariski closed $\mathbf{Q}$-defined real subgroups of $\mathrm{GL}_n$. The geometric arguments that these groups are finitely presented should (with enough work) yield explicit presentations. $\endgroup$
    – YCor
    Mar 19 at 15:11
  • $\begingroup$ @HJRW Finitely generated, a systematic procedure that outputs generators. $\endgroup$
    – Jim
    Mar 19 at 16:12
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    $\begingroup$ @Jim Well, if you don't use any properties of $\operatorname{GL}(n, \mathbb{Z})$, then virtually every problem you can pose about computing normalisers will be undecidable (this can be extracted from the Baumslag-Boone-Neumann strengthening of Adian-Rabin). $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Mar 19 at 22:51

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Here’s a strategy that sometimes works but is likely hopeless in general. Suppose $G$ acts on a complex $K$ with the property that each finite subgroup of $G$ fixes a point. For example, if the complex $K$ happens to enjoy the CAT(0) property, this will be satisfied.

In this situation, suppose $H$ is a finite subgroup. It has a fixed subspace $X \subset K$. Each $g \in G$ sends the fixed-point set $X$ for $H$ to the fixed-point set for $gHg^{-1}$, so if $g$ normalizes $H$, then $g$ actually preserves $X$, and if $X$ has the property that $H$ is its pointwise stabilizer, the converse holds; namely, if $g$ preserves $X$, then $g$ normalizes $H$.

Anyway, there are standard techniques to attempt to compute a presentation of a group from its action on a simply connected complex, so if $X$ enjoys this property, then you might attempt to compute a presentation for $N_G(H)$ this way.

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