# Is there a general method for computing finitely generated normalizers?

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.

• 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?
– HJRW
Mar 19 at 7:14
• If your input consists just of finite presentations of the group and subgroup then the problem is almost certainly theoretically undecidable. Mar 19 at 8:15
• @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.
– YCor
Mar 19 at 15:11
• @HJRW Finitely generated, a systematic procedure that outputs generators.
– Jim
Mar 19 at 16:12
• @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). Mar 19 at 22:51

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.