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.



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


