# Normalizers in arithmetic groups

This is a question about the class of arithmetic groups. I am using the definition in Serre's survey: $$\Gamma$$ is arithmetic if it can be embedded into $$G_\mathbb{Q}$$ for some algebraic subgroup $$G \subset GL_n$$ defined over $$\mathbb{Q}$$ so that $$\Gamma$$ is commensurable with $$G_\mathbb{Z}$$. This question came up when looking at groups of homotopy classes of homotopy automorphisms, which are known to be arithmetic groups for simply-connected finite CW-complexes by work of Sullivan.

Suppose that I have an arithmetic group $$\Gamma$$ and finite subgroup $$H \subset \Gamma$$. Is it true that the normalizer $$N(H) \subset \Gamma$$ is again an arithmetic group? Of course the interesting case is when $$N$$ is not normal.

• I'm not sure there's a single universally accepted definition of "arithmetic group", could you provide a definition? Do you consider $\mathrm{SL}_n(\mathbf{Z})\ltimes\mathbf{Z}^n$ as arithmetic group? Do you consider a Zariski-dense finitely generated free subgroup of infinite index of $\mathrm{SL}_2(\mathbf{Z})$ as an arithmetic group?
– YCor
Feb 5 at 15:32
• Yes, I consider both to be arithmetic (see e.g. Example (7) of Serre's survey). Feb 5 at 15:44
• The normalizer of $N(H)$ is a $\mathbf{Q}$-defined subgroup (for every finite subgroup $H$ of $\mathrm{GL}_n(\mathbf{Q})$), and so is its intersection with $G$.
– YCor
Feb 5 at 15:49
• So that means the answer to my question is "yes" with Serre's definition, correct? Feb 5 at 15:53
• @JHM H is finite, so in particular arithmetic. The OP asked whether N(H) is arithmetic, not whether it is an arithmetic subgroup of G. Feb 5 at 19:12

The definition of "arithmetic groups" depends essentially on the definition of group of $$\mathbb{Z}$$ points of a linear algebraic group scheme $$\textbf{G}$$. The group of $$\mathbb{Z}$$ points needs be made precise. Up to finite index, we can say the group of $$\mathbb{Z}$$ points is the stabilizer of a $$\mathbb{Z}$$-module $$\Lambda\approx \mathbb{Z}^n$$, e.g. $$\Gamma=PGL(\mathbb{Z}^n)$$. Compare Platonov-Rapinchuk's "Algebraic Groups and Number Theory", pp.172--173, especially Proposition 4.2. All the standard arithmetic groups have this form ($$PGL, Sp, O$$, etc.).
We find the answer to your question is "yes" in a certain sense, if we take the lattice-stabilizer definition of arithmetic groups. In this case, we can identify $$Z_\Gamma H$$ as the stabilizer of a $$\mathbb{Z}[H]$$-lattice:
We assume that $$\Gamma$$ is defined as the stabilizer of a lattice $$\Lambda$$ in some representation $$V$$ of $$G$$ defined over $$\mathbb{Q}$$. It is enough to show the centralizer $$Z_\Gamma H$$ is "arithmetic", since the indices $$[Z_\Gamma H : N_\Gamma H]$$ and $$[Z_G H : N_G H]$$ are finite whenever $$H$$ is finite.
If $$H$$ is finite subgroup of $$\Gamma$$, then the lattice, that is, $$\mathbb{Z}$$-module $$\Lambda$$ defining $$\Gamma$$ naturally becomes a $$\mathbb{Z}[H]$$-module. Likewise the $$\mathbb{Q}$$-vector space $$V$$ defining the representation $$G$$ naturally becomes a $$\mathbb{Q}[H]$$-module. Then $$Z_\Gamma H$$ is "arithmetic" in that $$Z_\Gamma H$$ consists of all $$\mathbb{Q}[H]$$-linear automorphisms which stabilize the $$\mathbb{Z}[H]$$-module $$\Lambda$$. Thus the $$\mathbb{Z}[H]$$-module $$\Lambda$$ becomes the lattice of "integer points" inside the $$\mathbb{Q}[H]$$-module $$V$$ defining the arithmeticity of $$Z_\Gamma H$$. Even more we see $$Z_\Gamma H$$ is basically the "$$\mathbb{Z}$$-points" of $$Z_G H$$, i.e. those $$\mathbb{Q}[H]$$-automorphisms which stabilize the lattice $$\Lambda$$.