Subgroups of $GL_n(\mathbb Z)$ with finite coinvariants - MathOverflow

Subgroups of $GL_n(\mathbb Z)$ with finite coinvariants

2012-07-27T12:17:43Z

Is there a finite index torsion-free subgroup $G$ of $GL_n(\mathbb Z)$, where $n\ge 3$, such that the coinvariants group $\mathbb Z^n_G$ is finite?

Here $G$ acts on $\mathbb Z^n$ in the standard way, and $\mathbb Z^n_G$ by definition is the quotient of $\mathbb Z^n$ by the subgroup generated by the set $\{gz-z: g\in G, z\in\mathbb Z^n\}$ .

If $G=GL_n(\mathbb Z)$, then $\mathbb Z^n_G$ is finite because G contains $-I_n$, but then $GL_n(\mathbb Z)$ isn't torsion-free.

Answer by Wilberd van der Kallen for Subgroups of $GL_n(\mathbb Z)$ with finite coinvariants

2012-07-27T14:19:20Z

For any finite index subgroup $G$ there is a nonzero integer $m$ so that $G$ contains the elementary matrices $e_{ij}(m)$ that have ones on the diagonal, $m$ at the $(i,j)$ entry and zeroes elsewhere. So the span of $\{gz-z: g\in G, z\in\mathbb Z^n\}$ contains all multiples of $m$ and ${\mathbb Z}^n_G$ is finite. Now just take a torsion-free finite index subgroup $G$.

Edit (by Igor Belegradek): I add some detail for my records. If $e_r$ is a vector in the standard basis, then it is easy to write $me_r$ in the form $v-e_{ij}(m)v$ for some $i\neq j$ and $v\in\mathbb Z^n$. It remains to show that for any finite index subgroup $G$ of $GL_n(\mathbb Z)$ all such $e_{ij}(m)$ lie in $G$ for some $m$. By making $G$ smaller we can assume it is normal and of index $k$. Since $e_{ij}(m)=e_{ij}(1)^m$, it suffices to assume that $k$ divides $m$.

Answer by Andy Putman for Subgroups of $GL_n(\mathbb Z)$ with finite coinvariants

2012-07-27T17:27:23Z

van der Kallen gave a nice answer for the situation at hand, but I thought I'd give a somewhat more general one. The question is equivalent to showing that $(\mathbb{R}^n)_G=0$ for some finite-index torsion-free subgroup $G$ of $\text{GL}_n(\mathbb{Z})$. The representation $\mathbb{R}^n$ is a nontrivial irreducible representation of $\text{SL}_n(\mathbb{R})$, so this follows from the following more general result (just take $\Gamma$ to be a torsion-free lattice, for instance the level $3$ principle congruence subgroup).

LEMMA : Let $V$ be a nontrivial finite-dimensional irreducible representation of $\text{SL}_n(\mathbb{R})$ over $\mathbb{R}$ and let $\Gamma$ be any lattice in $\text{SL}_n(\mathbb{R})$ . Then $V_{\Gamma} = 0$.

To see this, observe that using the Borel density theorem (which says that $\Gamma$ is Zariski dense in $\text{SL}_n(\mathbb{R})$ ), we can get that $V$ is also a nontrivial irreducible $\Gamma$-representation. Now, $V_{\Gamma} = V/K$ where $K$ is spanned by the set $\{x-g(x)\text{ $|$ }x \in V, g \in \Gamma\}$. Clearly $K$ is a nontrivial $\Gamma$-subrepresentation of $V$, so by the irreducibility of $V$ we must have $K=V$.