Subgroups of $\mathrm{GL}(n,\mathbb{Z})$ which are not finitely generated

The group $\mathrm{GL}(n,\mathbb{Z})$ is finitely generated: take for example diagonal matrices, permutations and one elementary matrix (upper triangular). Are there some simple / nice examples of non-finitely generated subgroups? If possible, for $n$ not too big.

Solution 1:

For $n \ge 2$, $GL(n,\mathbb{Z})$ contains a subgroup isomorphic to the rank 2 free group $F_2$ (this is a very simple case of the Tits alternative). The commutator subgroup of $F_2$ is not finitely generated.

Solution 2:

I want to use the same idea as Lee Mosher, based around the well-known free subgroup, but give a more explicit example (that is, the generators have a nicer form). In the free group $F(a, b)$ the set $\{a^kba^k:k\in\mathbb{Z}\}$ freely generates a free group on countably many generators (for a proof, see here). So, the matrices of the following form generate a non-finitely generated group of $GL(2,\mathbb{Z})$. $$ \left( \begin{array}{cc} 1&0\\ 2&1 \end{array} \right)^k \left( \begin{array}{cc} 1&2\\ 0&1 \end{array} \right) \left( \begin{array}{cc} 1&0\\ 2&1 \end{array} \right)^k $$ These matrices are easily seen to have the following form. $$ \left( \begin{array}{cc} 4k+1&2\\ 4k(2k+1)&4k+1 \end{array} \right) $$

As an aside, $\operatorname{GL}(n, \mathbb{Z})$ also contains finitely generated groups which are not finitely presentable. I cannot think of an obvious proof of this fact, and indeed my reasoning requires some serious results! The first result is about small cancellation groups. Small cancellation groups are groups with presentations where the relators do not interact very much with one another (any cancellation between the relators is "small"). It is a recent result, using some rather complicated machinery, that such groups are linear over $\mathbb{Z}$, so given a small cancellation group $G$ there exists some $n$ such that $G$ embeds into $\operatorname{GL}(n, \mathbb{Z})$. It is then a result of Rips that there exist small cancellation groups which contain finitely generated subgroups which are not finitely presentable: see Rips' paper Subgroups of small cancellation groups.