TITLE: Finitely presented group containing every $\mathrm{GL}_n(\mathbb{Z})$
QUESTION [15 upvotes]: Does there exist a concrete example of a finitely presented group that contains an isomorphic copy of $\operatorname{GL}_n(\mathbb{Z})$ for every $n\in\mathbb{N}$? I think the Higman embedding theorem implies such a group must exist, but probably not in an especially constructive way, so I'm curious if there's a concrete example. I'm also especially interested in whether there exists a type $F_\infty$ example. (I'm also generally curious if there are any interesting implications of such a group existing.)

REPLY [3 votes]: Here is a justification of why there is a general principle behind the construction. (This a formal description of what I had in mind when writting my comment.)
Let $S$ be a set and $R$ a subset of the formal union $\bigcup_{k \geq 1} S^k \backslash \{ (s_1, \ldots, s_k) \mid \exists i \neq j, s_i=s_j\}$. For every $r \in R \cap S^k$, fix a word $w_r(x_1, \ldots, x_k)$ of $k$ variables on a fixed alphabet $\{x_1,x_2, \ldots\}$. Now, define the group $G$ by the presentation
$$G(S,R,w_\cdot):= \langle S \mid w_r(r)=1, \ r \in R \rangle.$$
Next, assume that a group $H$ acts on $S$ by preserving $R$ (i.e. $hr \in R$ for all $r \in R$ and $h \in H$) and $w_\cdot$ (i.e. $w_{hr}=w_r$ for all $h \in H$ and $r \in R$). An element of $H$ permutes the generators of $G$, inducing a permutation of the words of generators, and finally an automorphism of $G$. Let's consider the associated semidirect product $G \rtimes H$.
Fact: The group $G \rtimes H$ is finite presented if the following conditions are satisfied:

$H$ is finitely presented;
$H$ acts on $S^n \backslash \{ (s_1, \ldots, s_n) \mid \exists i \neq j, s_i=s_j\}$ with finitely many orbits for every $n \leq \max \{ k \mid R \cap S^k \neq \emptyset\}$;
for every $s \in S$, the $H$-stabiliser of $s$ is finitely generated.

So far, I have just rephrased in a more general framework what we already said. The point I want to emphasize here is that many "big" groups can be embedded in finitely presented groups from this point of view.
Example 1: If $H \curvearrowright S$, the permutational wreath product $\mathbb{Z}/n\mathbb{Z} \wr_S H$ is an example. Here, $R=S \cup S^2 \backslash \mathrm{thick \ diagonal}$ and $w_s=x_1^n$, $w_{(r,s)}=[x_1,x_2]$.
This shows that $\bigoplus_\mathbb{N} \mathbb{Z}/n\mathbb{Z}$ embeds in a finitely presented group.
Example 2: An example I like is $\mathrm{Sym}_\mathrm{fin}(X) \rtimes H$ from an action $H \curvearrowright X$ (where $\mathrm{Sym}_\mathrm{fin}$ denotes the group of finitely supported permutations). Here, $S = X^2 \backslash \mathrm{thick \ diagonal}$, the action $H \curvearrowright S$ is induced by $H \curvearrowright X$, $R=S \cup S^2 \backslash \mathrm{thick \ diagonal} \cup \{ ((a,b),(b,c),(a,c)) \}$, and $w_s=x_1^2$, $w_{(r,s,t)} = x_1x_2x_1x_3$, $w_{(r,s)}=[x_1,x_2]$ if $r \cap s = \emptyset$ and $(x_1x_2)^3$ otherwise.
So $\mathrm{Sym}_\mathrm{fin}(\mathbb{N})$ embeds in a finitely presented group. (This also follows from the construction of Houghton groups.)
Example 3: More generally, the construction applies very well for families defined by (labelled) graphs, such as Coxeter and Artin groups. For instance, one can show that $\mathcal{B}_\infty$, the group of finitely supported braids with infinitely many strands, embeds in a finitely presented group. (This also follows from the construction of braided versions of Thompson's groups.)
Probably this can be done for other generalisations of braid groups (e.g. virtual braid groups, loop braid groups, twin groups).
Example 4: As already described by Yves, Steinberg groups can be also used, answering the main question of this discussion.