What are the cohomological dimensions of ${\rm Aut}(F_n)$, ${\rm Out}(F_n)$, ${\rm SL}_n(\mathbb{Z})$ over the rationals ℚ and integers ℤ? $\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}\DeclareMathOperator{\cd}{cd}\DeclareMathOperator\SL{SL}$For a group $G$, the cohomological dimension of $G$ over the ring $R$, denoted by $\cd_R(G)$, is given by
$$\cd_R(G)=\max\{n : H^n(G;M)\neq 0 \hspace{1mm} \mbox{for some} \hspace{1mm}  RG\mbox{-module }M\}$$
However, $\Aut(F_n)$ is a discrete group of automorphisms of a free group with $n$ generators. The quotient by inner automorphisms is the outer automorphism group of a free group, denoted by $\Out(F_n)$. I would like to know the numbers $\cd_{\mathbb{Q}}\bigl(\Aut(F_n)\bigr)$,  $\cd_{\mathbb{Q}}\bigl(\Out(F_n)\bigr)$,  $\cd_{\mathbb{Q}}\bigl(\SL_n(\mathbb{Z})\bigr)$ and $\cd_{\mathbb{Z}}\bigl(\Aut(F_n)\bigr)$,  $\cd_{\mathbb{Z}}\bigl(\Out(F_n)\bigr)$,  $\cd_{\mathbb{Z}}\bigl(\SL_n(\mathbb{Z})\bigr)$.
P.S.  I have googled these numbers, but I mostly found ‘virtual cohomological numbers’.
 A: All the answers are contained in the comments by now, but let me compile everything together to be more official.
The easy part is that for all three groups, $\textrm{cd}_\mathbb{Z}$ is infinite, since the groups contain torsion.
As for $\textrm{cd}_\mathbb{Q}$, the answers are that $\textrm{cd}_\mathbb{Q}(\textrm{Aut}(F_n))=2n-2$, $\textrm{cd}_\mathbb{Q}(\textrm{Out}(F_n))=2n-3$, and $\textrm{cd}_\mathbb{Q}(\textrm{SL}_n(\mathbb{Z}))=n(n-1)/2$.
Here are a few more details: In each case, this value equals the virtual cohomological dimension of the group. The vcd is always an upper bound, so the point is that for all these groups it is also a lower bound, which can be shown by exhibiting some subgroup whose $\textrm{cd}_{\mathbb{Q}}$ is known to equal the vcd; call this a witness subgroup. For $\textrm{Aut}(F_n)$, it easy to see that it contains $\mathbb{Z}^{2n-2}$ as a witness (just consider all the automorphisms that fix $x_1$ and send each $x_i$ ($i\ne 1$) to $x_1^{m_i}x_i x_1^{n_i}$ for some $m_i,n_i\in\mathbb{Z}$). Similarly, for $\textrm{Out}(F_n)$ we have $\mathbb{Z}^{2n-3}$ as a witness. Finally, for $\textrm{SL}_n(\mathbb{Z})$, the subgroup of strictly upper triangular matrices is a witness.
