Higher cohomology for trivial module for finite groups of Lie type

Question

Is anything known about the cohomology past $\mathrm{H}^1$ and $\mathrm{H}^2$ for the trivial module for a finite group of Lie type in cross characteristic?

For the moment I just care about $\dim \mathrm{H}^n(G,k)$ for $n \geq 3$, $G$ a finite group of Lie type defined in characteristic $p$ and $k$ a field with $\operatorname{char} k = r \neq p$.

I'm also only looking at the rank 1 finite simple groups, so $\operatorname{PSL}_2(p^n)$, $\operatorname{PSU}_3(p^n)$, ${}^2\!\operatorname{G}_2(3^{2n+1})$ and $\operatorname{Sz}(2^{2n+1}) = {}^2\!\operatorname{B}_2(2^{2n+1})$. I know the answer for $\operatorname{PSL}_2(q)$ but have not found anything for the others.

Even just knowing the dimensions of $\mathrm{H}^3$ would be very useful in this case.

Answer 1

Lots is known about the cross-characteristic case: it is the same characteristic case that is the difficult one. The method used was introduced by Quillen, who worked out the full answer for $GL_n$. I think that the best reference for the general results is the book by Fiedorowicz and Priddy `Homology of classical groups over finite fields and their associated infinite loop spaces' Springer LNM volume 674. This won't help you with the twisted groups where there isn't a characteristic zero analogue though.