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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.

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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.

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  • $\begingroup$ Thanks - this could be what I want. I'll have to track it down in the library since the google books preview doesn't seem to have quite enough and all the talk of loop spaces is very confusing. (Sorry for the late accept, had to figure out whether it seemed right or not) $\endgroup$ May 24, 2019 at 18:24

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