Let $p$ be a prime and $F$ algebraic closer of $F_p$. I want to know if it is possible to construct family of groups $\{G_i\}_{i=1}^{\infty}$ and a family of simple modules $V_i$ over $F[G_i]$ of large dimension i.e., $dim V_i \geq |G_i|^{0.01}$ such that $H^1(G_i,V_i)$ is non trivial?
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This isn't a complete answer, but I believe it is possible by the Kan-Thurston Theorem, which says that every path connected space has the (co)homology of a $K(G,1)$. You can build a space with prescribed nontrivial cohomology (with $V_i$-coefficients), and then that must be isomorphic to the cohomology of some $BG_i$-space (hence group $G_i)$.
The reason I am not going to say this is a complete answer, is because due to the construction of the theorem, there might be a small problem getting the desired coefficients. The paper is entitled Every Connected Space has the Homology of a $K(\pi,1)$, by Kan and Thurston.
answered Oct 14, 2011 at 18:26
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$\begingroup$ Thanks for your answer. Do you know what is the size of the groups G_i? How it depends on the dimension of $V_i$? $\endgroup$ Commented Oct 15, 2011 at 17:13
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$\begingroup$ No clue. But it works for all local coefficient systems, so I don't think the dimension of $V_i$ should matter. There is perhaps other papers that analyze properties of this construction. $\endgroup$ Commented Oct 15, 2011 at 21:11