I have some very basic understanding of group cohomology and simplicial cohomology, both from different contexts, but I am unable to understand how the two are related. Let me explain a simple instance.
That is, suppose we have a (discrete, finitely generated) group $\Gamma$ and its action on a space $V$, we can define $H^1(\Gamma, V)$. For instance, lets assume $V$ is a Hilbert space $\mathcal{H}$), and $\Gamma$ acts by unitary transformations, so that we have a unitary representation $\rho:\Gamma \to \mathcal{U}(\mathcal{H})$. We can now define $H^1(\Gamma,\mathcal{H})$, also denoted $H^1(\Gamma,\rho)$ in this case.
On the other hand, for a simplicial complex $X$, we can use cochains as functions to $\mathbb{R}$ say, to define the cohomology group $H^1(X,\mathbb{R})$.
Broadly, my question is:
Can we obtain $H^1(\Gamma,\rho)$ as simplicial cohomology, by making $\Gamma$ act on some simplicial complex $X$? Is the group cohomology realized combinatorially this way?
I am aware of something to this effect, namely equivariant cohomology. But I am still not clear on what exactly should the simplicial complex look like, what should the action be, and why $H^1(\Gamma, \rho)=H^1(X,\mathbb{R})$? Is there a simple way of intuitively understanding the correspondence?