Group cohomology with compact support Let $\Gamma$ be a discrete group, $V$ a left $\Gamma$-module. One can define the 
groups $H^i(\Gamma,V)$ ($i=0,1,2,\dots$) in many ways, and then prove their equivalence: as derived functors
of the functor of $\Gamma$-invariants; as the homology of an explicit complex of cochains;
or as the usual (Steenrod's) cohomology of the local system $\tilde V$ attached to $V$ on the classifying space $B\Gamma$ of $\Gamma$: $H^i(\Gamma,V) = H^i(B\Gamma,\tilde V)$.
Yet I know of only one definition of group cohomology with compact support. One defines
$H^i_c(\Gamma,V)$ as $H^i_c(B\Gamma,\tilde V)$. 

Are there other ways to define group cohomology with compact support, with no reference to the classifying space? is there in particular a definiton with an explicit complex of cochains?  

Of course, any reference would be welcome.
Giving an explicit description in terms of a complex of cochains might be difficult in general, but I would be happy to have one in the following well-known, overstudied example:
When $\Gamma$ is a congruence subgroup of $SL_2({\bf Z})$. In this case, one finds in the litterature something close to what I am asking: an explicit description in terms of cochains of the "parabolic cohomology group" $H^1_p(\Gamma,V)$ defined as the image of the natural map $H^1_c(\Gamma,V) \rightarrow H^1(\Gamma,V)$. One shows, under mild assumptions on $\Gamma$, that $H^1_p(\Gamma,V) = Z^1_p(\Gamma,V)/B_1(\Gamma,V)$ where $Z_1(\Gamma,V)$ is the subgroup of the group of cocycles $Z^1(\Gamma,V)=\{u:\Gamma \rightarrow V,\  u(gg')=u(g)+gu(g')\}$ that satisfy $u(p) \in (p-1) V$ for all parabolic elements $p \in \Gamma$. (cf for example Hida, inv. math. 63). Now that's only a description of the $H^1_p$, while the $H^1_c$ is (slightly) bigger. And a similar description of the 
$H^2_c$ would be handy as well, when computing cup-products. So is it possible to give such a description? Is it done somewhere is the litterature?
 A: Please see page 352 (in the Appendix) of Hida's book "Elementary theory of L-functions and Eisenstein series".
A: I ve got the same problem... nevertheless I found a short description by Kurt Haberland - Perioden von Modulformen in einer Variablen und Gruppenkohomologie 1, section 2.1... if u ve got any problems with german... just tell me... 
A: This response is a little late, but I have thought about the same
question recently.  I don't think there is a way to define
cohomology with compact support in a purely group theoretic way.
The problem is that compact cohomology will distinguish between
multiple cusps, but
cocycles can only capture one cusp.  
Let $\Gamma$ be a torsion free congruence subgroup of $SL_2(\mathbb{Z})$.
Let $X=\mathbb{H}/\Gamma$ and assume that there is more than one cusp.
Let $E$ be a real vector space on which 
$\Gamma$ acts.  Denote by $H^1_!(X, \widetilde{E})$ the image of 
$H_c^1(X, \widetilde{E})$ in $H^1(X, \widetilde{E})$.  Our Eichler-Shimura
style map 
$$ H^1_!(X, \widetilde{E}) \to H^1(\Gamma, E)$$
sends the cohomology class of the compactly supported 
$1$-form $\omega$ to the 1-cocycle
$$ \gamma \in \Gamma \to \int_{z_0}^{\gamma(z_0)} \omega.$$
Let $(\omega, z_0)$ denote the corresponding $1$-cocycle.
Here we have $z_0\in \mathbb{H} \cup \mathbb{P}^1(\mathbb{Q})$ and 
a different
choice of $z_0'$ will differ by the 1-coboundary 
$$ \gamma \in \Gamma \to (\gamma - 1) \int_{z_0}^{z_0'} \omega. $$
\
Let $a_1,...,a_r$ be coset representatives of 
$\mathbb{P}^1(\mathbb{Q}) / \Gamma$.   .  Let $\omega_1$ and $\omega_2$
be compactly supported $1$-forms that represent the same cohomology 
class of $H^1_!(X, \widetilde{E})$.  Then $\omega_1-\omega_2 = du$
where $du$ has compact support.  In particular, we see that $u$
must be constant around the cusps in $X$.  We may view $u$ as
a $E$ valued function on $\mathbb{H}\cup \mathbb{P}^1(\mathbb{Q})$ 
with $\gamma(u(z))= u(\gamma(z))$.
Being constant around the cusp $a_i$ means that $u(a_i)$ is fixed by 
the parabolic subgroup $P_{a_i}$ that fixes $a_i$.  That is to say
$u(a_i) \in E^{P_{a_i}}$.  
We want to write $H_c^1(X, \widetilde{E})$ in a purely group theoretic
manner.  This means we want to write $H_c^1(X, \widetilde{E})$ as
some quotient that looks like
$$\frac{\text{a special set of 1-cocycles}}
{\text{a special set of 1-coboundaries}}. $$
Any such isomorphism should send $\omega$ to the $1$-cocycle
$(\omega, a_1)$.  The cusp we choose should not matter.
Assigning a $1$-cocycle to $\omega$ is already problematic!
It will not necessarily be injective.
For any choice of $c_i\in H^0(P_{a_i},E)$ for each
$i$,
we may construct a bump function that is constantly $c_i$ around
$a_i$.  This bump function will can be defined to be 
$\gamma(c_i)$ around $\gamma(a_i)$
and therefore will lift to a $0$-form (not compact) on $X$.
The $1$-cocycle $(du, a_i)$ (note that this is actually a $1$-coboundary)
is
$$\gamma \to (\gamma - 1)u(a_i). $$
The form $du$ will be compact.  If we choose $c_1$ to be $0$,
then the 1-cocycle $(du, a_1)$ will be trivial.  However the form
$du$ will not be trivial.  
The issue is that a $1$-cocycle can only keep track of one
cusp.  When we allow the exact forms $du$ where $u$ is not
compact, we only need to look at one cusp and thus a $1$-cocycle
is able to retain all the information of the cohomology class.
The solution of course is to simultaneously consider all cusps.
The result are modular symbols.
