Group cohomology with compact support

Question

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?

Answer 1

Please see page 352 (in the Appendix) of Hida's book "Elementary theory of L-functions and Eisenstein series".

Answer 2

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...

Answer 3

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

$$ \gamma \in \Gamma \to (\gamma - 1) \int_{z_0}^{z_0'} \omega. $$

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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.