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Let $SL(2,{\mathbb R})$ act on ${\mathbb R}^2$ by matrix multiplication.

What is known about group cohomology $H^*(SL(2,{\mathbb R}),{\mathbb R}^2)$?

And about $$H^*(\Gamma,{\mathbb R}^2)$$ for a cocompact lattice $$\Gamma\subset SL(2,{\mathbb R})?$$

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    $\begingroup$ You are looking at $SL_2(\mathbb R)$ as a discrete group? $\endgroup$
    – Mariano Suárez-Álvarez
    Nov 18, 2014 at 19:14
  • $\begingroup$ Yes. Actually, I am more interested in the second question (cocompact lattices). $\endgroup$
    – ThiKu
    Nov 19, 2014 at 0:42

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It is zero. This is an application of the "centre kills" trick, which I will state in homology.

Trick. Let $M$ be a $G$-module for which there is an element $z$ in the centre of $G$ which acts as $-1$ on $M$. Then $2H_*(G;M)=0$.

In your situation the homology is a real vector space, so if multiplication by 2 kills it then it is already dead. You can then use Universal Coefficients to get the result in cohomology, or just work out the details of the trick in cohomology.

The proof is as follows. For any element $z \in G$, the map $$m \vert g_1 \vert g_2 \vert \cdots \vert g_n \longmapsto m \cdot z \vert g_1^z \vert g_2^z \vert \cdots \vert g_n^z$$ on the bar complex is i) a chain map, and ii) chain homotopic to the identity, so induces the identity map on homology. But if $z$ has the properties described then this map is equal to $-1$.

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    $\begingroup$ a contrario the cohomology of cocompact subgroups of $PSL_2(\mathbf R)$ with coefficients in $2\times 2$ symmetric matrices can be non-trivial. Example: for the group of isochronous automorphisms of the form $-86x_0^2+x_1^2+x_2^2$, the $H^1$ has dimension 10. $\endgroup$
    – few_reps
    Nov 19, 2014 at 19:57
  • $\begingroup$ Can you give a reference? $\endgroup$
    – ThiKu
    Nov 20, 2014 at 12:03
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I will answer only for cohomology in degree 1, as I am not sure for higher degrees.

On the other hand I think that what I am saying remains true when R^2 is replaced by any finite dimensional representation of SL_2(R).

For the first cohomology of a lattice with values in a finite dimensional representation, the key word is Eichler-Shimura isomorphism. These cohomology groups can be nonzero, see

http://www.ams.org/journals/tran/1987-300-02/S0002-9947-1987-0876476-0/S0002-9947-1987-0876476-0.pdf

or

http://www.math.u-psud.fr/~labourie/preprints/pdf/propsurf.pdf

and the (original) references there. For the cohomology group H^1(SL_2(R),V) with V a finite dimensional SL_2(R)-module, I think it is always zero. I don't have a reference on the top of my head. But if you understand the Eichler-Shimura isomorphism for lattices, it should give a proof of vanishing for SL_2(R).

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  • $\begingroup$ Thank you for the hints to the literature. After looking up the references it seems to me that for cocompact lattices $H^*(\Gamma,E)=0$ is implied by Theorem VII.6.7 in Borel-Wallach's "Contiinuous Cohomology". $\endgroup$
    – ThiKu
    Nov 19, 2014 at 3:09
  • $\begingroup$ Namely, if E is an irreducible finite-dimensional representation, whose highest weight is not preserved by the Cartan Involution, then $H^*(\Gamma,E)=0$. $\endgroup$
    – ThiKu
    Nov 19, 2014 at 3:12
  • $\begingroup$ Also in the cocompact case? $\endgroup$
    – ThiKu
    Nov 19, 2014 at 3:13
  • $\begingroup$ After identification $PSL(2,\R)=SO(2,1)$ one can apply Raghunathan's paper link.springer.com/article/10.1007%2FBF02867433 It says that $H^1$ of a representation of a cocompact lattice $\Gamma\subset SO(2,1)$ that arises as restriction of a rep. of SO(2,1) must vanish except if the heighest weight is an integer multiple of the heighest weight of the (3-dimensional) standard representation. $\endgroup$
    – ThiKu
    Nov 19, 2014 at 5:02
  • $\begingroup$ In my case, the heighest weight of the 2-dimensional representation is half of the heighest weight of the 3-dimensional standard representation, so it is not an integer multiple and $H^1$ should vanish. Am I mixing up something? $\endgroup$
    – ThiKu
    Nov 19, 2014 at 5:04

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