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What is known about the group homology of $\mathrm{GL}_2(\mathbb{R})$ with real coefficients and what are strategies to compute it (or at least some groups for low degrees)? Here I want to consider the general linear group with the discrete topology.

There seems to be a lot of research on at first glance more complicated situations (i.e. non-field coefficients and replacing $\mathbb{R}$ by finite fields or more complicated rings), so I was wondering what was known about this seemingly simpler case.

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    $\begingroup$ You are aware of Milnor's theorem "On the homology of Lie groups made discrete"? $\endgroup$
    – Ryan Budney
    Commented Jan 30 at 19:05
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    $\begingroup$ @RyanBudney just to be clear I don't think Milnor's paper solves this problem -- in the rat/real coeffs case he just focuses on showing that the map to group homology with the usual topology is zero in a range. $\endgroup$
    – Kevin Casto
    Commented Jan 30 at 19:19
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    $\begingroup$ @KevinCasto I'm not asserting it solves the problem but it provides an enormous amount of insight. The question-asker does not make it clear he is aware, so I thought I would point out a result that gives structure. $\endgroup$
    – Ryan Budney
    Commented Jan 30 at 21:39
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    $\begingroup$ @RyanBudney I‘m aware of that. But for the context I have in mind, the fact that the map from cohomology of the topological group to the cohomology of the discrete group being zero is kind of the point. $\endgroup$
    – ThorbenK
    Commented Jan 31 at 6:40

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