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What is the current state of knowledge of the group homology of $SL_2(k[t,t^{-1}])$?

I am mostly interested in the case $k$ is algebraically closed of characteristic zero. The most recent work I am aware of is these two papers of Knudson from 1996-7:

http://www.ams.org/mathscinet-getitem?mr=1375567

http://www.ams.org/mathscinet-getitem?mr=1443493

Has there been any more work in the time since?

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There are two papers that could be interesting to you. First, there is a paper of Kevin Hutchinson:

  • K. Hutchinson. On the low-dimensional homology of ${\rm SL}_2(k[t,t^{-1}])$. J. Algebra 425 (2015), 324–366.

He uses the amalgamation sequence from Knudson's paper and computes the boundary maps to show that second and third homology split (in a way similar to the fundamental theorem of K-theory) - one summand is homology of ${\rm SL}_2(k)$ and the other summand is something else: $K^{\rm MW}_1(k)$ for degree $2$ and an oriented scissors congruence group for degree $3$.

This splitting (with $\mathbb{Z}[1/2]$-coefficients) is generalized to all degrees in

  • M. Wendt. Homology of ${\rm SL}_2$ over function fields I: parabolic subcomplexes. to appear in J. Reine Angew. Math. arXiv:1404.5825

Proposition 6.11 shows that the homology of ${\rm SL}_2(k[T,T^{-1}])$ splits as a copy of homology of ${\rm SL}_2(k)$ and the remainder in degree $i$ is given by groups of oriented configurations of $i+1$ points on the projective line. The computations in the latter paper are not done using Knudson's amalgamation sequence, but rather by computing the quotient of a product of two trees (for the two valuations at $T=0$ and $T=\infty$) by the action of ${\rm SL}_2(k[T,T^{-1}])$. While this is now a 2-dimensional space (and so more complicated than the tree you have to deal with when using the amalgamation sequence), the stabilizers come out nicer (making the analysis and homology computations for the stabilizers easier than in Knudson's paper).

However, as far as I know, there is not much that can be said about these groups of configurations of points on the projective line in general. The degree 3 groups (4 points on $\mathbb{P}^1$) can be "computed" over $\overline{\mathbb{Q}}$ via K-theory and Borel regulators but even over $\mathbb{C}$ they would be unknown; and the situation is worse for $\geq 5$ points.

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