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# Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

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1435-5345
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Volume 2018, Issue 739

# Homology of SL2 over function fields I: Parabolic subcomplexes

Matthias Wendt
Published Online: 2015-09-23 | DOI: https://doi.org/10.1515/crelle-2015-0047

## Abstract

The present paper studies the group homology of the groups ${\mathrm{SL}}_{2}\left(k\left[C\right]\right)$ and ${\mathrm{PGL}}_{2}\left(k\left[C\right]\right)$, where $C=\overline{C}\setminus \left\{{P}_{1},\mathrm{\dots },{P}_{s}\right\}$ is a smooth affine curve over an algebraically closed field k. It is well known that these groups act on a product of trees and the quotients can be described in terms of certain equivalence classes of rank two vector bundles on the curve $\overline{C}$. There is a natural subcomplex consisting of cells with suitably non-trivial isotropy group. The paper provides explicit formulas for the equivariant homology of this “parabolic subcomplex”. These formulas also describe group homology of ${\mathrm{SL}}_{2}\left(k\left[C\right]\right)$ above degree s, generalizing a result of Suslin in the case $s=1$.

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Published Online: 2015-09-23

Published in Print: 2018-06-01

This work has partially been supported by the Alexander-von-Humboldt-Stiftung.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 739, Pages 159–205, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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