Abstract
The present paper studies the group homology of the groups
{\operatorname{SL}_{2}(k[C])}
and
{\operatorname{PGL}_{2}(k[C])}
, where
{C=\overline{C}\setminus\{P_{1},\dots,P_{s}\}}
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
{\operatorname{SL}_{2}(k[C])}
above degree s, generalizing a result of Suslin in the case
{s=1}
.