Joachim Schwermer

The integral homology ofSL 2 andPSL 2 of euclidean imaginary quadratic integers

One way to study the cohomology of a group T of finite virtual cohomological dimension is to find a finite dimensional contractible space X on which F acts properly (such a space X always exists by [18], 1-7), and to then analyze the action, In this paper we want to consider the arithmetic groups SL2(0) and PSL2(0) = SL2(O)/±I where 0 is the ring of integers in an imaginary quadratic number field fc; the classical choice of X in this case is hyperbolic three-space H, i.e., the associated symmetric space SL2(C)/SU(2). As early as 1892 Bianchi [2] exhibited fundamental domains for the action of PSL2(0) on H for some small values of the discriminant. The space H has also turned out to be very useful in studying the relation between automorphic forms associated to SL2(0) and the cohomology of SL2(O) (cf. [12], [10]), and in studying the topology of certain hyperbolic 3-manifolds (cf, [25]). However, this choice of X is inconvenient for actual explicit computations of the cohomology of r = (P)SL2(0) with integral coefficients because the dimension of H is three, whereas the virtual cohomological dimension of F is two, indicating that it may be possible for F to act properly on a contractible space of dimension two; in addition, the quotient FH is not compact. A more useful space X for our purposes is given by work of Mendoza [14], which we recall in §3; using Minkowskis reduction theory (cf. §2), he constructs a f-invariant 2-dimensional deformation retract I(fe) of H such that the quotient of I(fc) by any subgroup of F of finite index is compact; I(k) is endowed with a natural CW structure such that the action of F is cellular and the quotient FI(k) is a finite CW-complex. The main object of this paper is to show how this construction can be used to completely determine the integral homology groups of PSL2(0). This is done by analyzing a spectral sequence which relates the homology of PSL2(0) to the homology of the quotient space PSL2(0)I(k) and the homology of the stabilizers of the cells (cf. [5], VII). We will confine our computations to the cases where 0 is a euclidean ring, i.e., @-@_d is the ring of integers in fc = Q(V-d) for d = 1,2,3,7 and 11. We will write out in detail the case d = 2 (cf. §5), which contains