A note on the homotopy type of Omega BSL3(Z)((2)over-cap)

It is known that for p-perfect groups G of finite virtual cohomological dim ension and finite type mod-p cohomology, the p-completed classifying space BG(p boolean AND) has the property that Omega BG(p)(boolean AND) is a retra ct of the loop space on a simply-connected, F-p-finite, p-complete space. I n this note we consider a particular example where this theorem applies, na mely we study the homotopy type of BSL3(Z)(2)(boolean AND). It particular m e analyse Omega BSt(3)(Z)(2)(boolean AND), a double cover of Omega BSL3(Z)( 2)(boolean AND), and obtain a splitting theorem for it in terms of 2-primar y Moore spaces and fibres of degree 2(r) maps on spheres. We also give a fo rmula for the Poincare series of H*(Omega B Gamma(p)(boolean AND);F-p) for a general group Gamma,(:) as above, in terms of possibly simpler components . This formula is used to calculate the mod-2 homology of Omega B Gamma(2)( boolean AND) for Gamma = SL3(Z) or St(3)(Z) as modules over a certain tenso r subalgebra.