The cohomology groups of the outer Whitehead automorphism group of a free product

Let Gamma = G(1) * G(2) * ... * G(n) be an n-fold free product. For g(i) is an element of G(i) and g(k) is an element of G(k), let alpha(gj)(i) is an element of Aut(Gamma) be the automorphism induced by alpha(gj)(i)(g(k)) = {(gk if k = i,) (ggjk if k not equal i) where (gj)(gk) means conjugating g(k) by g(j)). Then alpha(gj)(i) is a Whitehead automorphism, and the group generated by all alpha(gj)(i) is known at the Whitehead automorphism group of Gamma, denoted Wh(Gamma). In this paper, we calculate H*(Wh(Gamma)). The main tools used are the action of Wh(Gamma) on a space constructed by McCullough-Miller and the equivariant spectral sequence. We also specialize to the case of Gamma a free Coxeter group, where further analysis allows us to determine a presentation for the cohomology ring.