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If X is a finite CW-complex that is a Postnikov piece, then X is a K(G, 1). GUIDO’S BOOK OF CONJECTURES 45 By the theorem mentioned above, if X is a finite CW-complex that is a Postnikov piece, then the higher homotopy groups of X are Q-vector spaces. Hence, it seems that the fate of this conjecture will depend on our ability to find finite CW-complexes whose higher homotopy groups are nonzero Q-vector spaces. Do you know any? It would be very exciting to generalize Serre’s old theorem along these lines, if the conjecture were true.

519 (2000), 143–153. 21E. Guentner and J. Kaminker, Exactness and the Novikov conjecture, Topology 41 (2002), no. 2, 411–418. 22N. Ozawa, Amenable actions and exactness for discrete groups, C. R. Acad. Sci. Paris S´er. I Math. 330 (2000), no. 8, 691–695. 23J. Brodzki, G. A. Niblo, N. J. OA/0603621. 36 GUIDO’S BOOK OF CONJECTURES 15. Michelle Bucher-Karlsson and Anders Karlsson Volumes of ideal simplices in Hilbert’s geometry and symmetric spaces Dear Guido, We, the two authors of this note, first met each other in a weekly seminar that you organized at ETHZ and our first joint publication was in fact an outgrowth of one of those very appreciated seminars.

Michelle Bucher-Karlsson and Anders Karlsson Volumes of ideal simplices in Hilbert’s geometry and symmetric spaces Dear Guido, We, the two authors of this note, first met each other in a weekly seminar that you organized at ETHZ and our first joint publication was in fact an outgrowth of one of those very appreciated seminars. It is therefore an extra pleasure for us to jointly contribute to the present volume. We learnt a lot from you in our years in Zurich and maybe you can help us once more with the following geometric question we are interested in: Let X be a bounded convex domain in Rn endowed with its natural Hilbert’s metric d, where d(x, y) equals the logarithm of the projective cross-ratio of x and y, and the endpoints of the chord through x and y.