The cohomology ring of the sapphires that admit the Sol geometry

Abstract

Let G be the fundamental group of a sapphire that admits the Sol geometry and is not a torus bundle. We determine a finite free resolution of Z over ZG and calculate a partial diagonal approximation for this resolution. We also compute the cohomology rings H * (G; A) for A = Z and A = Z p for an odd prime p, and indicate how to compute the groups H * (G; A) and the multiplicative structure given by the cup product for any system of coefficients A.