We investigate cohomogeneity-1 metrics whose principal orbit is an Aloff-Wallach space SU(3)/U (1). In particular, we are interested in metrics whose holonomy is contained in Spin (7). Complete metrics of this kind which are not product metrics have exactly one singular orbit. We prove classification results for metrics on tubular neighbourhoods of various singular orbits. Since the equation for the holonomy reduction has only few explicit solutions, we make use of power series techniques. In order to prove the convergence and the smoothness near the singular orbit, we apply methods developed by Eschenburg and Wang. As a by-product of these methods, we find many new examples of Einstein metrics of cohomogeneity 1.

a b s t r a c tWe classify all cohomogeneity-one manifolds with principal orbit Q 1,1,1 (1)) whose holonomy is contained in Spin (7).Various metrics with different kinds of singular orbits can be constructed by our methods. It turns out that the holonomy of our metrics is automatically SU(4) and that they are asymptotically conical. Moreover, we investigate the smoothness of the metrics at the singular orbit.

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a b s t r a c tWe show that K 3 surfaces with non-symplectic automorphisms of prime order can be used to construct new compact irreducible G 2 -manifolds. This technique was carried out in detail by Kovalev and Lee for non-symplectic involutions. We use the Chen-Ruan orbifold cohomology to determine the Hodge diamonds of certain complex threefolds, which are the building blocks for this approach.

Abstract. Let S be a K3 surface that admits a non-symplectic automorphism ρ of order 3. We divide S × P 1 by ρ × ψ where ψ is an automorphism of order 3 of P 1 . There exists a threefold ramified cover of a partial crepant resolution of the quotient that is a Calabi-Yau orbifold. We compute the Euler characteristic of our examples and obtain values ranging from 30 to 219.