Multiple complexes and gaps in Farrell cohomology

Let Γ be a group of finite virtual cohomological dimension n. In this paper, we show that Γ has a multiple complex, i.e. Γ has a projective resolution that is the tensor product of γ complexes which are resolutions of some period after n-steps, where γ is the maximum of the p-ranks of the finite elementary abelian p-subgroups of Γ. This is an extension of the results of Benson and Carlson (Math. Z. 195 (1987) 221) and Benson and Greenless (J. Algebra 192 (1997) 678). Finally, we present a result on gaps in Farrell cohomology using a multiple complex. This result is a generalization of the result of Benson et al. (J. Algebra 131 (1990) 40) and Juan-Pineda (J. Pure and Appl. Algebra 111 (1996) 213).