Approximate finiteness properties of infinite groups

A discrete group Γ is said to be of type F(n) if and only if there is a classifying complex BΓ with finite n-skeleton. We show that Γ is of type F(n) if Γ admits a regular cellular action on a CW complex Y such that Y is homotopy equivalent to a complex with finite (n−1)-skeleton, YΓ is a complex with finite n-skeleton, and for each p-cell σ of Y(0⩽p⩽n) the stabilizer Γσ is of type F(n−p).