The Action Dimension of Artin Groups

The action dimension of a discrete group G is the minimum dimension of a contractible manifold, which admits a proper G-action. In this dissertation, we study the action dimension of general Artin groups. The main result is that the action dimension of an Artin group with the nerve L of dimension n for n 6= 2 is less than or equal to (2n+1) if the Artin group satisfies the K(π, 1)-Conjecture and the top cohomology group of L with Z-coefficients is trivial. For n = 2, we need one more condition on L to get the same inequality; that is the fundamental group of L is generated by r elements where r is the rank of H1(L,Z). We prove our theorem by constructing an aspherical manifold of dimension 2n + 1 which has the Artin group as its fundamental group. AMS classification numbers. Primary: 20F36, 20F65 Secondary: 32S22