The action dimension of a discrete group

Colloquim joint with Simons Semester

A space is called a K(G,1) if its fundamental group is G and if its universal cover is contractible.  The geometric  dimension of G is the smallest dimension of a model for K(G,1) by a cell complex.  Its action dimension is the smallest dimension of a model for K(G,1) by a manifold.  In many examples there is natural model for K(G,1)  by a manifold and it can be shown that in these cases the action dimension equals the dimension of the natural model by manifold. The method for computing the action dimension involves a classical  obstruction of van Kampen for embedding a cell complex into a Euclidean space of some dimension.  I will discuss the Action Dimension Conjecture which relates the action dimension to two well-known conjectures in geometric group theory:  the Singer Conjecture on the vanishing of L^2 -Betti numbers of G when the model for K(G,1) is a closed manifold and the Euler Characteritic Conjecture on the sign of the Euler charactersistic in such a case.