ABSTRACT
In Chapter 4, the central dimension of a linear group G is introduced. The central dimension of G is the dimension of the quotient space A/ζG(A) denoted by centdim F (G). Simple properties of the central dimension are obtained. Dually, the augmentation dimension of G is the dimension of the commutator subspace [G, A], denoted by augdim F (G) and simple properties of augmentation dimension are obtained. The bulk of the chapter is concerned with the intimate relationship between these two dimensions, which corresponds roughly to the relationship between the central quotient group and the commutator subgroup. We explore the question of when the finiteness of centdim F (G) implies that of augdim F (G). Unlike with abstract groups, there are examples when centdim F (G) is finite, but augdim F (G) is finite. However if the finite section p-rank r of G is finite (where p must be the characteristic of F in the non-zero characteristic case) and if d = centdim F (G) is finite, then augdim F (G) is also finite and bounded in terms of d and r. Generalizations of this result are then explored, as is the dual situation when augdim F (G) is finite and the question arises as to the size of centdim F (G).
