Towards Dimension Expanders over Finite Fields

In this paper we study the problem of explicitly constructing a dimension expander: Let Fⁿ be the n dimensional linear space over the field F. Find a small (ideally constant) set of linear transformations from Fⁿ to itself {A_i}_iisinI such that for every linear subspace V C Fⁿ of dimension dim(V) < n/2 we have dim (Sigma_iisinIA_i(V)) ges(1+alpha)ldrdim(V), where alpha > 0 is some constant. In other words, the dimension of the subspace spanned by {A_i(V)}_iisinI should be at least (1 + alpha) ldr dim(V). For fields of characteristic zero Lubotzky and Zelmanov completely solved the problem by exhibiting a set of matrices, of size independent of n, having the dimension expansion property. In this paper we consider the finite field version of the problem and obtain the following results. 1. We give a constant number of matrices that expand the dimension of every subspace of dimension d < n/2 by a factor of (1 + 1/logn). 2. We give a set of O(log n) matrices with expanding factor of´ (1 + alpha), for some constant alpha > 0. Our constructions are algebraic in nature and rely on expanding Cayley graphs for the group Z/Zn and small-diameter Cayley graphs for the group SL/₂(p).