Exponential growth of product of matrices in {\rm SL}(2,{\mathbb{R}}).

In this paper we investigate the exponential growth of products of two matrices A,B \in {{\rm SL}(2,\mathbb{R})} . We prove, assuming A is a fixed hyperbolic matrix, that for Lebesgue almost every B, products of length n involving less than na, 0 [?] a < 1/2 matrices B are uniformly bounded from below by gn for some g > 1. [ABSTRACT FROM AUTHOR]