Diophantine properties of elements of SO(3)

A number α ∈ ℝ is diophantine if it is not well approximable by rationals, i.e. for some C, ε > 0 and any relatively prime p, q ∈ ℤ we have |αq - p| > Cq^-1-∈. It is well-known and is easy to prove that almost every α in ℝ is diophantine. In this paper we address a noncommutative version of the diophantine properties. Consider a pair A, B ∈ SO(3) and for each n ∈ ℤ₊ take all possible words in A, A^-1, B, and B^-1 of length n, i.e. for a multiindex script I sign = (i₁, j₁, . . . , i_m, j_m) define |script I sign| = ∑^m_k=1(|i_k|+|j_k|) = n and W_n(A, B) = {W_{script I sign}(A, B) = Aⁱ¹B^j1 . . . A^imB^jm|_{|script I sign|=n}. Gamburd-Jakobson-Sarnak [GJS] raised the problem: prove that for Haar almost every pair A, B ∈ SO(3) the closest distance of words of length n to the identity, i.e. s_{A, B}(n) = min_{|script I sign|=n} ||W_{script I sign}(A, B) - E||, is bounded from below by an exponential function in n. This is the analog of the diophantine property for elements of SO(3). In this paper we prove that s_{A, B}(n) is bounded from below by an exponential function in n². We also exhibit obstructions to a "simple" proof of the exponential estimate in n.