  We prove that any Parseval wavelet frame is the projection of an orthonormal
wavelet basis for a representation of the Baumslag-Solitar group $$BS(1,2)=<
u,t | utu^{-1}=t^2>.$$ We give a precise description of this representation in
some special cases, and show that for wavelet sets, it is related to symbolic
dynamics. We show that the structure of the representation depends on the
analysis of certain finite orbits for the associated symbolic dynamics. We give
concrete examples of Parseval wavelets for which we compute the orthonormal
dilations in detail; we show that there are examples of Parseval wavelet sets
which have infinitely many non-isomorphic orthonormal dilations.
