  We give an operator theoretic approach to the constructions of
multiresolutions as they are used in a number of basis constructions with
wavelets, and in Hilbert spaces on fractals. Our approach starts with the
following version of the classical Baumslag-Solitar relations $u t = t^2 u$
where $t$ is a unitary operator in a Hilbert space $\mathcal H$ and $u$ is an
isometry in $\mathcal H$. There are isometric dilations of this system into a
bigger Hilbert space, relevant for wavelets. For a variety of carefully
selected dilations, the ``bigger'' Hilbert space may be $L^2(\br)$, and the
dilated operators may be the unitary operators which define a dyadic wavelet
multiresolutions of $L^2(\br)$ with the subspace $\mathcal H$ serving as the
corresponding resolution subspace. That is, the initialized resolution which is
generated by the wavelet scaling function(s). In the dilated Hilbert space, the
Baumslag-Solitar relations then take the more familiar form $u t u^{-1} = t^2$.
  We offer an operator theoretic framework including the standard construction;
and we show how the representations of certain discrete semidirect group
products serve to classify the possibilities. For this we analyze and compare
several types of unitary representations of these semidirect products: the
induced representations in Mackey's theory, the wavelet representations on
$L^2(\br)$, the irreducible representation on the dual, the finite dimensional
representations, and the the regular representation.
