  A G-solenoid is a laminated space whose leaves are copies of a single Lie
group G, and whose transversals are totally disconnected sets. It inherits a
G-action and can be considered as dynamical system. Free Z^d-actions on the
Cantor set as well as a large class of tiling spaces possess such a structure
of G-solenoid. We show that a G-solenoid can be seen as a projective limit of
branched manifolds modeled on G. This allows us to give a topological
description of the transverse invariant measures associated with a G-solenoid
in terms of a positive cone in the projective limit of the dim(G)-homology
groups of these branched manifolds. In particular we exhibit a simple criterion
implying unique ergodicity. A particular attention is paid to the case when the
Lie group $G$ is the group of affine orientation preserving isometries of the
Euclidean space or its subgroup of translations.
