Projection theorems for linear-fractional families of projections

Marstrand's theorem states that applying a generic rotation to a planar set A before projecting it orthogonally to the x-axis almost surely gives an image with the maximal possible dimension min (1, dim A). We first prove, using the transversality theory of Peres-Schlag locally, that the same result holds when applying a generic complex linear-fractional transformation in PSL(2, C) or a generic real linear-fractional transformation in PGL(3, R). We next show that, under some necessary technical assumptions, transversality locally holds for restricted families of projections corresponding to one-dimensional subgroups of PSL(2, C) or PGL(3, R). Third, we demonstrate, in any dimension, local transversality and resulting projection statements for the families of closest-point projections to totally-geodesic subspaces of hyperbolic and spherical geometries.