  Let $G$ be a group acting properly and by isometries on a metric space $X$;
it follows that the quotient or orbit space $X/G$ is also a metric space. We
study the Vietoris-Rips and \v{C}ech complexes of $X/G$. Whereas (co)homology
theories for metric spaces let the scale parameter of a Vietoris-Rips or
\v{C}ech complex go to zero, and whereas geometric group theory requires the
scale parameter to be sufficiently large, we instead consider intermediate
scale parameters (neither tending to zero nor to infinity). As a particular
case, we study the Vietoris-Rips and \v{C}ech thickenings of projective spaces
at the first scale parameter where the homotopy type changes.
