Given a metric space $X$, the Vietoris-Rips complex VR$(X)$ is a classical simplicial complex obtained from $X$, and a group $G$ acting properly by isometries yields another metric space $X/G$ of the orbits of $X$ under $G$. There is a canonical way in which $G$ can act on VR$(X)$, so instead using the Vietoris-Rips metric thickening VR${}^m(X)$ allows a meaningful comparison between VR${}^m(X)/G$ and VR${}^m(X/G)$ as metric spaces.

This talk will survey a variety of properties which a group action on a metric space can have with some examples, and culminate with a discussion of the strong $r$-diameter action, which guarantees that under certain scale parameters VR${}^m(X)/G\simeq$ VR${}^m(X/G)$. I also discuss a strictly weaker condition and present some open questions concerning the connection between the two. Finally, I will briefly mention some analogous results for the Cech metric thickening.