# Terminology about G- simplicial complexes

For a simplicial complex $$X$$ with an action of a discrete group $$G$$, we can impose the following condition, namely that if $$g\in G$$ stabilizes a given simplex $$\sigma\subseteq X$$, then $$g:\sigma\to\sigma$$ is in fact the identity map. This condition is nice because it implies that the quotient $$X/G$$ is glued out of simplices (rather than simplices modulo a finite group of permutations of their vertices).

Is there a standard term for this condition (or any related one) on the action $$G\curvearrowright X$$?

• This condition appears in several of my papers, and despite looking I have not found a standard term for it. One potentially useful way of thinking about it is that it is equivalent to saying that you can turn your simplicial complex into a semisimplicial set with a $G$-action. – Andy Putman May 27 '19 at 14:45

I've never stumbled across a standard one in all of the literature, though I've definitely appreciated a remark by Ken Brown in his group cohomology bible: "The hypothesis that $$G_\sigma$$ fixes $$\sigma$$ pointwise is not very restrictive in practice. In the case of a simplicial action, for example, it can always be achieved by passage to the barycentric subdivision."
Some references have the pointwise-fixed assumption baked into the definition of a $$G$$-CW-complex, but not all. Ken Brown's book doesn't, and he calls a $$G$$-CW-complex "admissible" if we include the pointwise-fixed constraint. This doesn't seem to be standard though. (Now as for definite nonstandard terminology, Farb-Margalit's book "A Primer on Mapping Class Groups" calls it a group action "without rotations".)