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 $G\curvearrowright X$?

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    $\begingroup$ 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. $\endgroup$ – 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".)

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  • $\begingroup$ My vote is to start using "admissible (simplicial) action" often. $\endgroup$ – Chris Gerig Jun 2 '19 at 18:33

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