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."

Now as for nonstandard terminology, Farb-Margalit's book "A Primer on Mapping Class Groups" calls it a group action "without rotations". And 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.

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".)

No, at least I've never stumbled across one in most of the literature and all of the expected 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."

Now as for nonstandard terminology, Farb-Margalit's book "A Primer on Mapping Class Groups" calls it a group action "without rotations" (I'm not in favor). And some references have the pointwise-fixed assumption baked into the definition of a $G$-CW-complex, but not all.

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."

Now as for nonstandard terminology, Farb-Margalit's book "A Primer on Mapping Class Groups" calls it a group action "without rotations". And 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.