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I'd like to know the $G$-equivariant mapping class groups of the torus --- by which I mean the groups of connected components of the groups of $G$-equivariant diffeomorphisms, $$ \pi_0\big( \mathrm{Diff}(T^2)^G \big) $$ ---, as the $G$-action ranges over the point group actions of the 16 non-trivial wallpaper groups.

Equivalently these would be called the orbifold mapping class groups of the flat 2-orbifolds $T^2 /\!\!/ G$.

One would hope this is a basic result citable from the literature, but maybe it's not. I guess one way to approach the computation is via the orbifold generalization of the Dehn-Nielsen-Baer theorem (as in Zieschang 1973), which partly reduces the question to the computation of the orbifold fundamental groups $\pi_1^{\mathrm{orb}}(T^2/\!\!/G)$ (to which the previous comment seems to apply, too).

Is anything more concrete already known?

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    $\begingroup$ Mapping class groups of hyperbolic orbifolds were studied by MacLachlan--Harvey (Proc. LMS, 1975), who proved that they are naturally isomorphic to the mapping class groups of the corresponding marked surfaces. §9 of Zieschang's paper addresses the Euclidean case directly, although it would take a little work to identify the resulting groups from Zieschang's descriptions. $\endgroup$ Commented Aug 26 at 7:20
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    $\begingroup$ Another important comment: the 17 wallpaper groups include non-orientable examples (eg the Klein bottle). But Zieschang's paper that you cite is only stated for the orientable case. More generally, mapping class groups in the non-orientable case are much more subtle: for instance, homotopy implies isotopy fails. (To see why, just think about pushing the boundary of a Möbius band across its core.) $\endgroup$ Commented Aug 26 at 7:30
  • $\begingroup$ Thanks. I also made an elementary check, for the p3 action of $\mathbb{Z}/3$ on $T^2$: The equivariant mapping class group contains $\mathrm{Sym}(3)$ as a subgroup, generated by rigid translation along one of the fixed point vectors and by $\pi$-rotation. (And maybe $\mathrm{Sym}(3)$ exhausts it, but not sure yet). Now the orbifold quotient $T^2/(\mathbb{Z}/3)$ is the 2-sphere with 3-cone points, and $\mathrm{Sym}(3)$ is also the ordinary MCG of the 3-punctured 2-sphere. $\endgroup$ Commented Aug 26 at 10:35

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