Sphere decomposition for finite subgroups of 3-manifold mapping class groups

arXiv:2610.31798v1 Announce Type: new Abstract: For a closed, oriented 3-manifold $M$, we show that any finite subgroup of its mapping class group preserves an isotopy class of filling sphere system on $M$. This is an isotopy version of the "equivariant sphere theorem" of Meeks-Yau. As part of our analysis, we establish a bijection between simplices of the Guirardel--Levitt outer space for $\pi_1(M)$ and filling sphere systems in $M$, generalizing an argument of Hatcher. We also compute the image of $\mathrm{Homeo}(M)\to\mathrm{Out}(\pi_1M)$ for any such $M$.

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