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

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For a closed, oriented 3-manifold, we show any finite subgroup of its mapping class group preserves an isotopy class of filling sphere system. This is an isotopy version of the “equivariant sphere theorem” of Meeks-Yau. As part of our analysis, we establish a bijection between filling sphere systems and simplices of the Guirardel-Levitt outer space for the fundamental group, generalizing an argument of Hatcher.