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

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 $\mathrm{Homeo}(M)\to \mathrm{Out}({\pi }_{1}M)$ for any such $M$.

arXiv comment25 pages