A note on the Nielsen realization problem for connected sums of $S^2\times S^1$

Abstract

We consider finite group-actions on 3-manifolds ${\mathcal{H}}_g$ obtained as the connected sum of $g$ copies of $S^2\times S^1$, with free fundamental group $F_g$ of rank $g$. We prove that, for $g>1$, a finite group of diffeomorphisms of ${\mathcal{H}}_g$ inducing a trivial action on homology is cyclic. As a consequence, no non-cyclic subgroup of the twist subgroup of the mapping class group of ${\mathcal{H}}_g$ (generated by Dehn twists along embedded 2-spheres) can be realized by diffeomorphisms (in the sense of the Nielsen realization problem). We also discuss when a finite subgroup of the outer automorphism group $\mathrm{Out}(F_g)$ of the fundamental group of ${\mathcal{H}}_g$ can be realized by a group of diffeomorphisms of ${\mathcal{H}}_g$.

I added an Erratum to the previous version (which appeared in Rend. Istit. Mat. Univ. Trieste 53, 2021)

References in corpus (1)

• Embedded minimal surfaces