Classification of aut-fixed subgroups in free-abelian times surface groups

In this paper, we are concerned with the direct product $G=\pi_1(\Sigma)\times \Z^k$ for $\Sigma$ a compact orientable surface with negative Euler characteristic, and give a complete classification of its fixed subgroups of automorphisms. As a corollary, we show that $G$ contains, up to isomorphism, infinitely many fixed subgroups of automorphisms if and only if $k\geq 2$, which is a contrast to that of hyperbolic groups. As an application on Nielsen fixed point theory, we provide a family of aspherical manifolds without Jiang's Bound Index Property. Moreover, we also give some results on the fixed subgroups of the direct product $H\times \Z^k$ for $H$ a non-elementary torsion-free hyperbolic group.