Extending finite free actions of surfaces (Q6570075)

A finite group \(G\) of homeomorphisms of a closed orientable surface \(\mathcal{S}\) \textit{extends} (or \textit{bounds}) if there is a compact orientable 3-manifold \(M\) with boundary \(\mathcal{S}\) such that the action of \(G\) on \(\mathcal{S}\) extends to \(M\); it \textit{extends to a handlebody} if \(M\) can be chosen to be a handlebody (these are the \textit{completely reducible actions} of \textit{B. P. Zimmermann} [``Large finite group actions on surfaces: Hurwitz groups, maximal reducible and maximal handlebody groups, bounding and non-bounding actions'', Preprint, \url{arXiv:2110.11050}]. For example, many Hurwitz actions of maximal possible order \(84(g-1)\) of a surface of genus \(g > 1\) extend (even geometrically as groups of isometries of hyperbolic 3-manifolds with totally geodesic boundary) but, considering fixed point sets, no Hurwitz action extends to a handlebody (in fact, these actions are \textit{irreducible}). The situation is more difficult for \textit{free actions} where one cannot use fixed point sets; \textit{E. G. Samperton} [C. R., Math., Acad. Sci. Paris 360, 161--167 (2022; Zbl 1486.57032)] gave the first examples of free actions on a surface which do not extend.\N\NIn an old paper by \textit{M. Reni} and \textit{B. Zimmermann} [Proc. Am. Math. Soc. 124, No. 9, 2877--2887 (1996; Zbl 0868.57018)], the question is discussed whether a free action of a finite group on a surface which extends to a 3-manifold extends also to a handlebody. In the present paper, the first examples of free actions on a surface are constructed which extend to a 3-manifold but do not extend to a handlebody. Samperton proved in the paper cited above that, if the Bogomolov multiplier \(B_0(G)\) of \(G\) is trivial (a quotient of the Schur multiplier \(H_2(G)\)), then every free action of \(G\) extends (on the other hand, if \(B_0(G) \ne 0\) and \(G\) contains no dihedral subgroup then some free action of \(G\) does not extend). As a consequence, if \(B_0(G) = 0\) and there is no \textit{Schottky system of loops} (i.e., extending to a handlebody) for a free action of \(G\) on a surface then the action of \(G\) extends but it does not extend to a handlebody; this is the approach of the present paper. An interesting question remaining here is whether an extending action on a hyperbolic surface extends also geometrically to a hyperbolic 3-manifold with totally geodesic boundary (which should be true).