Bounding and nonbounding finite group actions on surfaces
In the study of transformation groups on 2-dimensional closed surfaces, a basic problem is which finite group actions on closed surfaces extend to compact 3-manifolds. This is equivalent to asking which finite group actions on closed surfaces bound. The author of this paper considers the problem when the group is a linear fractional group \(\text{PSL}(2, p^n)\) and the action is orientation-preserving. Theorem 1. Let \(p^n\) be a prime power such that either \(p\equiv 1\mod 4\), or \(n\) is even or \(p=2\). Then every action of a linear fractional group \(\text{PSL}(2, p^n)\) on a closed orientable surface bounds. For all other values of \(p^n\), each group \(\text{PSL}(2, p^n)\) admits nonbounding actions on closed connected surfaces. A geometric version of Theorem 1 is also given: Theorem 2. Let \(p^n\) be a prime power such that \(p>3\) and either \(p\equiv 1\mod 4\) or \(n\) is even. Then each indecomposable isometric action of \(\text{PSL}(2, p^n)\) on a hyperbolic surface bounds geometrically. In particular, all Hurwitz and genus actions of such groups bound geometrically. In addition, the author computes the 2-dimensional equivariant bordism group \(\Omega_2(\text{PSL}(2, 7))\). Theorem 3. The 2-dimensional bordism group \(\Omega_2(\text{PSL}(2, 7))\) of the group \(\text{PSL}(2, 7)\) is infinite cyclic, generated by the Hurwitz action of \(\text{PSL}(2, 7)\) on Klein's quartic.
scientific article; zbMATH DE number 2092799