Bounding and nonbounding finite group actions on surfaces

Abstract

We consider the problem of which finite orientation-preserving group actions on closed surfaces extend to compact 3-manifolds. A solution is known for cyclic, dihedral and Abelian groups. In the present paper, we consider actions of the linear fractional groups PSL(2, p n ). Our main results imply that, for primes p≡1 mod 4 , all actions of the groups PSL(2, p n ) bound compact 3-manifolds. In particular, we show that all isometric Hurwitz and genus actions of these groups on hyperbolic surfaces bound geometrically, i.e., extend isometrically to compact hyperbolic 3-manifolds with totally geodesic boundary. On the other hand, for all primes p≡3 mod 4 there exist nonbounding actions of PSL(2, p n ).