Extending finite group actions from surfaces to handlebodies

Abstract

Abstract. We show that every action of a finite dihedral group on a closed orientable surface F extends to a 3-dimensional handlebody V, with ∂V = F. In the case of a finite abelian group G, we give necessary and sufficient conditions for a G-action on a surface to extend to a compact 3-manifold, or, equivalently in this case, to a 3-dimensional handlebody; in particular all (fixed-point) free actions of finite abelian groups extend to handlebodies. This is no longer true for free actions of arbitrary finite groups: we give a procedure which allows us to construct free actions of finite groups on surfaces which do not extend to a handlebody. We also show that the unique Hurwitz action of order 84(g − 1) of P SL(2, 27) on a surface F of genus g = 118 does not extend to any compact 3-manifold M with ∂M = F, thus resolving the only case of Hurwitz actions of type P SL(2, q) of low order which remained open in an earlier paper (Math. Proc. Cambridge Philos. Soc. 117 (1995), 137-151).

Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds

“…We note that some of the results of [4] were shown previously by Hidalgo [7], and Reni and Zimmerman [13]. Specifically, [7,13] both show that every orientation-preserving action (not necessarily free) of an abelian group on a surface extends to an action on a handlebody, and [13] shows the same for dihedral group actions.…”

“…A number of other related results can be found in [13]. For instance, building on [6], they show that the only 84(g − 1) Hurwitz actions of type PSL(2, q) with 7 ≤ q < 1000 that do not extend to any 3-manifold occur when q = 7 and q = 27.…”

Genus one 1-bridge knots and Dunwoody manifolds

“…Let now G n be the cyclic group of order n generated by the homeomorphism ρ n on T n . The action of G n on T n extends to both the handlebodies U n and U ′ n (see [29]), and hence to the 3-manifold M. Let B 1 (resp. B ′ 1 ) be a disc properly embedded in U n (resp.…”

Maximum orders of cyclic and abelian extendable actions on surfaces

“…In the following, contents in §3.1 can be found in [1]. Lemma 3.6 can be found in [3]; Lemma 3.7 and 3.8 rely on [1, 2, 5, 10, 12]: [5] for the orbifold having isolated singular points, and [1,2,10,12] for the orbifold whose singular set has dimension at least 1; Lemma 3.9, 3.14 and 3.15 can be found in [7,15]; Lemma 3.16 can be found in [9].…”