# Extending finite group actions from surfaces to handlebodies

@inproceedings{Reni1996ExtendingFG, title={Extending finite group actions from surfaces to handlebodies}, author={Marco Reni and Bruno Zimmermann}, year={1996} }

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…

## 13 Citations

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

- MathematicsComptes Rendus. Mathématique
- 2022

We provide the first known example of a finite group action on an oriented surface T that is free, orientation-preserving, and does not extend to an arbitrary (in particular, possibly non-free)…

Structure of Whittaker groups and applications to conformal involutions on handlebodies

- Mathematics
- 2010

Extending periodic automorphisms of surfaces to 3-manifolds

- Mathematics
- 2020

Let $G$ be a finite group acting on a connected compact surface $\Sigma$, and $M$ be an integer homology 3-sphere. We show that if each element of $G$ is extendable over $M$ with respect to a fixed…

Maximum orders of cyclic and abelian extendable actions on surfaces

- Mathematics
- 2013

Let $\Sigma_g (g>1)$ be a closed surface embedded in $S^3$. If a group $G$ can acts on the pair $(S^3, \Sigma_g)$, then we call such a group action on $\Sigma_g$ extendable over $S^3$.
In this paper…

Genus one 1-bridge knots and Dunwoody manifolds*

- Mathematics
- 2000

In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (possibly S 3 ), branched over genus one 1-bridge knots. As a consequence, we…

Automorphism groups of Schottky type

- Mathematics
- 2005

A group H of (conformal/anticonformal) automorphisms of a closed Riemann surface S of genus g ≥ 2 is said of Schottky type if there is a Schottky uniformization of S for which it lifts. We observe…

Every finite group action on a compact 3-manifold preserves infinitely many hyperbolic spatial graphs

- Mathematics
- 2014

We consider symmetries of spatial graphs in compact 3-manifolds described by smooth finite group actions. This paper provides a method for constructing an infinite family of hyperbolic spatial graphs…

A Note on Surfaces Bounding Hyperbolic 3-Manifolds

- Mathematics
- 2004

Abstract.We consider the problem of whether a given hyperbolic surface occurs as the totally geodesic boundary of a compact hyperbolic 3-manifold (as some or as the only boundary component). We…

## References

SHOWING 1-10 OF 22 REFERENCES

Extending finite group actions on surfaces to hyperbolic 3-manifolds

- Mathematics
- 1995

Let G be a finite group of orientation preserving isometrics of a closed orientable hyperbolic 2-manifold F g of genus g > 1 (or equivalently, a finite group of conformal automorphisms of a closed…

On Schottky groups with automorphisms.

- Mathematics
- 1994

We consider a closed Riemann surface S and a group H of conformal automorphisms of S . We seek a Schottky uniformization (Ω, G, π: Ω → S) of the surface S with the property that every element of H…

Remarks on the cobordism group of surface diffeomorphisms

- Mathematics
- 1982

In this note we sketch our computation of the group A 2 of cobordism classes of orientation-preserving diffeomorphisms of closed, oriented surfaces. See Sect. 2 for precise definitions. This…

The genus of PSl2 (q).

- Mathematics
- 1987

In [3] we computed the genus of the linear fractional group PS/2(p), where p*z5 is a prime. That is, we determined the least integer g such that PSl2(p) could occur äs an effective group of…

Surfaces and the second homology of a group

- Mathematics
- 1987

LetG be a group andK(G, 1) an Eilenberg—MacLane space, i.e. π1(K(G,1))≅G, πi(K(G,1))=0,i≠1. We give a purely algebraic proof that the second homology groupH2(G)=H2(G,ℤ)≅H2(K(G,1)) is isomorphic to…

Handlebody orbifolds and Schottky uniformizations of hyperbolic 2-orbifolds

- Mathematics
- 1995

The retrosection theorem says that any hyperbolic or Riemann surface can be uniformized by a Schottky gro-up. We generalize this theorem to the case of hyperbolic 2-orbifolds by giving necessary and…

Chapter VIII The Equivariant Loop Theorem for Three-Dimensional Manifolds and a Review of the Existence Theorems for Minimal Surfaces

- Mathematics
- 1984

Generators and Relations for Discontinuous Groups

- Mathematics
- 1991

This paper contains the Bass-Serre theory generalizing free products with amalgamation and HNNextensions; the structure of finite extensions of free groups and applications to finite group actions on…

Surfaces and Planar Discontinuous Groups

- Mathematics
- 1980

Free groups and graphs.- 2-Dimensional complexes and combinatorial presentations of groups.- Surfaces.- Planar discontinuous groups.- Automorphisms of planar groups.- On the complex analytic theory…