Group Actions on Handlebodies

McCullough, Darryl; Miller, Andy; Zimmermann, Bruno

doi:10.1112/plms/s3-59.2.373

Cited by 55 publications

(46 citation statements)

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“…In [6] an algebro-geometrical theory of finite group actions on handlebodies has been developed; we recall some terminology from it, adopting a slightly different point of view (see also [8]). Let G be a finite group acting orientation-preservingly on a handlebody V. Let D be a 2-dimensional properly embedded disk in V such that ∂D = D ∩ ∂V is a nontrivial closed curve on F = ∂V.…”

Section: Finite Group Actions On Handlebodiesmentioning

confidence: 99%

Extending finite group actions from surfaces to handlebodies

Reni¹,

Zimmermann²

1996

Proc. Amer. Math. Soc.

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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).

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Section: Finite Group Actions On Handlebodiesmentioning

confidence: 99%

Extending finite group actions from surfaces to handlebodies

Reni¹,

Zimmermann²

1996

Proc. Amer. Math. Soc.

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“…By the Equivariant Loop Theorem [9], or by [4], there exists an essential compressing disc Remark 2.2. Orientation-preserving group actions on handlebodies were extensively studied in [7] using orbifold techniques. To extend some of that approach to actions with orientation-reversing elements, one could define silvered orbifold compression bodies using arbitrary 2-orbifolds with boundary in place of the F i and arbitrary discal 3-orbifolds in place of the W i , and attaching them together using orbifold 1-handles of the form D × I where D is a discal 2-orbifold.…”

Section: Silvered Compression Bodies and The Standard Constructionmentioning

confidence: 99%

Orientation-reversing involutions on handlebodies

Kalliongis

McCullough

1996

Trans. Amer. Math. Soc.

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Abstract. The observation that the quotient orbifold of an orientationreversing involution on a 3-dimensional handlebody has the structure of a compression body leads to a strong classification theorem, and general structure theorems. The structure theorems decompose the action along invariant discs into actions on handlebodies which preserve the I-fibers of some I-bundle structure. As applications, various results of R. Nelson are proved without restrictive hypotheses. IntroductionThroughout this paper h will be an orientation-reversing involution on a 3-dimensional orientable handlebody H. Moreover, h is assumed to be tame, so the action of h on an invariant ball about a fixed point must be equivalent either to the antipodal map on the 3-ball, or to a reflection across a 2-disc in the 3-ball, or (precisely when the fixed point lies in ∂H) to a reflection across a half 2-disc in a half 3-ball. The fixed point set fix(h) is thus a union of finitely many isolated fixed points and finitely many 2-manifolds properly imbedded in H. It is known (Theorem 6.1 of [2]) that the 2-dimensional components are incompressible, although this will be seen in passing when we examine the structure of h, and is immediate from the structure theory we will develop.A simple type of involution occurs when h is a product h 1 × h 2 on A × I, where A is an orientable 2-manifold with boundary, each h i is either the identity or an involution, and exactly one of the h i is orientation-reversing. Also, when H is the orientable I-bundle over a nonorientable 2-manifold with boundary, one has similar examples. By gluing such actions together along invariant discs in the boundary, one constructs more complicated examples. Our main structure results, Theorems 5.1 and 5.6, show that all orientation-reversing involutions are built up in this way, and furnish descriptions of the component actions in terms of the fixed-point data of h and the genus of H. The actions on the pieces in these decompositions are sometimes but not always unique up to equivalence; the conditions under which uniqueness holds and the extent to which it fails are fully analyzed in auxiliary theorems. As corollaries of the main structure results, one obtains the following simple characterizations of fiber-preserving involutions in terms of the fixed point data.

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“…Of central importance in deriving this characterization is the identification of the quotient of any such action as an orbifold handlebody (Proposition 1.1). As described in [MMZ,§3], an (orientable) orbifold handlebody is formed by gluing together orbifold 0-handles (orientable 3-orbifolds covered by B3) and orbifold 1-handles (products with / of orientable 2-orbifolds covered by B2) using a core graph of groups (r, G) which satisfies a set of normalized conditions (described below). The handlebody orbifold with core (r, G) is denoted by V(T, G).…”

Section: Equivalence and Strong Equivalence Of Actions On Handlebodiementioning

confidence: 99%

“…Suppose that the finite group G acts on Vg where g > 1. Then |G| < 12(f/ -1) [Z,MMZ,§7]. This implies that, up to isomorphism, there are only finitely many groups which act on Vg.…”

Section: Equivalence and Strong Equivalence Of Actions On Handlebodiementioning

confidence: 99%

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Equivalence and Strong Equivalence of Actions on Handlebodies

Kalliongis¹,

Miller²

1988

Transactions of the American Mathematical Society

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ABSTRACT. An algebraic characterization is given for the equivalence and strong equivalence classes of finite group actions on 3-dimensional handlebodies. As one application it is shown that each handlebody whose genus is bigger than one admits only finitely many finite group actions up to equivalence. In another direction, the algebraic characterization is used as a basis for deriving an explicit combinatorial description of the equivalence and strong equivalence classes of the cyclic group actions of prime order on handlebodies with genus larger than one. This combinatorial description is used to give a complete closed-formula enumeration of the prime order cyclic group actions on such handlebodies.

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Group Actions on Handlebodies

Cited by 55 publications

References 0 publications

Extending finite group actions from surfaces to handlebodies

Extending finite group actions from surfaces to handlebodies

Orientation-reversing involutions on handlebodies

Equivalence and Strong Equivalence of Actions on Handlebodies

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