Finite groups acting on 3-manifolds and cyclic branched coverings of knots

@article{Mecchia2009FiniteGA,
  title={Finite groups acting on 3-manifolds and cyclic branched coverings of knots},
  author={M. Mecchia},
  journal={arXiv: Geometric Topology},
  year={2009}
}
  • M. Mecchia
  • Published 2009
  • Mathematics
  • arXiv: Geometric Topology
We are interested in finite groups acting orientation-preservingly on 3-manifolds (arbitrary actions, ie not necessarily free actions). In particular we consider finite groups which contain an involution with nonempty connected fixed point set. This condition is satisfied by the isometry group of any hyperbolic cyclic branched covering of a strongly invertible knot as well as by the isometry group of any hyperbolic 2-fold branched covering of a knot in the 3-sphere. In the paper we give a… Expand
3 Citations
Finite groups acting on 3-manifolds and cyclic branched coverings of knots
We are interested in finite groups acting orientation-preservingly on 3–manifolds (arbitrary actions, ie not necessarily free actions). In particular we consider finite groups which contain anExpand
Finite group actions and cyclic branched covers of knots in $\mathbf{S}^3$
We show that a hyperbolic $3$-manifold can be the cyclic branched cover of at most fifteen knots in $\mathbf{S}^3$. This is a consequence of a general result about finite groups of orientationExpand
Bounds for fixed points on hyperbolic 3-manifolds
Abstract For a compact (without boundary) hyperbolic n -manifold M with n ≥ 4 , we show that there exists a finite bound B such that for any homeomorphism f : M → M and any fixed point class F of f ,Expand

References

SHOWING 1-10 OF 38 REFERENCES
Finite Simple Groups Acting on 3-Manifolds and Homology Spheres
Summary. - Any finite group admits actions on closed 3-manifolds, and in particular free actions. For actions with fixed points, assumptions on the type of the fixed point sets of elementsExpand
Hyperbolic 3-manifolds as cyclic branched coverings
Abstract. There is an extensive literature on the characterization of knots in the 3-sphere which have the same 3-manifold as a common n-fold cyclic branched covering, for some integer $ n \ge 2 $.Expand
On finite simple groups acting on integer and mod 2 homology 3-spheres
We prove that the only finite non-abelian simple groups G which possibly admit an action on a Z2-homology 3-sphere are the linear fractional groups PSL(2,q), for an odd prime power q (and theExpand
On finite groups acting on ℤ2-homology 3-spheres
Abstract.We consider finite groups G admitting orientation-preserving actions on homology 3-spheres (arbitrary, i.e. not necessarily free actions), concentrating on the case of nonsolvable groups. ItExpand
How hyperbolic knots with homeomorphic cyclic branched coverings are related
Abstract We determine the exact geometric relation between two hyperbolic knots K and K ′ such that the n -fold cyclic branched covering of K coincides with the m -fold cyclic branched covering of KExpand
On the classification of finite groups acting on homology 3-spheres
In previous work we showed that the only .nite nonabelian simple group acting by di.eomorphisms on a homology 3-sphere is the alternating or dodecahedral group A5. Here we characterize .niteExpand
GROUPS WHICH ACT ON Su WITHOUT FIXED POINTS.
An equivalent formulation of the problem is the following. Which groups can occur as the fundamental groups of manifolds Mn for which the universal covering space A?n is homeomorphic to Sn. MoreExpand
On hyperelliptic involutions of hyperbolic 3-manifolds
By classical results, a compact Riemann surface, or equivalently an orientable closed hyperbolic 2-manifold, has at most one hyperelliptic involution (i.e. the quotient by the involution is theExpand
Knots whose branched cyclic coverings have periodic homology
Let Mk be the A-fold branched cyclic covering of a (tame) knot of S1 in S3. Our main result is that the following statements are equivalent: (1) Hx(Mk) is periodic with period n, i.e.Expand
On $\pi$-hyperbolic knots with the same 2-fold branched coverings
Abstract. We consider the following problem from the Kirby's list (Problem 3.25): Let K be a knot in $S^3$ and M(K) its 2-fold branched covering space. Describe the equivalence class [K] of K in theExpand
...
1
2
3
4
...