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Maximal actions of finite 2-groups on - homology 3-spheres
No full textIt is known that a finite 2-group acting on a Z_2-homology 3-sphere has at most ten conjugacy classes of involutions; the action of groups with the maximal number of conjugacy classes of involutions is strictly related to some questions concerning the
representation of hyperbolic 3-manifolds as 2-fold branched coverings of knots.
Using a low-dimensional approach we classify these maximal actions both from an algebraic and from a geometrical point of view
How hyperbolic knots with homeomorphic cyclic branched coverings are related
No full textAbstractWe 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 K′. If m and n are not powers of two the solution of the problem is known (complete solution for the case m=n, partial results for the case n different from m). In the present work, we give a complete solution to the problem for branching orders which are powers of two, and thus in particular also for the most basic case of 2-fold branched coverings
On finite simple groups acting on integer and mod 2 homology 3-spheres
No full textWe prove that the only finite non-abelian simple groups G which possibly admit an action on a
Z_2-homology 3-sphere are the linear fractional groups PSL(2, q), for an odd prime power q (and the dodecahedral
group A_5 isomorphic to PSL(2, 5) in the case of an integer homology 3-sphere), by showing that G has dihedral
Sylow 2-subgroups and applying the Gorenstein–Walter classification of such groups. We also discuss the
minimal dimension of a homology sphere on which a linear fractional group PSL(2, q) acts
The number of knots and links with the same 2-fold branched covering
No full textWe give a nearly complete solution of the problem of how many different knots and links in the 3-sphere, and more generally in homology 3-spheres, can have the same hyperbolic 3-manifold as their common 2-fold branched covering. This number depends on the number of components of the links. We show that the best possible upper bound is 9 for knots and for 2-component links, and 3 for links with more than two components
On quotient orbifolds of hyperbolic 3-manifolds of genus two
Full text linkWe analyse the orbifolds that can be obtained as quotients of genus two hyperbolic 3-manifolds by their orientation preserving isometry groups. The genus two hyperbolic 3-manifolds are exactly the hyperbolic 2-fold branched coverings of 3-bridge links. If the 3-bridge link is a knot, we prove that the underlying topological space of the quotient orbifold is either the 3-sphere or a lens space and we describe the combinatorial setting of the singular set for each possible isometry group. In the case of 3-bridge links with two or three components, the situation is more complicated and we show that the underlying topological space is the 3-sphere, a lens space or a prism manifold. Finally we present an infinite family of hyperbolic 3-manifolds that are simultaneously the 2-fold branched covering of three inequivalent knots, two with bridge number three and the third one with bridge number strictly greater than three
On finite simple and nonsolvable groups acting on homology 4-spheres.
No full textThe only finite nonabelian simple group acting on a
homology 3-sphere - necessarily non-freely - is the dodecahedral group A_5 isomorphic to PSL(2,5) (in analogy, the only
finite perfect group acting freely on a homology 3-sphere is the binary dodecahedral group SL(2,5)). In the present
paper we show that the only finite simple groups acting on a homology 4-sphere, and in particular on the 4-sphere, are the alternating or linear fractional groups groups
A_5 isomorphic to PSL(2,5) and A_6 isomorphic to PSL(2,9). From this we deduce a short list of groups
which contains all finite nonsolvable groups
admitting an action on a homology 4-spheres
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