Wyniki wyszukiwania

1

On finite groups acting on a connected sum of 3-manifolds S² × S¹

100%

Zimmermann B.

Fundamenta Mathematicae

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2014

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tom 226

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nr 2

131-142

EN

Let $H_{g}$ denote the closed 3-manifold obtained as the connected sum of g copies of S² × S¹, with free fundamental group of rank g. We prove that, for a finite group G acting on $H_{g}$ which induces a faithful action on the fundamental group, there is an upper bound for the order of G which is quadratic in g, but there does not exist a linear bound in g. This implies then a Jordan-type bound for arbitrary finite group actions on $H_{g}$ which is quadratic in g. For the proofs we develop a calculus for finite group actions on $H_{g}$, by codifying such actions by handle-orbifolds and finite graphs of finite groups.

2

Cyclic branched coverings and homology 3-spheres with large group actions

100%

Zimmermann B.

Fundamenta Mathematicae

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2004

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tom 184

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nr 1

343-353

EN

We show that, if the covering involution of a 3-manifold M occurring as the 2-fold branched covering of a knot in the 3-sphere is contained in a finite nonabelian simple group G of diffeomorphisms of M, then M is a homology 3-sphere and G isomorphic to the alternating or dodecahedral group 𝔸₅ ≅ PSL(2,5). An example of such a 3-manifold is the spherical Poincaré sphere. We construct hyperbolic analogues of the Poincaré sphere. We also give examples of hyperbolic ℤ₂-homology 3-spheres with PSL(2,q)-actions, for various small prime powers ,q. We note that the groups PSL(2,q), for odd prime powers ,q, are the only candidates for being finite nonabelian simple groups which possibly admit actions on ℤ₂-homology 3-spheres (but the exact classification remains open).

3

On finite groups of isometries of handlebodies in arbitrary dimensions and finite extensions of Schottky groups

63%

Mecchia M., Zimmermann B.

Fundamenta Mathematicae

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2015

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tom 230

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nr 3

237-249

EN

It is known that the order of a finite group of diffeomorphisms of a 3-dimensional handlebody of genus g > 1 is bounded by the linear polynomial 12(g-1), and that the order of a finite group of diffeomorphisms of a 4-dimensional handlebody (or equivalently, of its boundary 3-manifold), faithful on the fundamental group, is bounded by a quadratic polynomial in g (but not by a linear one). In the present paper we prove a generalization for handlebodies of arbitrary dimension d, uniformizing handlebodies by Schottky groups and considering finite groups of isometries of such handlebodies. We prove that the order of a finite group of isometries of a handlebody of dimension d acting faithfully on the fundamental group is bounded by a polynomial of degree d/2 in g if d is even, and of degree (d+1)/2 if d is odd, and that the degree d/2 for even d is best possible. This implies analogous polynomial Jordan-type bounds for arbitrary finite groups of isometries of handlebodies (since a handlebody of dimension d > 3 admits S¹-actions, there does not exist an upper bound for the order of the group itself).

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Zimmermann B.

1 Mecchia M.

1 2015

1 2014

1 2004

On finite groups acting on a connected sum of 3-manifolds S² × S¹

Cyclic branched coverings and homology 3-spheres with large group actions

On finite groups of isometries of handlebodies in arbitrary dimensions and finite extensions of Schottky groups