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1

Maximal actions of finite 2-groups on ℤ₂-homology 3-spheres

100%

Mecchia M.

Fundamenta Mathematicae

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2004

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tom 184

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nr 1

205-221

EN

It is known that a finite 2-group acting on a ℤ₂-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.

2

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).

3

On finite groups acting on acyclic low-dimensional manifolds

51%

Guazzi A., Mecchia M., Zimmermann B.

Fundamenta Mathematicae

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2011

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tom 215

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nr 3

203-217

EN

We consider finite groups which admit a faithful, smooth action on an acyclic manifold of dimension three, four or five (e.g. Euclidean space). Our first main result states that a finite group acting on an acyclic 3- or 4-manifold is isomorphic to a subgroup of the orthogonal group O(3) or O(4), respectively. The analogous statement remains open in dimension five (where it is not true for arbitrary continuous actions, however). We prove that the only finite nonabelian simple groups admitting a smooth action on an acyclic 5-manifold are the alternating groups 𝔸₅ and 𝔸₆, and deduce from this a short list of finite groups, closely related to the finite subgroups of SO(5), which are the candidates for orientation-preserving actions on acyclic 5-manifolds.

Ograniczanie wyników

3 Fundamenta Mathematicae

3 Mecchia M.

2 Zimmermann B.

1 Guazzi A.

1 2015

1 2011

1 2004

Maximal actions of finite 2-groups on ℤ₂-homology 3-spheres

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

On finite groups acting on acyclic low-dimensional manifolds