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Some results and conjectures on finite groups acting on homology spheres
, 2005
"... Abstract. This is a note based on a talk given in the Workshop on geometry and topology of 3-manifolds, Novosibirsk, 22-26 August 2005. We consider the class of finite groups, which admit arbitrary, i.e. not necessarily free actions on integer and mod 2 homology spheres, with an emphasis on the 3- a ..."
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Abstract. This is a note based on a talk given in the Workshop on geometry and topology of 3-manifolds, Novosibirsk, 22-26 August 2005. We consider the class of finite groups, which admit arbitrary, i.e. not necessarily free actions on integer and mod 2 homology spheres, with an emphasis on the 3- and 4-dimensional cases. We recall some classical results and present some recent progress as well as new results, open problems and the emerging conjectural picture of the situation. We are interested in the class of finite groups, and in particular in finite non-solvable and simple groups, which admit actions on integer and mod 2 homology spheres (arbitrary, i.e. not necessarily free actions), with an emphasis on the 3- and 4-dimensional case. We present some classical results, some recent progress as well as new results, open problems and the emerging conjectural picture of the situation. 1. Basic problem. Which finite groups G admit orientation-preserving smooth actions on certain classes of manifolds: spheres Sn, integer homology spheres, mod 2 homology spheres (i.e., homology with coefficients in the integers Z2 mod 2). We consider only orientation-preserving, faithful, but not necessarily free actions (in general, the free case is classical, the main new results presented concern nonfree actions). Particular emphasis will be on dimension three. We note that every finite group admits a free action on a rational homology 3-sphere [4]. Also, any finite group admits a faithful orthogonal action on a sphere (by choosing a linear faithful representation); on the other hand, the classes of groups admitting free actions on integer or mod 2 homology spheres are very restricted. The most important single case is that of the 3-sphere. If an action of a finite group G on S3 is nonfree then, by Thurston’s orbifold geometrization theorem, it is Zimmermann, B.P., Some results and conjectures on finite groups acting on ho-mology spheres.
Finite simple groups acting on 3-manifolds and homology spheres
- Rend. Istit. Mat. Univ. Trieste
, 2002
"... Summary.- Any finite group admits actions on closed 3-manifolds, and in particular free actions. For actions with fixed points, as-sumptions on the type of the fixed point sets of elements drasti-cally reduce the types of the possible groups. Concentrating on the basic case of finite simple groups w ..."
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Summary.- Any finite group admits actions on closed 3-manifolds, and in particular free actions. For actions with fixed points, as-sumptions on the type of the fixed point sets of elements drasti-cally reduce the types of the possible groups. Concentrating on the basic case of finite simple groups we show in the present pa-per that, if some involution of a finite simple group G acting orientation-preservingly on a closed orientable 3-manifold has nonempty connected fixed point set, then G is isomorphic to a projective linear group PSL(2, q), and thus of a very restricted type. The question was motivated by our work on the possible types of isometry groups of hyperbolic 3-manifolds occuring as cyclic branched coverings of knots in the 3-sphere. We char-acterize also finite groups which admit actions on Z2-homology spheres, generalizing corresponding results for integer homology spheres. 1.
TOPOLOGICAL SYMMETRY GROUPS OF GRAPHS IN 3-MANIFOLDS
, 2012
"... We prove that for every closed, connected, orientable, irreducible 3-manifold there exists an alternating group An which is not the topological symmetry group of any graph embedded in the manifold. We also show that for every finite group G there is an embedding Γ of some graph in a hyperbolic rati ..."
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We prove that for every closed, connected, orientable, irreducible 3-manifold there exists an alternating group An which is not the topological symmetry group of any graph embedded in the manifold. We also show that for every finite group G there is an embedding Γ of some graph in a hyperbolic rational homology 3-sphere such that the topological symmetry group of Γ is isomorphic to G.
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"... arXiv version: fonts, pagination and layout may vary from GTM published version Finite groups acting on 3–manifolds and cyclic branched coverings of knots MATTIA MECCHIA We are interested in finite groups acting orientation-preservingly on 3–manifolds (arbitrary actions, ie not necessarily free acti ..."
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arXiv version: fonts, pagination and layout may vary from GTM published version Finite groups acting on 3–manifolds and cyclic branched coverings of knots MATTIA MECCHIA 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 S 3. In the paper we give a characterization of nonsolvable groups of this type. Then we consider some possible applications to the study of cyclic branched coverings of knots and of hyperelliptic diffeomorphisms of 3–manifolds. In particular we analyze the basic case of two distinct knots with the same cyclic branched covering.
IMBEDDINGS OF FREE ACTIONS ON HANDLEBODIES
, 2001
"... Abstract. Fix a free, orientation-preserving action of a finite group G on a 3-dimensional handlebody V. Whenever G acts freely preserving orientation on a connected 3-manifold X, there is a G-equivariant imbedding of V into X. There are choices of X closed and Seifert-fibered for which the image of ..."
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Abstract. Fix a free, orientation-preserving action of a finite group G on a 3-dimensional handlebody V. Whenever G acts freely preserving orientation on a connected 3-manifold X, there is a G-equivariant imbedding of V into X. There are choices of X closed and Seifert-fibered for which the image of V is a handlebody of a Heegaard splitting of X. Provided that the genus of V is at least 2, there are similar choices with X closed and hyperbolic.
Hyperbolic isometries versus symmetries of links
, 2006
"... We prove that every finite group is the orientation-preserving isometry group of the complement of a hyperbolic link in the 3-sphere. ..."
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We prove that every finite group is the orientation-preserving isometry group of the complement of a hyperbolic link in the 3-sphere.
On minimal actions of finite groups on Euclidean spaces and spheres
, 812
"... Abstract. We prove that the minimal dimension of a faithful, smooth action of a finite group on a Euclidean space coincides with the minimal dimension of a faithful, linear action of the group (i.e., with the minimal dimension of a faithful, real, linear representation), for various classes of metac ..."
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Abstract. We prove that the minimal dimension of a faithful, smooth action of a finite group on a Euclidean space coincides with the minimal dimension of a faithful, linear action of the group (i.e., with the minimal dimension of a faithful, real, linear representation), for various classes of metacyclic, linear fractional, symmetric and alternating groups. We prove the analogous result also for actions on (homology) spheres. 1.
on homology
, 2008
"... minimal actions of linear fractional and finite simple groups ..."
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Links, Symmetry and Groups
"... Preface I would like to thank my supervisor Craig Hodgson for his patience and guidance, Arun Ram for his help and support and my friends and family for their support. iii ..."
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Preface I would like to thank my supervisor Craig Hodgson for his patience and guidance, Arun Ram for his help and support and my friends and family for their support. iii
