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Malnormal subgroups and Frobenius groups: basics and examples

Pierre de la Harpe, Claude Weber (2014)

Confluentes Mathematici

Malnormal subgroups occur in various contexts. We review a large number of examples, and compare the general situation to that of finite Frobenius groups of permutations.In a companion paper [18], we analyse when peripheral subgroups of knot groups and 3 -manifold groups are malnormal.

Mapping class group and the Casson invariant

Bernard Perron (2004)

Annales de l’institut Fourier

Using a new definition of the second and third Johsnon homomorphisms, we simplify and extend the work of Morita on the Casson invariant of homology-spheres defined by Heegard splittings. In particular, we calculate the Casson invariant of the homology-sphere obtained by gluing two handlebodies along a homeomorphism of the boundary belonging to the Torelli subgroup.

Mapping class group of a handlebody

Bronisław Wajnryb (1998)

Fundamenta Mathematicae

Let B be a 3-dimensional handlebody of genus g. Let ℳ be the group of the isotopy classes of orientation preserving homeomorphisms of B. We construct a 2-dimensional simplicial complex X, connected and simply-connected, on which ℳ acts by simplicial transformations and has only a finite number of orbits. From this action we derive an explicit finite presentation of ℳ.

Matrix factorizations and link homology

Mikhail Khovanov, Lev Rozansky (2008)

Fundamenta Mathematicae

For each positive integer n the HOMFLYPT polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.

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

Mattia Mecchia (2004)

Fundamenta Mathematicae

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.

Metabelian groups acting on compact Riemann surfaces.

G. Gromadzki (1995)

Revista Matemática de la Universidad Complutense de Madrid

A metabelian group G acting as automorphism group on a compact Riemann surface of genus g ≥ 2 has order less than or equal to 16(g-1). We calculate for which values of g this bound is achieved and on these cases we calculate a presentation of the group G.

Minimal degree sequence for 2-bridge knots

Prabhakar Madeti, Rama Mishra (2006)

Fundamenta Mathematicae

We discuss polynomial representations for 2-bridge knots and determine the minimal degree sequence for all such knots. We apply the connection between rational tangles and 2-bridge knots.

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