Showing papers on "Torsion-free abelian group published in 2013"
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TL;DR: For a genus g surface with one boundary component, S, the Torelli group is the group of orientation preserving homeomorphisms of S that induce the identity on homology as discussed by the authors.
Abstract: For a oriented genus g surface with one boundary component, S, the Torelli group is the group of orientation preserving homeomorphisms of S that induce the identity on homology. The Magnus representation of the Torelli group represents the action on F/F" where F=pi_1(S) and F" is the second term of the derived series. We show that the kernel of the Magnus representation, Mag(S), is highly non-trivial and has a rich structure as a group. Specifically, we define an infinite filtration of Mag(S) by subgroups, called the higher order Magnus subgroups, M_k(S). We develop methods for generating nontrivial mapping classes in M_k(S) for all k and g>1. We show that for each k the quotient M_k(S)/M_{k+1}(S) contains a subgroup isomorphic to a lower central series quotient of free groups E(g-1)_k/E(g-1)_{k+1}. Finally We show that for g>2 the quotient M_k(S)/M_{k+1}(S) surjects onto an infinite rank torsion free abelian group. To do this, we define a Johnson-type homomorphism on each higher order Magnus subgroup quotient and show it has a highly non-trivial image.
5 citations
TL;DR: In this paper, it was shown that the ring of endomorphisms of a reduced torsion-free weakly transitive Abelian group is commutative if and only if the automata are bounded right-nil-potent.
Abstract: It is proved that if all the endomorphisms of a reduced torsion-free weakly transitive Abelian group are bounded right-nilpotent, then its ring of endomorphisms is commutative. The ring of endomorphisms of a torsion-free Abelian group with periodic group of automorphisms and Engel ring of endomorphisms is also commutative.
5 citations
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TL;DR: In this paper, the authors classify Jordan $G$-tori, where G$ is any torsion-free abelian group, using the Zelmanov prime structure theorem, such a class divides into three types, namely, the Hermitian type, the Clifford type and the Albert type.
Abstract: We classify Jordan $G$-tori, where $G$ is any torsion-free abelian group. Using the Zelmanov prime structure theorem, such a class divides into three types, namely, {the Hermitian type, the Clifford type and the Albert type.} We concretely describe Jordan $G$-tori of each type.
2 citations
01 Jan 2013
TL;DR: In this paper, it was shown that any finite sumset over a torsion-free abelian group is isomorphic to a sum set over the integers, including the case of equality.
Abstract: In this chapter, we prove two very basic results concerning finite sumsets over a torsion-free abelian group. First, we show that any finite sumset over a torsion-free abelian group is isomorphic to a sumset over the integers. Second, we give the basic lower bound for finite sumsets over a torsion-free group, including characterizing the case of equality.