The Experts below are selected from a list of 303 Experts worldwide ranked by ideXlab platform
Gerhard Rosenberger - One of the best experts on this subject based on the ideXlab platform.
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Torsion Free Commutator Subgroups of Generalized Coxeter Groups
Results in Mathematics, 2005Co-Authors: Rubén A. Hidalgo, Gerhard RosenbergerAbstract:In this note we consider generalized Coxeter groups and we study the problem of when their Commutator Subgroup is torsion free. As a consequence we describe all (i) Coxeter groups, (ii) triangle groups and (iii) index two orientation preserving Subgroups of the finite co-volume hyperbolic Coxeter tetrahedra, for which the Commutator Subgroup is torsion free.
Johnson Jonathan - One of the best experts on this subject based on the ideXlab platform.
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Two-Bridge Knots and Residual Torsion-Free Nilpotence
2020Co-Authors: Johnson JonathanAbstract:The residual torsion-free nilpotence of the Commutator Subgroup of a knot group has proven to be an important property with applications to ribbon concordance and bi-orderability. In a lecture, Mayland stated that a two-bridge knot group has a Commutator Subgroup which is a union of an ascending chain of parafree groups. In this case, a result of Baumslag would imply that the Commutator Subgroups of two-bridge knot groups are residually torsion-free nilpotent. However, the transcript of Mayland's lecture does not contain a complete proof, and later, Mayland and Murasugi stated that the transcript is "seriously marred by misprints and minor errors." This paper completes the proof of Mayland's assertion which had remained unfinished. This proof makes use of a modified version of a graph theoretic construction of Hirasawa and Murasugi in order to understand the structure of the Commutator Subgroup of a two-bridge knot group.Comment: 39 pages, 16 figure
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Residual Torsion-Free Nilpotence, Bi-Orderability and Pretzel Knots
2020Co-Authors: Johnson JonathanAbstract:The residual torsion-free nilpotence of the Commutator Subgroup of a knot group has played a key role in studying the bi-orderability of knot groups. A technique developed by Mayland provides a sufficient condition for the Commutator Subgroup of a knot group to be residually-torsion-free nilpotent using work of Baumslag. In this paper, we apply Mayland's technique to several genus one pretzel knots and a family of pretzel knots with arbitrarily high genus. As a result, we obtain a large number of new examples of knots with bi-orderable knot groups. These are the first examples of bi-orderable knot groups for knots which are not fibered or alternating.Comment: 19 pages, 4 figure
Rubén A. Hidalgo - One of the best experts on this subject based on the ideXlab platform.
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Torsion Free Commutator Subgroups of Generalized Coxeter Groups
Results in Mathematics, 2005Co-Authors: Rubén A. Hidalgo, Gerhard RosenbergerAbstract:In this note we consider generalized Coxeter groups and we study the problem of when their Commutator Subgroup is torsion free. As a consequence we describe all (i) Coxeter groups, (ii) triangle groups and (iii) index two orientation preserving Subgroups of the finite co-volume hyperbolic Coxeter tetrahedra, for which the Commutator Subgroup is torsion free.
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kleinian groups with common Commutator Subgroup
Complex Variables and Elliptic Equations, 1995Co-Authors: Rubén A. HidalgoAbstract:In this paper, we obtain a certain rigidity property for Kleinian groups. This asserts that if F and G (both non-elementary torsion-free Fuchsian groups) and [F F] = [G G],then F= G (all equalities are taken in the sense of Mobius transformations). This result is connected to Torelli's theorem (for closed Riemann surfaces) and can be a step in understanding an equivalent for more general type of Riemann surfaces.
R V Skuratovskii - One of the best experts on this subject based on the ideXlab platform.
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minimal generating set of the Commutator Subgroup of sylow 2 Subgroups of alternating group and its structure
Researches in Mathematics and Mechanics, 2019Co-Authors: R V SkuratovskiiAbstract:The size of a minimal generating set for the Commutator Subgroup of Sylow 2-Subgroups of alternating group is found. The structure of Commutator Subgroup of Sylow 2-Subgroups of the alternating group ${A_{{2^{k}}}}$ is investigated. It is shown that $(Syl_2 A_{2^k})^2 = Syl'_2 A_{2^k}, \, k>2$. It is proved that the Commutator length of an arbitrary element of the iterated wreath product of cyclic groups $C_{p_i}, \, p_i\in \mathbb{N}$ equals to 1. The Commutator width of direct limit of wreath product of cyclic groups is found. This paper presents upper bounds of the Commutator width $(cw(G))$ of a wreath product of groups. A recursive presentation of Sylows $2$-Subgroups $Syl_2(A_{{2^{k}}})$ of $A_{{2^{k}}}$ is introduced. As a result the short proof that the Commutator width of Sylow 2-Subgroups of alternating group ${A_{{2^{k}}}}$, permutation group ${S_{{2^{k}}}}$ and Sylow $p$-Subgroups of $Syl_2 A_{p^k}$ ($Syl_2 S_{p^k}$) are equal to 1 is obtained. A Commutator width of permutational wreath product $B \wr C_n$ is investigated. An upper bound of the Commutator width of permutational wreath product $B \wr C_n$ for an arbitrary group $B$ is found.
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Minimal generating set of Sylow 2-Subgroups Commutator Subgroup of alternating group. Commutator width in Sylow $p$-Subgroups of alternating, symmetric groups and in the wreath product of groups
arXiv: Group Theory, 2019Co-Authors: R V SkuratovskiiAbstract:The size of minimal generating set for Commutator of Sylow 2-Subgroup of alternating group was found. Given a permutational wreath product of finite cyclic groups sequence we prove that the Commutator width of such groups is 1 and we research some properties of its Commutator Subgroup. It was shown that $(Syl_2 A_{2^k})^2 = Syl'_2 (A_{2^k}), \, k>2$. A new approach to presentation of Sylow 2-Subgroups of alternating group ${A_{2^{k}}}$ was applied. As a result the short proof that the Commutator width of Sylow 2-Subgroups of alternating group ${A_{2^{k}}}$, permutation group ${S_{2^{k}}}$ and Sylow $p$-Subgroups of $Syl_2 A_{p^k}$ ($Syl_2 S_{p^k}$) are equal to 1 was obtained. Commutator width of permutational wreath product $B \wr C_n$ were investigated. It was proven that the Commutator length of an arbitrary element of Commutator of the wreath product of cyclic groups $C_{p_i}, \, p_i\in \mathbb{N} $ equals to 1. The Commutator width of direct limit of wreath product of cyclic groups are found. As a corollary, it was shown that the Commutator width of Sylows $p$-Subgroups $Syl_2(S_{p^{k}})$ of symmetric $S_{p^{k}}$ and alternating groups $A_{p^{k}}$ $p \geq 2$ are also equal to 1. A recursive presentation of Sylows $2$-Subgroups $Syl_2(A_{2^{k}})$ of $A_{2^{k}}$ was introduced. The structure of Sylows $2$-Subgroups Commutator of symmetric and alternating groups were investigated. For an arbitrary group $B$ an upper bound of Commutator width of $C_p \wr B$ was founded.
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The Commutator and centralizer description of Sylow 2-Subgroups of alternating and symmetric groups
arXiv: Group Theory, 2017Co-Authors: R V SkuratovskiiAbstract:Given a permutational wreath product sequence of cyclic groups of prime order we research a Commutator width of such groups and some properties of its Commutator Subgroup. Commutator width of Sylow 2-Subgroups of alternating group $A_{2^{k}}$, permutation group $S_{2^k}$ and $C_p \wr B$ were founded. The result of research was extended on Subgroups $(Syl_2 {A_{2^k}})'$, $p>2$. The paper presents a construction of Commutator Subgroup of Sylow 2-Subgroups of symmetric and alternating groups. Also minimal generic sets of Sylow 2-Subgroups of $A_{2^k}$ were founded. Elements presentation of $(Syl_2 {A_{2^k}})'$, $(Syl_2 {S_{2^k}})'$ was investigated. We prove that the Commutator width \cite {Mur} of an arbitrary element of a discrete wreath product of cyclic groups $C_p$ is 1. Key words: wreath product of group, Commutator width of $p$-Sylow Subgroups, Commutator Subgroup, centralizer Subgroup, semidirect product.
Witold Tomaszewski - One of the best experts on this subject based on the ideXlab platform.
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A Basis of Bachmuth Type in the Commutator Subgroup of a Free Group
Canadian Mathematical Bulletin, 2003Co-Authors: Witold TomaszewskiAbstract:AbstractWe show here that the Commutator Subgroup of a free group of finite rank poses a basis of Bachmuth's type.