scholarly journals 2-Generator arithmetic Kleinian groups III

2002 ◽  
Vol 90 (2) ◽  
pp. 161
Author(s):  
M. D. E. Conder ◽  
C. Maclachlan ◽  
G. J. Martin ◽  
E. A. O'Brien
Keyword(s):  
Conjugacy Classes ◽  
Kleinian Groups ◽  
The Other ◽  
Commensurability Class

This paper forms part of the program to identify all the 2-generator arithmetic Kleinian groups. Here we identify all conjugacy classes of such groups with one generator parabolic and the other generator elliptic. There are exactly $14$ of these and exactly $5$ Bianchi groups in their commensurability class, namely $\mathrm{PSL}(2,{\mathcal O}_d)$ for $d=1,2,3,7$ and $15$. This complements our earlier identification of the $4$ arithmetic Kleinian groups generated by two parabolic elements.

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

The Poset of Conjugacy Classes and Decomposition of Products in the Symmetric Group

1992 ◽  
Vol 35 (2) ◽  
pp. 152-160 ◽  
Author(s):  
François Bédard ◽  
Alain Goupil
Keyword(s):  
Symmetric Group ◽  
Group Algebra ◽  
Conjugacy Classes ◽  
The Other ◽  
Linear Combinations ◽  
Graded Poset

AbstractThe action by multiplication of the class of transpositions of the symmetric group on the other conjugacy classes defines a graded poset as described by Birkhoff ([2]). In this paper, the edges of this poset are given a weight and the structure obtained is called the poset of conjugacy classes of the symmetric group. We use weights of chains in the posets to obtain new formulas for the decomposition of products of conjugacy classes of the symmetric group in its group algebra as linear combinations of conjugacy classes and we derive a new identity involving partitions of n.

scholarly journals Groups with finitely many conjugacy classes of subgroups with large subnormal defect

1995 ◽  
Vol 37 (1) ◽  
pp. 69-71 ◽  
Author(s):  
Howard Smith
Keyword(s):  
Positive Integer ◽  
Free Group ◽  
Conjugacy Classes ◽  
The Other ◽  
Chief Factor ◽  
Simple Groups ◽  
Finitely Generated ◽  
Finite Image ◽  
Chief Factors ◽  
Locally Graded Groups

Given a group G and a positive integer k, let vk(G) denote the number of conjugacy classes of subgroups of G which are not subnormal of defect at most k. Groups G such that vkG) < ∝ for some k are considered in Section 2 of [1], and Theorem 2.4 of that paper states that an infinite group G for which vk(G) < ∝ (for some k) is nilpotent provided only that all chief factors of G are locally (soluble or finite). Now it is easy to see that a group G whose chief factors are of this type is locally graded, that is, every nontrivial, finitely generated subgroup F of G has a nontrivial finite image (since there is a chief factor H/K of G such that F is contained in H but not in K). On the other hand, every (locally) free group is locally graded and so there is in general no restriction on the chief factors of such groups. The class of locally graded groups is a suitable class to consider if one wishes to do no more than exclude the occurrence of finitely generated, infinite simple groups and, in particular, Tarski p-groups. As pointed out in [1], Ivanov and Ol'shanskiĭ have constructed (finitely generated) infinite simple groups all of whose proper nontrivial subgroups are conjugate; clearly a group G with this property satisfies v1(G) = l. The purpose of this note is to provide the following generalization of the above-mentioned theorem from [1].

Solution of Brauer’s k ⁢ ( B ) k(B) -conjecture for π-blocks of π-separable groups

2018 ◽  
Vol 30 (4) ◽  
pp. 1061-1064
Author(s):  
Benjamin Sambale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Semidirect Product ◽  
Solvable Groups ◽  
Conjugacy Classes ◽  
The Other ◽  
Other Hand ◽  
A Finite Group

Abstract Answering a question of Pálfy and Pyber, we first prove the following extension of the {k(GV)} -problem: Let G be a finite group and let A be a coprime automorphism group of G. Then the number of conjugacy classes of the semidirect product {G\rtimes A} is at most {\lvert G\rvert} . As a consequence, we verify Brauer’s {k(B)} -conjecture for π-blocks of π-separable groups which was proposed by Y. Liu. This generalizes the corresponding result for blocks of p-solvable groups. We also discuss equality in Brauer’s Conjecture. On the other hand, we construct a counterexample to a version of Olsson’s Conjecture for π-blocks which was also introduced by Liu.

2-generator arithmetic Kleinian groups

1999 ◽  
Vol 1999 (511) ◽  
pp. 95-117 ◽  
Author(s):  
C Maclachlan ◽  
G.J Martin
Keyword(s):  
Finite Order ◽  
Conjugacy Classes ◽  
Kleinian Groups ◽  
Triangle Groups

Abstract We show that among the infinitely many conjugacy classes of finite co-volume Kleinian groups generated by two elements of finite order, there are only finitely many which are arithmetic. In particular there are only finitely many arithmetic generalized triangle groups. This latter result generalizes a theorem of Takeuchi.

scholarly journals A reduction theorem for perfect locally finite minimal non-FC groups

1999 ◽  
Vol 41 (1) ◽  
pp. 81-83 ◽  
Author(s):  
FELIX LEINEN
Keyword(s):  
Conjugacy Class ◽  
Proper Subgroup ◽  
Conjugacy Classes ◽  
The Other ◽  
Reduction Theorem ◽  
Locally Finite ◽  
Open Question ◽  
Finite Conjugacy

A group G is said to be a minimal non-FC group, if G contains an infinite conjugacy class, while every proper subgroup of G merely has finite conjugacy classes. The structure of imperfect minimal non-FC groups is quite well-understood. These groups are in particular locally finite. At the other end of the spectrum, a perfect locally finite minimal non-FC group must be a p-group. And it has been an open question for quite a while now, whether such groups exist or not.

scholarly journals ON CONJUGACY CLASSES OF THE KLEIN SIMPLE GROUP IN CREMONA GROUP

2016 ◽  
Vol 59 (2) ◽  
pp. 395-400
Author(s):  
HAMID AHMADINEZHAD
Keyword(s):  
Simple Group ◽  
Three Dimensional ◽  
Conjugacy Classes ◽  
The Other ◽  
Pezzo Surface ◽  
Del Pezzo Surface ◽  
Cremona Group ◽  
Del Pezzo

AbstractWe consider countably many three-dimensional PSL2($\mathbb{F}$7)-del Pezzo surface fibrations over ℙ1. Conjecturally, they are all irrational except two families, one of which is the product of a del Pezzo surface with ℙ1. We show that the other model is PSL2($\mathbb{F}$7)-equivariantly birational to ℙ2×ℙ1. Based on a result of Prokhorov, we show that they are non-conjugate as subgroups of the Cremona group Cr3(ℂ).

Conjugacy Classes of Finite Subgroups of Kleinian Groups

1991 ◽  
Vol 113 (1) ◽  
pp. 179 ◽  
Author(s):  
Mark Feighn ◽  
Geoffrey Mess
Keyword(s):  
Conjugacy Classes ◽  
Kleinian Groups ◽  
Finite Subgroups

scholarly journals Covering Theorems for FINASIGS VIII—almost all conjugacy classes in An have exponent ≤4

1978 ◽  
Vol 25 (2) ◽  
pp. 210-214 ◽  
Author(s):  
J. L. Brenner
Keyword(s):  
Conjugacy Class ◽  
Conjugacy Classes ◽  
The Other ◽  
Alternating Group ◽  
Other Hand ◽  
Covering Theorems ◽  
Image Position ◽  
Almost All

AbstractThe product of two subsets C, D of a group is defined as . The power Ce is defined inductively by C0 = {1}, Ce = CCe−1 = Ce−1C. It is known that in the alternating group An, n > 4, there is a conjugacy class C such that CC covers An. On the other hand, there is a conjugacy class D such that not only DD≠An, but even De≠An for e<[n/2]. It may be conjectured that as n ← ∞, almost all classes C satisfy C3 = An. In this article, it is shown that as n ← ∞, almost all classes C satisfy C4 = An.

Smale solenoid attractors and affine Hirsch foliations

2017 ◽  
Vol 39 (2) ◽  
pp. 531-553 ◽  
Author(s):  
BIN YU
Keyword(s):  
Close Relationships ◽  
Orbit Space ◽  
Conjugacy Classes ◽  
The Other ◽  
List Type ◽  
Orbit Spaces ◽  
Stable And Unstable Manifolds ◽  
Unstable Manifolds ◽  
The One ◽  
Structurally Stable

The main purpose of this paper is to study north–south Smale solenoid diffeomorphisms on$3$-manifolds by using affine Hirsch foliations. A north–south Smale solenoid diffeomorphism$f$on a closed$3$-manifold$M$is a diffeomorphism whose non-wandering set is composed of a Smale solenoid attractor$\unicode[STIX]{x1D6EC}_{a}$and a Smale solenoid repeller$\unicode[STIX]{x1D6EC}_{r}$. The key observation is that a north–south Smale solenoid diffeomorphism$f$automatically induces two non-isotopically leaf-conjugate affine Hirsch foliations${\mathcal{H}}^{s}$and${\mathcal{H}}^{u}$on the orbit space of the wandering set of$f$(abbreviated to thewandering orbit spaceof$f$) by the stable and unstable manifolds of$\unicode[STIX]{x1D6EC}_{a}$and$\unicode[STIX]{x1D6EC}_{r}$, respectively. Under this viewpoint, we build some close relationships between north–south Smale solenoid diffeomorphisms and Hirsch manifolds (the closed$3$-manifolds admitting two non-isotopically leaf-conjugate affine Hirsch foliations).∙On the one hand, the union of the wandering orbit spaces is nearly in one-to-one correspondence with the union of Hirsch manifolds.∙On the other hand, surprisingly, the topology of the wandering orbit space (Hirsch manifold) is nearly a complete invariant of north–south Smale solenoid diffeomorphisms up to semi-global conjugacy.Moreover, as applications, we consider several more concrete questions. For instance, we prove that every diffeomorphism in many semi-global conjugacy classes of north–south Smale solenoid diffeomorphisms are not structurally stable.

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