Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups

1987 ◽  
Vol 102 (2) ◽  
pp. 251-257 ◽  
Author(s):  
C. MacLachlan ◽  
A. W. Reid
Keyword(s):  
Kleinian Groups ◽  
Fuchsian Groups ◽  
Arithmetic Groups ◽  
Quaternion Algebras ◽  
Isomorphism Classes ◽  
Similar Characterization ◽  
One To One

Arithmetic Fuchsian and Kleinian groups can all be obtained from quaternion algebras (see [2,12]). In a series of papers ([8,9,10,11]), Takeuchi investigated and characterized arithmetic Fuchsian groups among all Fuchsian groups of finite covolume, in terms of the traces of the elements in the group. His methods are readily adaptable to Kleinian groups, and we obtain a similar characterization of arithmetic Kleinian groups in §3. Commensurability classes of Kleinian groups of finite co-volume are discussed in [2] and it is shown there that the arithmetic groups can be characterized as those having dense commensurability subgroup. Here the wide commensurability classes of arithmetic Kleinian groups are shown to be approximately in one-to-one correspondence with the isomorphism classes of the corresponding quaternion algebras (Theorem 2) and it easily follows that there are infinitely many wide commensurability classes of cocompact Kleinian groups, and hence of compact hyperbolic 3-manifolds.

scholarly journals On free ring extensions of degreen

1981 ◽  
Vol 4 (4) ◽  
pp. 703-709
Author(s):  
George Szeto
Keyword(s):  
Galois Extension ◽  
Quaternion Algebra ◽  
Ring Extension ◽  
Quaternion Algebras ◽  
Free Ring ◽  
Azumaya Algebra ◽  
Ring Extensions ◽  
One To One ◽  
Generalized Quaternion

Nagahara and Kishimoto [1] studied free ring extensionsB(x)of degreenfor some integernover a ringBwith 1, wherexn=b,cx=xρ(c)for allcand somebinB(ρ=automophism of  B), and{1,x…,xn−1}is a basis. Parimala and Sridharan [2], and the author investigated a class of free ring extensions called generalized quaternion algebras in whichb=−1andρis of order 2. The purpose of the present paper is to generalize a characterization of a generalized quaternion algebra to a free ring extension of degreenin terms of the Azumaya algebra. Also, it is shown that a one-to-one correspondence between the set of invariant ideals ofBunderρand the set of ideals ofB(x)leads to a relation of the Galois extensionBover an invariant subring underρto the center ofB.

Two-generator arithmetic Fuchsian groups. II

1992 ◽  
Vol 111 (1) ◽  
pp. 7-24 ◽  
Author(s):  
C. Maclachlan ◽  
G. Rosenberger
Keyword(s):  
Quaternion Algebra ◽  
Single Point ◽  
Number Fields ◽  
Conjugacy Classes ◽  
Fuchsian Groups ◽  
Arithmetic Groups ◽  
Quaternion Algebras ◽  
Triangle Groups ◽  
Maximal Orders ◽  
Or Groups

An arithmetic Fuchsian group is necessarily of finite covolume and so of the first kind. From the structure theorem for finitely generated Fuchsian groups those of the first kind which can be generated by two elements are triangle groups, groups of signature (1;q;0) or (1; ; 1) or groups of signature (0;2,2,2,e;0) where e is odd 6. It is known that there are finitely many conjugacy classes of arithmetic groups with these signatures or indeed with any fixed signature 3, 15. In the case of non-cocompact groups, the arithmetic groups are conjugate to groups commensurable with the classical modular group and are easily determined. For the other groups described above the space of conjugacy classes of all such Fuchsian groups of fixed signature can be described in terms of the traces of a pair of generating elements and their product. In the case of triangle groups this space is a single point. This description has been utilized to determine all classes of triangle groups 13 and groups of signature (1;q;0) which are arithmetic 15. In this paper we determine all classes of groups of signature (0;2,2,2,e;0) with e odd which are arithmetic. The techniques involving traces used so profitably in 15 are not so fruitful in this case. Consequently we have in the main resorted to a quite different method which does not rely on having a precise description of the space of conjugacy classes and hence could be applicable to groups other than those which have rank 2. Extensive use is made of results of Borell on the structure of arithmetic Fuchsian groups which require detailed information on the number fields defining the quaternion algebras. Consequently, the results are given in terms of the quaternion algebra which is determined by its defining field and ramification set, and maximal orders in that quaternion algebra.

scholarly journals A characterization of the weighted version of McEliece–Rodemich–Rumsey–Schrijver number based on convex quadratic programming

2015 ◽  
Vol 07 (04) ◽  
pp. 1550050
Author(s):  
Carlos J. Luz
Keyword(s):  
Quadratic Programming ◽  
Discrete Math ◽  
Convex Quadratic Programming ◽  
Stability Number ◽  
Weighted Version ◽  
Similar Characterization ◽  
Number Formula ◽  
Weighted Stability ◽  
Theta Number

For any graph [Formula: see text] Luz and Schrijver [A convex quadratic characterization of the Lovász theta number, SIAM J. Discrete Math. 19(2) (2005) 382–387] introduced a characterization of the Lovász number [Formula: see text] based on convex quadratic programming. A similar characterization is now established for the weighted version of the number [Formula: see text] independently introduced by McEliece, Rodemich, and Rumsey [The Lovász bound and some generalizations, J. Combin. Inform. Syst. Sci. 3 (1978) 134–152] and Schrijver [A Comparison of the Delsarte and Lovász bounds, IEEE Trans. Inform. Theory 25(4) (1979) 425–429]. Also, a class of graphs for which the weighted version of [Formula: see text] coincides with the weighted stability number is characterized.

scholarly journals On quaternion algebras over the composite of quadratic number fields

2021 ◽  
Vol 56 (1) ◽  
pp. 63-78
Author(s):  
Vincenzo Acciaro ◽  
◽  
Diana Savin ◽  
Mohammed Taous ◽  
Abdelkader Zekhnini ◽  
...  
Keyword(s):  
Number Fields ◽  
Complete Characterization ◽  
Quadratic Number ◽  
Quaternion Algebras ◽  
Quadratic Number Fields

Let p and q be two positive prime integers. In this paper we obtain a complete characterization of division quaternion algebras HK(p, q) over the composite K of n quadratic number fields.

27.—Generating Sets for Fuchsian Groups

1975 ◽  
Vol 72 (4) ◽  
pp. 317-326 ◽  
Author(s):  
J. H. H. Chalk ◽  
B. G. A. Kelly
Keyword(s):  
Quadratic Forms ◽  
Fundamental Region ◽  
Fuchsian Groups ◽  
Quaternion Algebras ◽  
Generating Sets ◽  
Complete Set ◽  
Explicit Bounds

SynopsisFor a class of Fuchsian groups, which includes integral automorphs of quadratic forms and unit groups of indefinite quaternion algebras, it is shown that the geometry of a suitably chosen fundamental region leads to explicit bounds for a complete set of generators.

scholarly journals Siegel modular forms and theta series attached to quaternion algebras

1991 ◽  
Vol 121 ◽  
pp. 35-96 ◽  
Author(s):  
Siegfried Böcherer ◽  
Rainer Schulze-Pillot
Keyword(s):  
Modular Forms ◽  
Orthogonal Group ◽  
Automorphic Forms ◽  
Theta Series ◽  
Siegel Modular Forms ◽  
Theta Correspondence ◽  
Quaternion Algebras ◽  
Metaplectic Group

The two main problems in the theory of the theta correspondence or lifting (between automorphic forms on some adelic orthogonal group and on some adelic symplectic or metaplectic group) are the characterization of kernel and image of this correspondence. Both problems tend to be particularly difficult if the two groups are approximately the same size.

scholarly journals Structure of perfect rings

1970 ◽  
Vol 2 (1) ◽  
pp. 117-124 ◽  
Author(s):  
Vlastimil Dlab
Keyword(s):  
Present Note ◽  
Matrix Representation ◽  
Simple Characterization ◽  
One To One ◽  
The Way

In the present note, we offer a simple characterization of perfect rings in terms of their components and socle sequences, which is subsequently used to establish a one-to-one correspondence between perfect rings and certain finite additive categories. This correspondence is effected by means of a matrix representation, which describes the way in which perfect rings are built from local perfect rings.

scholarly journals On generalized quaternion algebras

1980 ◽  
Vol 3 (2) ◽  
pp. 237-245 ◽  
Author(s):  
George Szeto
Keyword(s):  
Automorphism Group ◽  
Commutative Ring ◽  
Galois Extension ◽  
Ring Extension ◽  
Separable Extension ◽  
Quaternion Algebras ◽  
Generalized Quaternion

LetBbe a commutative ring with1, andG(={σ})an automorphism group ofBof order2. The generalized quaternion ring extensionB[j]overBis defined byS. Parimala andR. Sridharan such that (1)B[j]is a freeB-module with a basis{1,j}, and (2)j2=−1andjb=σ(b)jfor eachbinB. The purpose of this paper is to study the separability ofB[j]. The separable extension ofB[j]overBis characterized in terms of the trace(=1+σ)ofBover the subring of fixed elements underσ. Also, the characterization of a Galois extension of a commutative ring given by Parimala and Sridharan is improved.

Sign in / Sign up

Export Citation Format

Share Document