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 Transfer of quadratic forms and of quaternion algebras over quadratic field extensions

2018 ◽  
Vol 111 (2) ◽  
pp. 135-143
Author(s):  
Karim Johannes Becher ◽  
Nicolas Grenier-Boley ◽  
Jean-Pierre Tignol
Keyword(s):  
Quadratic Forms ◽  
Quadratic Field ◽  
Quaternion Algebras ◽  
Field Extensions

scholarly journals Explicit Bounds for Hermite Polynomials in the Oscillatory Region

2000 ◽  
Vol 3 ◽  
pp. 307-314 ◽  
Author(s):  
William H. Foster ◽  
Ilia Krasikov
Keyword(s):  
Quadratic Forms ◽  
Hermite Polynomials ◽  
Polynomial Inequalities ◽  
Explicit Bounds

AbstractWe apply a method of positive quadratic forms based on polynomial inequalities to establish sharp explicit bounds on the envelope of Hermite polynomials in the oscillatory region |x| < (2k – 3/2)1/2.

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 Abstract theory of packings and coverings. I.

1959 ◽  
Vol 4 (2) ◽  
pp. 92-95 ◽  
Author(s):  
A. M. Macbeath
Keyword(s):  
Convex Body ◽  
Complex Plane ◽  
Quadratic Forms ◽  
Lattice Points ◽  
Fuchsian Groups ◽  
Linear Groups ◽  
Abstract Theory ◽  
Basic Ideas ◽  
Groups Of Transformations

The aim of this note is to examine the basic ideas underlying Minkowski's theorem on lattice points in a symmetrical convex body and related results of Blichfeldt, and to indicate how these can be generalized. Theorems analogous to Minkowski's, on the automorphisms of quadratic forms and other linear groups and on Fuchsian groups of transformations in the complex plane, have been obtained by Siegel [6] and Tsuji [7], Generalizations which include these are due to Chabauty [2] and Santalo [5].

Groups of units of zero ternary quadratic forms

1981 ◽  
Vol 88 (1-2) ◽  
pp. 141-157 ◽  
Author(s):  
Colin Maclachlan
Keyword(s):  
Finite Index ◽  
Quadratic Forms ◽  
Modular Group ◽  
Conjugacy Classes ◽  
Rational Integer ◽  
Fuchsian Groups ◽  
Integer Coefficients ◽  
Ternary Quadratic Forms

SynopsisThe groups of units of indefinite ternary quadratic forms with rational integer coefficients contain subgroups of index two which are isomorphic to Fuchsian groups and which, for zero forms, are commensurable with the classical modular group. This is used to obtain a family of forms whose groups are representatives of the conjugacy classes of maximal groups associated with zero forms. The signatures of the groups of the forms in this family are determined and it is shown that the group associated to any zero form is isomorphic to a subgroup of finite index in the group of one of three particular forms. This last result should be compared with the corresponding result by Mennicke on non-zero forms.

scholarly journals Addendum: "Chain equivalences for symplectic bases, quadratic forms and tensor products of quaternion algebras"

2015 ◽  
Vol 14 (07) ◽  
pp. 1591001
Author(s):  
Adam Chapman
Keyword(s):  
Quadratic Forms ◽  
Tensor Products ◽  
Quaternion Algebras

We show how the chain lemma for quaternion algebras over fields of F ≠ 2 and [Formula: see text] can be strengthened so that up to two slots are changed at a time.

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