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Here we determine the arithmetic data i.e. the totally real number field and the set of ramified places of the defining quaternion algebra, of all those commensurability classes of arithmetic Fuchsian groups which contain a surface group of genus 2, i.e. a group of signature (2;– –).
In this paper a formula is derived for the number of conjugacy classes of cyclic subgroups of finite order in those arithmetic Fuchsian groups which are of minimal covolume in their commensurability class. The formula is entirely in terms of the number theoretic data defining the commensurability class of the arithmetic group so that, in particular, any two such groups of minimal covolume in the class, will be isomorphic.
Achalasia is a motility disorder of the oesophagus that typically presents with dysphagia, regurgitation and chest pain. A rare presenting symptom is stridor. A case of previously treated achalasia re-presenting with stridor is described and associated imaging presented.
E. Bujalance, Universidad National de Educación a Distancia, Madrid,A. F. Costa, Universidad National de Educación a Distancia, Madrid,E. Martínez, Universidad National de Educación a Distancia, Madrid
This article is an extended version of the lecture given at UNED in July 1998 on “Topics on Riemann surfaces and Fuchsian groups” to mark the 25th anniversary of UNED.
The object of that lecture was to motivate the definition of arithmetic Fuchsian groups from the special and very familiar example of the classical modular group. This motivation proceeded via quaternion algebras and the lecture ended with the definition of arithmetic Fuchsian groups in these terms. This essay will go a little beyond that to indicate how the number theoretic data defining an arithmetic Fuchsian group can be used to determine geometric and group-theoretic information. No effort is made here to investigate other approaches to arithmetic Fuchsian groups via quadratic forms or to discuss and locate these groups in the general theory of discrete arithmetic subgroups of semi-simple Lie groups. Thus the horizons of this article are limited to giving one method of introducing an audience familiar with the ideas of Fuchsian groups and Riemann surfaces to the interesting special subclass of arithmetic Fuchsian groups.
A Fuchsian group is a discrete subgroup of SL(2,ℝ) or of PSL(2,ℝ) = SL(2,ℝ)/ < −I >. We will frequently employ the usual abuse of notation by writing elements of PSL(2,ℝ) as matrices, while strictly they are only determined up to sign.
Lateral thinking in biomimetic materials chemistry has permitted chemists to create fascinating structures that mimic the biomaterials optimized by Nature. The integration of organic and inorganic chemistry at multiple length scales gives optimal performance characteristics to biomaterials, such as bone. In a similar fashion, lateral thinking in our lab has enabled us to consolidate the chemistry of inorganic surfactant-templated mesoporous materials with the organic-inorganic hybrid structure of amorphous xerogels. A new class of materials, periodic mesoporous organosilicas (PMOs), has emerged that marries organic and solid-state chemistry in the channels of hexagonally ordered mesoporous materials. Various organic and organometallic groups may be integrated into the framework, creating materials with novel, tunable properties. Surfactant can be solvent-extracted or ion-exchanged to create a high surface area PMO with the framework and the organic group intact. This renders the organic groups accessible for reaction to give a new type of “chemistry of the channels”.
Let P be a polyhedron in H3
of finite volume such that the group Γ(P) generated
by reflections in the faces of P is a discrete subgroup of
Isom H3. Let Γ+(P) denote the
subgroup of index 2 consisting entirely of orientation-preserving isometries so that
Γ+(P) is a Kleinian group of finite covolume.
Γ+(P) is called a polyhedral group.
As discussed in  and 
for example (see §2 below), associated to a Kleinian
group Γ of finite covolume is a pair (AΓ, kΓ)
which is an invariant of the
commensurability class of Γ; kΓ is a number field
called the invariant trace-field, and
AΓ is a quaternion algebra over kΓ.
It has been of some interest recently (cf. [13, 16])
to identify the invariant trace-field and quaternion algebra associated to a Kleinian
group Γ of finite covolume since these are closely related to the geometry and
topology of H3/Γ.
In this paper we give a method for identifying these in the case of polyhedral
groups avoiding trace calculations. This extends the work in 
and  on
arithmetic polyhedral groups. In §6 we compute the invariant trace-field and
quaternion algebra of a family of polyhedral groups arising from certain triangular
prisms, and in §7 we give an application of this calculation to construct closed
hyperbolic 3-manifolds with ‘non-integral trace’.
For a fixed finite group G, the numbers Ng
of equivalence classes of
orientation-preserving actions of G on closed orientable surfaces
Σg of genus g can be encoded by a generating
When equivalence is determined by the
isomorphism class of the quotient orbifold Σg/G,
we show that the generating function is rational. When equivalence is topological
conjugacy, we examine the cases
where G is abelian and show that the generating function is again
rational in the
cases where G is cyclic.
Extending earlier work, we establish the finiteness of the number
two-generator arithmetic Kleinian
groups with one generator parabolic and the other either parabolic or elliptic.
We also identify all the
arithmetic Kleinian groups generated by two parabolic elements. Surprisingly,
there are exactly 4 of these,
up to conjugacy, and they are all torsion free.
A generalised triangle group has a presentation of the form
where R is a cyclically reduced word involving both x and y. When R = xy, these classical triangle groups have representations as discrete groups of isometrics of S2, R2, H2 depending on
In this paper, for other words R, faithful discrete representations of these groups in Isom +H3 = PSL(2, C) are considered with particular emphasis on the case R = [x, y] and also on the relationship between the Euler characteristic χ and finite covolume representations.
In the nineteenth century, Hurwitz  and Wiman  obtained bounds for the order of the automorphism group and the order of each automorphism of an orientable and unbordered compact Klein surface (i. e., a compact Riemann surface) of topological genus g s 2. The corresponding results of bordered surfaces are due to May, , . These may be considered as particular cases of the general problem of finding the minimum topological genus of a surface for which a given finite group G is a group of automorphisms. This problem was solved for cyclic and abelian G by Harvey  and Maclachlan , respectively, in the case of Riemann surfaces and by Bujalance , Hall  and Gromadzki  in the case of non-orientable and unbordered Klein surfaces. In dealing with bordered Klein surfaces, the algebraic genus—i. e., the topological genus of the canonical double covering, (see Alling-Greenleaf )—was minimized by Bujalance- Etayo-Gamboa-Martens  in the case where G is cyclic and by McCullough  in the abelian case.
This book is a collection of articles addressing a range of topics in the theory of discrete groups; for the most part the papers represent talks delivered at a conference held at the University of Birmingham in January 1991 to mark the retirement of A. M. Macbeath from his chair at the University of Pittsburgh.
The central theme of the volume is the study of groups from a geometric point of view. Of course, the geometric aspect takes many forms; thus one may find a group which operates on Euclidean or hyperbolic space rubbing shoulders with free groups or some generalisation studied with the help of graphs or homological algebra. Groups with presentation also relate to abstract or formal geometry: in recent years the study of groups which act on trees has brought algebraic structures back into the geometric fold, and the seminal idea that a countable group itself carries a geometric essence has reinforced this return to geometry within rather than through group theory, giving Klein's Programme a fresh cutting edge.
A major part of group theory today relates directly to explicit algebraic or geometric objects — permutation groups, Coxeter groups and discrete subgroups of Lie groups are prominent — and one of the strengths of this field lies in the wealth of fascinating interactions with complex analysis and low dimensional topology. The serious study of discrete groups via combinatorial techniques began in hyperbolic space with Poincare and Dehn, and the reader will find here many echoes of their original ideas and interests.
This book constitutes the proceedings of a conference held at the University of Birmingham to mark the retirement of Professor A. M. Macbeath. The papers represent up-to-date work on a broad spectrum of topics in the theory of discrete group actions, ranging from presentations of finite groups through the detailed study of Fuchsian and crystallographic groups, to applications of group actions in low dimensional topology, complex analysis, algebraic geometry and number theory. For those wishing to pursue research in these areas, this volume offers a valuable summary of contemporary thought and a source of fresh geometric insights.
A Fuchsian group is a discrete subgroup of PSL2(R) and two such groups Γ1,Γ2 are commensurable if and only if the intersection Γ1 ∩ Γ2 is of finite index in both Γ1 and Γ2. In this paper we determine when two two-generator Fuchsian groups of finite covolume are commensurable and in addition, the relationship between two such groups, by obtaining part of the lattice of the commensurability class which contains one representative from each conjugacy class, in PGL2(R), of two-generator groups. By the result of Margulis (see e.g. [Z]) that a non-arithmetic commensurability class contains a unique maximal member, the non-arithmetic cases reduce to a compilation of earlier results on determining which two-generator Fuchsian groups occur as subgroups of finite index in other two-generator Fuchsian groups [Sc], [S2], [R]. For the arithmetic cases, all two-generator arithmetic Fuchsian groups have been determined [T2], [T4], [MR] and one can immediately read off when two such groups are commensurable from the structure of the corresponding quaternion algebra (see e.g. [T3]). The relationship between such groups is more difficult to determine and we utilise structure theorems for arithmetic Fuchsian groups [B], [V]. The relationship between arithmetic triangle groups was determined in [T3].
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.