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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;– –).
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 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.
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.
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.