Fuchsian Groups Research Articles

Character ratios, representation varieties and random generation of finite groups of Lie type

We use character theory of finite groups of Lie type to establish new results on representation varieties of Fuchsian groups, and also on probabilistic generation of groups of Lie type.

Fundamental Inequality for Hyperbolic Coxeter and Fuchsian Groups Equipped with Geometric Distances

Abstract We prove that the hitting measure is not equivalent to the Lebesgue measure for a large class of nearest-neighbour random walks on hyperbolic reflection groups and Fuchsian groups with regular Dirichlet polygons.

Hausdorff dimension of non-conical limit sets

Geometrically infinite Kleinian groups have non-conical limit sets with the cardinality of the continuum. In this paper, we construct a geometrically infinite Fuchsian group such that the Hausdorff dimension of the non-conical limit set equals zero. For finitely generated, geometrically infinite Kleinian groups, we prove that the Hausdorff dimension of the non-conical limit set is positive.

Riemann–Roch isometries in the non-compact orbifold setting

We generalize work of Deligne and Gillet–Soulé on a functorial Riemann–Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces of finite type \Gamma\backslash\mathbb H , for \Gamma\subset\mathrm {PSL}_{2}(\mathbb R) a Fuchsian group of the first kind, equipped with the Poincaré metric. This metric is singular at cusps and elliptic fixed points, and the original results of Deligne and Gillet–Soulé do not apply to this setting. Our theorem relates the determinant of cohomology of the trivial sheaf, with an explicit Quillen type metric defined in terms of the Selberg zeta function of \Gamma , to a metrized version of the tautological line bundle in the theory of moduli spaces of pointed orbicurves, and the self-intersection bundle of a suitable twist of the canonical sheaf \omega . We make use of surgery techniques through Mayer–Vietoris formulas for determinants of Laplacians, in order to reduce to explicit evaluations of such for model hyperbolic cusps and cones. We carry out these computations, which are of independent interest for theoretical physics, and can be adapted to other geometries. We go on to derive an arithmetic Riemann–Roch formula in the realm of Arakelov geometry, which applies in particular to integral models of modular curves with elliptic fixed points. This vastly extends previous work of the first author, whose deformation-theoretic methods were limited to torsion free Fuchsian groups. As an application, we treat in detail the case of the modular curve X(1) , and we show the interesting arithmetic content of the metrized \Psi line bundles. From this, we answer the longstanding question of evaluating the Selberg zeta special value Z'(1, \mathrm {PSL}_2 (\mathbb Z)) . The result is expressed in terms of logarithmic derivatives of Dirichlet L functions. In the analogy between Selberg zeta functions and Dedekind zeta functions of number fields, this formula can be seen as the analytic class number formula for Z(s, \mathrm {PSL}_2 (\mathbb Z)) . The methods developed in this article were conceived so that they afford several variants, such as the determinant of cohomology of a flat unitary vector bundle with finite monodromies at cusps. Our work finds its place in the program initiated by Burgos–Kramer–Kühn of extending arithmetic intersection theory to singular Hermitian vector bundles.

A symmetric Markov coding and the ergodic theorem for actions of Fuchsian Groups

A symmetric Markov coding and the ergodic theorem for actions of Fuchsian Groups

Existence of Smooth Epimorphism from Fuchsian Group to the Group of Automorphisms of compact Riemann surface to the point group of Carbon Tetrachloride

Existence of Smooth Epimorphism from Fuchsian Group to the Group of Automorphisms of compact Riemann surface to the point group of Carbon Tetrachloride

Modular Dedekind symbols associated to Fuchsian groups and higher-order Eisenstein series

Let E(z, s) be the non-holomorphic Eisenstein series for the modular group $${\mathrm{SL}}(2,{{\mathbb {Z}}})$$. The classical Kronecker limit formula shows that the second term in the Laurent expansion at $$s=1$$ of E(z, s) is essentially the logarithm of the Dedekind eta function. This eta function is a weight 1/2 modular form and Dedekind expressed its multiplier system in terms of Dedekind sums. Building on work of Goldstein, we extend these results from the modular group to more general Fuchsian groups $${\Gamma }$$. The analogue of the eta function has a multiplier system that may be expressed in terms of a map $$S:{\Gamma }\rightarrow {{\mathbb {R}}}$$ which we call a modular Dedekind symbol. We obtain detailed properties of these symbols by means of the limit formula. Twisting the usual Eisenstein series with powers of additive homomorphisms from $${\Gamma }$$ to $${{\mathbb {C}}}$$ produces higher-order Eisenstein series. These series share many of the properties of E(z, s) though they have a more complicated automorphy condition. They satisfy a Kronecker limit formula and produce higher-order Dedekind symbols $$S^*:{\Gamma }\rightarrow {{\mathbb {R}}}$$. As an application of our general results, we prove that higher-order Dedekind symbols associated to genus one congruence groups $${\Gamma }_0(N)$$ are rational.

On the remainder in the weighted length spectrum for strictly hyperbolic Fuchsian groups

In this paper, we consider the remainder in a weighted form of the length spectrum for compact Riemann surfaces of genus greater than or equal to two. Earlier, we conducted a similar research where we applied the Cauchy residue theorem over two different square boundaries, one of which intersected the corresponding critical line, and some, quite complex estimates for the logarithmic derivative of the associated zeta functions of Selberg and Ruelle. The main goal of this paper is to achieve the same length spectrum with the same remainder as in our previous study, but in a much simpler way.

Intersecting limit sets of Kleinian subgroups and Susskind’s question

We construct a non-elementary Fuchsian group that admits two non-elementary subgroups with trivial intersection and whose radial limit sets intersect non-trivially. This negatively answers a question of Perry Susskind (1989) that was stated as a conjecture by James W. Anderson (2014).

On Some Higher Order Counting Functions for PSL(2,R)

This paper is devoted to some counting functions of level one and level three in the case of quotient space generated by some strictly hyperbolic Fuchsian group and the upper half-plane. Each of the functions is represented as a sum of some explicit part plus the error term. The explicit part is indexed over singularities of the corresponding Selberg zeta function. In particular, the obtained error term is not larger than O ( x 3/4) . The method applied in this paper follows traditional approach for achieving the error terms in the case of locally symmetric spaces of real rank one. In order to establish an analogy with the classical case, we consider the counting functions divided by x and x3, respectively.

A Prime Geodesic Theorem of Gallagher Type for Riemann Surfaces

We consider a cofinite Fuchsian group of the first kind with finitely many inequivalent parabolic elements and a unitary multiplier system of an arbitrary weight on it. Through the Gallagher–Koyama approach to the prime geodesic theorem on the corresponding noncompact hyperbolic surface, we reduce the exponent in the error term from $$\frac{3}{4}$$ to $$\frac{7}{10}$$ outside a set of finite logarithmic measure. Recent advances in results of the latter type and the methods applied are briefly discussed.

Etudes of the resolvent

Based on the notion of the resolvent and on the Hilbert identities, this paper presents a number of classical results in the theory of differential operators and some of their applications to the theory of automorphic functions and number theory from a unified point of view. For instance, for the Sturm–Liouville operator there is a derivation of the Gelfand–Levitan trace formula, and for the one-dimensional Schrödinger operator a derivation of Faddeev’s formula for the characteristic determinant and the Zakharov– Faddeev trace identities. Recent results on the spectral theory of a certain functional-difference operator arising in conformal field theory are then presented. The last section of the survey is devoted to the Laplace operator on a fundamental domain of a Fuchsian group of the first kind on the Lobachevsky plane. An algebraic scheme is given for proving analytic continuation of the integral kernel of the resolvent of the Laplace operator and the Eisenstein–Maass series. In conclusion there is a discussion of the relationship between the values of the Eisenstein–Maass series at Heegner points and the Dedekind zeta-functions of imaginary quadratic fields, and it is explained why pseudo-cusp forms for the case of the modular group do not provide any information about the zeros of the Riemann zeta-function. Bibliography: 50 titles.

Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy

We initiate the study of Selberg zeta functions Z_{Gamma ,chi } for geometrically finite Fuchsian groups Gamma and finite-dimensional representations chi with non-expanding cusp monodromy. We show that for all choices of (Gamma ,chi ), the Selberg zeta function Z_{Gamma ,chi } converges on some half-plane in mathbb {C}. In addition, under the assumption that Gamma admits a strict transfer operator approach, we show that Z_{Gamma ,chi } extends meromorphically to all of mathbb {C}.

Eisenstein series twisted with non-expanding cusp monodromies

Let $$\Gamma $$ be a geometrically finite Fuchsian group and suppose that $$\chi :\Gamma \rightarrow {{\,\mathrm{GL}\,}}(V)$$ is a finite-dimensional representation with non-expanding cusp monodromy. We show that the parabolic Eisenstein series for $$\Gamma $$ with twist $$\chi $$ converges on some half-plane. Further, we develop Fourier-type expansions for these Eisenstein series.

Construction of Signal Sets From Quotient Rings of the Quaternion Orders Associated With Arithmetic Fuchsian Groups

This paper aims to construct signal sets from quotient rings of the quaternion over a real number field associated with the arithmetic Fuchsian group Γ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4g</sub> , where g is the genus of the associated surface. These Fuchsian groups consist of the edge-pairing isometries of the regular hyperbolic polygons (fundamental region) P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4g</sub> , which tessellate the hyperbolic plane D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . The corresponding tessellations are the self-dual tessellations {4g, 4g}. Knowing the generators of the quaternion orders which realize the edge-pairings of the polygons, the signal points of the signal sets derived from the quotient rings of the quaternion orders are determined. It is shown by examples the relevance of adequately selecting the ideal in the maximal order to construct the signal sets satisfying the property of geometrical uniformity. The labeling of such signals is realized by using the mapping by set partitioning concept to solve the corresponding Diophantine equations (extreme quadratic forms). Trellis coded modulation and multilevel codes whose signal sets are derived from quotient rings of quaternion orders are considered possible applications.

Transformation laws for generalized Dedekind sums associated to Fuchsian groups

We establish transformation laws for generalized Dedekind sums associated to the Kronecker limit function of non-holomorphic Eisenstein series and their higher-order variants. These results apply to general Fuchsian groups of the first kind, and examples are provided in the cases of the Hecke triangle groups, the Hecke congruence groups $\Gamma_0(N)$, and the non-congruence arithmetic groups $\Gamma_0(N)^+$.

On Infinitely generated Fuchsian groups of the Loch Ness monster, the Cantor tree and the Blooming Cantor tree

AbstractIn this paper, for a non-compact Riemman surface S homeomorphic to either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we give a precise description of an infinite set of generators of a Fuchsian group Γ &lt; PSL(2, ℝ), such that the quotient space ℍ/Γ is a hyperbolic Riemann surface homeomorphic to S. For each one of these constructions, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.

Existence of Smooth Epimorphism from a Fuchsian Group to the point Group of Sulfur-Hexafloride

Existence of Smooth Epimorphism from a Fuchsian Group to the point Group of Sulfur-Hexafloride

Geodesic intersections and isoxial Fuchsian groups.

The set of axes of hyperbolic elements in a Fuchsian group depends on the commensurability class of the group. In fact, it has been conjectured that it determines the commensurability class and this has been verified for groups of the second kind by G. Mess and for arithmetic groups by D. Long and A. Reid. Here we show that the conjecture holds for almost all Fuchsian groups and explain why our method fails for arithmetic groups.

Lifting of Modular Forms

The existence and construction of vector-valued modular forms (vvmf) for any arbitrary Fuchsian group $\mathrm{G}$, for any representation $\rho:\mathrm{G} \longrightarrow \mathrm{GL}_{d}(\mathbb{C})$ of finite image can be established by lifting scalar-valued modular forms of the finite index subgroup $Ker(\rho)$ of $\mathrm{G}$. In this article vvmf are explicitly constructed for any admissible multiplier (representation) $\rho$, see Section 3 for the definition of admissible multiplier. In other words, the following question has been partially answered: For which representations $\rho$ of a given $\mathrm{G}$, is there a vvmf with at least one nonzero component ?