Fuchsian Groups Research Articles

Multifractal analysis of homological growth rates for hyperbolic surfaces

Abstract We perform a multifractal analysis of homological growth rates of oriented geodesics on hyperbolic surfaces. Our main result provides a formula for the Hausdorff dimension of level sets of prescribed growth rates in terms of a generalized Poincaré exponent of the Fuchsian group. We employ symbolic dynamics developed by Bowen and Series, ergodic theory and thermodynamic formalism to prove the analyticity of the dimension spectrum.

Large sieve inequalities for periods of Maass forms

For Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} a Fuchsian Group of the first kind, we obtain large sieve inequalities with weights the hyperbolic periods of Maass forms of even weight. This is inspired by work of Chamizo, who proved a large sieve inequality with weights values of Maass forms of weight 0. The motivation is applications in counting problems in Γ1\\Γ/Γ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _1 \\backslash \\Gamma /\\Gamma _2$$\\end{document}, where Γ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _1$$\\end{document}, Γ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _2$$\\end{document} are hyperbolic subgroups of Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}.

Proofs of ergodicity of piecewise Möbius interval maps using planar extensions

We give two results for deducing dynamical properties of piecewise Möbius interval maps from their related planar extensions. First, eventual expansivity and the existence of an ergodic invariant probability measure equivalent to Lebesgue measure both follow from mild finiteness conditions on the planar extension along with a new property “bounded non-full range” used to relax traditional Markov conditions. Second, the “quilting” operation to appropriately nearby planar systems, introduced by Kraaikamp and co-authors, can be used to prove several key dynamical properties of a piecewise Möbius interval map. As a proof of concept, we apply these results to recover known results on the well-studied Nakada α-continued fractions; we obtain similar results for interval maps derived from an infinite family of non-commensurable Fuchsian groups.

Geodesic cover of Fuchsian groups

Geodesic cover of Fuchsian groups

On the properties of hyperelliptic functions and isometries for Fuchsian groups isomorphism in $${\mathbb {D}}^2$$ and $${\mathbb {H}}^2$$

On the properties of hyperelliptic functions and isometries for Fuchsian groups isomorphism in $${\mathbb {D}}^2$$ and $${\mathbb {H}}^2$$

Reidemeister–Schreier rewriting process for matching uniform signal constellations to quotient groups of arithmetic Fuchsian groups

Reidemeister–Schreier rewriting process for matching uniform signal constellations to quotient groups of arithmetic Fuchsian groups

Zeta functions in higher Teichmüller theory

In this note we introduce zeta functions and L-functions for discrete and faithful representations of surface groups in PSL(d,R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{PSL}(d, \\mathbb R)$$\\end{document}, for d≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\ge 3$$\\end{document}. These are natural generalizations of the well known classical Selberg zeta function and L-function for Fuchsian groups, corresponding to the case d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}. We show that these complex functions have meromorphic extensions to the entire complex plane C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb C$$\\end{document}.

Computing equivalence classes of finite group actions on orientable surfaces

This paper focuses on the classification of classes of topological equivalence of finite group actions on Riemann surfaces. By the Riemann-Hurwitz bound, there are just finitely many groups that act conformally on a closed orientable surface Sg of genus g≥2. With each such action of a group G on Sg one can associate the fundamental group Γ=π(O) of the quotient orbifold O=Sg/G, isomorphic to a Fuchsian group determined completely by orbifold's signature. The Riemann existence theorem reduces the problem of the existence of an action of G on Sg to a purely group-theoretical problem of deciding whether there is an smooth epimorphism mapping the Fuchsian group Γ onto the group G. Using computer algebra systems such as Magma or GAP, together with the library of small groups, the generation of all finite group actions on a surface of fixed small genus g≥2 becomes almost a routine procedure. The difficult part is to determine the classes of these actions with respect to topological equivalence. To achieve this, one needs to investigate the action of the automorphism group of a Fuchsian group on the set of finite group actions on Sg with the corresponding signature. In this paper we derive several results on the topological equivalence of finite group actions on Riemann surfaces. As an application, we derive complete lists of finite group actions of genus g≤8 distinguished up to the topological equivalence.

Music of moduli spaces

A musical instrument, the plastic hormonica, is defined here as a birthday present for Dennis Sullivan, who pioneered and helped popularize the hyperbolic geometry underlying its construction. This plastic hormonica is based upon the Farey tessellation of the Poincaré disk decorated by its standard osculating horocycles centered at the rationals. In effect, one taps or holds points of another tessellation \tau with the same decorating horocycles to produce sounds depending on the fact that the lambda length of e\in\tau with this decoration is always an integer. Explicitly, tapping a decorated edge e\in\tau with lambda length \lambda produces a tone of frequency 440\, \xi^{\lambda-12N} , where \xi^{12}=2 and N is some fixed positive integer shift of octave. Another type of tap on edges of \tau is employed to apply flips, which may be equivariant for a Fuchsian group preserving \tau . Sounding the frequency for the edge after an equivariant flip, one can thereby audibly experience paths in Riemann moduli spaces and listen to mapping classes. The resulting chords, which arise from an ideal triangle complementary to \tau by sounding the frequencies of its frontier edges, correspond to a generalization of the classical Markoff triples, which are precisely the chords that arise from the once-punctured torus. In the other direction, one can query the genera of specified musical pieces.

The Sullivan dictionary and Bowen–Series maps

The Sullivan dictionary between Kleinian groups and rational dynamics describes striking similarities between the fields, both in terms of the objects of study as well as the techniques used. We give an expository account of a recent bridge between the two sides of the dictionary by describing a framework for combining a Fuchsian group with a complex polynomial into a single dynamical system on the Riemann sphere.

Pressure metrics for cusped Hitchin components

We study the cusped Hitchin component consisting of (conjugacy classes of) cusped Hitchin representations of a torsion-free geometrically finite Fuchsian group Γ into PSL(d,R). We produce Riemannian pressure metrics associated to the first fundamental weight and the first simple root. We produce a pressure path metric associated to the Hilbert length and describe its degeneracy.

Hitchin representations of Fuchsian groups

We survey the theory of Hitchin representations of closed surface groups into \mathsf{PSL}(d,\mathbb R) with a focus on their dynamical and geometric properties. We then describe recent extensions of this work to study Hitchin representations of co-finite area Fuchsian groups. The motivation for this recent work is a conjecture about the geometry of the augmented Hitchin component.

Exceptionally simple integrated correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supersymmetric Yang-Mills theory

Supersymmetric localisation has led to several modern developments in the study of integrated correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supersymmetric Yang-Mills (SYM) theory. In particular, exact results have been derived for certain integrated four-point functions of superconformal primary operators in the stress tensor multiplet which are valid for all classical gauge groups, SU(N), SO(N), and USp(2N), and for all values of the complex coupling, τ = θ/(2π) + 4πi/gYM2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_{YM}^2 $$\\end{document}. In this work we extend this analysis and provide a unified two-dimensional lattice sum representation valid for all simple gauge groups, in particular for the exceptional series Er (with r = 6, 7, 8), F4 and G2. These expressions are manifestly covariant under Goddard-Nuyts-Olive duality which for the cases of F4 and G2 is given by particular Fuchsian groups. We show that the perturbation expansion of these integrated correlators is universal in the sense that it can be written as a single function of three parameters, called Vogel parameters, and a suitable ’t Hooft-like coupling. To obtain the perturbative expansion for the integrated correlator with a given gauge group we simply need substituting in this single universal expression specific values for the Vogel parameters. At the non-perturbative level we conjecture a formula for the one-instanton Nekrasov partition function valid for all simple gauge groups and for general Ω-deformation background. We check that our expression reduces in various limits to known results and that it produces, via supersymmetric localisation, the same one-instanton contribution to the integrated correlator as the one derived from the lattice sum representation. Finally, we consider the action of the hyperbolic Laplace operator with respect to τ on the integrated correlators with exceptional gauge groups and derive inhomogeneous Laplace equations very similar to the ones previously obtained for classical gauge groups.

Shrinking target horospherical equidistribution via translated Farey sequences

For a certain diagonal flow on SL(d,Z)﹨SL(d,R) where d≥2, we show that any bounded subset (with measure zero boundary) of the horosphere or a translated horosphere equidistributes, under a suitable normalization, on a target shrinking into the cusp. This type of equidistribution is shrinking target horospherical equidistribution (STHE), and we show STHE for several types of shrinking targets. Our STHE results extend known results for d=2 and L﹨PSL(2,R) where L is any cofinite Fuchsian group with at least one cusp.The two key tools needed to prove our STHE results for the horosphere are a renormalization technique and Marklof's result on the equidistribution of the Farey sequence on distinguished sections. For our STHE results for translated horospheres, we introduce translated Farey sequences, develop some of their geometric and dynamical properties, generalize Marklof's result by proving the equidistribution of translated Farey sequences for the same distinguished sections, and use this equidistribution of translated Farey sequences along with the renormalization technique to prove our STHE results for translated horospheres.

Automorphic Forms and Holomorphic Functions on the Upper Half-plane

We define a set of holomorphic functions in terms of the Hauptmodul of a quotient Riemann surface and prove that these functions are holomorphic on the upper half-plane. It is also shown that these functions are automorphic forms of weight k with respect to a Fuchsian group.
 GANIT J. Bangladesh Math. Soc. 43.1 (2023) 036- 044

Discrete representations of finitely generated groups into PSL(2,R)${\rm PSL}(2,\mathbb {R})$

AbstractWe prove a factorization theorem for Fuchsian groups similar to those proved by Agol and Liu for 3‐manifold groups. As an application, we build Makanin–Razborov diagrams, which parameterize the collection of all discrete representations from an arbitrary but fixed finitely generated group to . We define a new class of groups called ‐discrete limit groups and then use the factorization theorem to obtain useful information about this class of groups.

On the Hyperbolic Bloch Transform

Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, we study the noncommutative Bloch transform of Fuchsian groups that we call the hyperbolic Bloch transform. First, we prove that the hyperbolic Bloch transform is injective and “asymptotically unitary” already in the simplest case—that is, when the Hilbert space is the regular representation of the Fuchsian group, $$\Gamma $$ . Second, when $$\Gamma \subset \textrm{PSU} (1, 1)$$ acts isometrically on the hyperbolic plane $$\mathbb {H}$$ and the Hilbert space is $$L^2 \left( \mathbb {H} \right) $$ , we define a modified, geometric Bloch transform that sends wave functions to sections of irreducible, flat, Hermitian vector bundles over $$\Sigma = \mathbb {H} / \Gamma $$ and transforms the hyperbolic Laplacian into the covariant one.

Algebra of hyperbolic band theory under magnetic field

We explore algebras associated with the hyperbolic band theory under a magnetic field for the first time. We define the magnetic Fuchsian group associated with a higher genus Riemann surface. By imposing the magnetic boundary conditions for the hyperbolic Bloch states, we construct the hyperbolic magnetic Bloch states and investigate their energy spectrum. We give a connection between such magnetic Bloch states and automorphic forms. Our theory is a general extension of the conventional algebra associated with the band theory defined on a Euclidean lattice/space into that of the band theory on a general hyperbolic lattice/Riemann surface.

Bergman space zero sets, modular forms, von Neumann algebras and ordered groups

A^2_{\alpha} will denote the weighted L^2 Bergman space. Given a subset S of the open unit disc we define \Omega(S) to be the infimum of \{s \mid \exists f \in A^2_{s-2}, \; f\neq 0, having S as its zero set \} . By classical results on Hardy space there are sets S for which \Omega(S)=1 . Using von Neumann dimension techniques and cusp forms we give examples of S where 1<\Omega(S)<\infty . By using a left order on certain Fuchsian groups we are able to calculate \Omega(S) exactly if \Omega (S) is the orbit of a Fuchsian group. This technique also allows us to derive in a new way well known results on zeros of cusp forms and indeed calculate the whole algebra of modular forms for \operatorname{PLS}_2 (\mathbb{Z}) .

On Bergman space zero sets. Editors’ comments on the article by Vaughan Jones

This article has two purposes. The first one is to describe what the Editors know on how and when Vaughan Jones worked on the subject of the article published in the same volume. The starting point, in the late 1980’s, was his fascination for a formula giving Murray–von Neumann dimensions of Hilbert spaces of unitary representations of Fuchsian groups. Over the years, he discovered surprising relations of these dimensions with other domains of mathematics. The second purpose is to expose with some details a subject which plays an important role in Jones’ article: the irreducible projective unitary representations of \operatorname{PSL}_2(\mathbf{R}) , which constitute a continuous family known as the discrete series, and which have interesting restrictions to various discrete subgroups of \operatorname{PSL}_2(\mathbf{R}) .