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We construct analogs of the classical Δ-function for quotients of the upper half plane 𝓗 by certain arithmetic triangle groups Γ coming from quaternion division algebras B. We also establish a relative integrality result concerning modular functions of the form Δ(αz)/Δ(z) for α in B⁺. We give two explicit examples at the end.

Arithmetic Fuchsian groups and the number of systoles.

Paul Schmutz (1996)

Mathematische Zeitschrift

Arithmetische Eigenschaften hyperbolischer Klassen elliptischer Modulgruppen.

Ulrich Christian (1991)

Mathematische Zeitschrift

Around the Borromean link.

José María Montesinos Amilibia (2008)

RACSAM

This is a survey of some consequences of the fact that the fundamental group of the orbifold with singular set the Borromean link and isotropy cyclic of order 4 is a universal kleinian group.

Chebyshev polynomials and the modulary group of level p

R.A. RANKIN (1954)

Mathematica Scandinavica

Classical projective geometry and arithmetic groups.

Mark McConnell (1991)

Mathematische Annalen

Closed geodesics, periods and arithmetic of modular forms.

Svetlana Katok (1985)

Inventiones mathematicae

Co-fibered products of algebraic curves

Fedor Bogomolov, Yuri Tschinkel (2009)

Open Mathematics

We give examples of failure of the existence of co-fibered products in the category of algebraic curves.

Cohomology of arithmetic subgroups of SL3 at infinity.

Joachim Schwermer, Ronnie Lee (1982)

Journal für die reine und angewandte Mathematik

Cohomology sets inside arithmetic groups

Stefan Kühnlein (2003)

Acta Arithmetica

Commutator subgroups of the extended Hecke groups $\bar{H} (λ_{q})$

Recep Şahin, Osman Bizim, I. N. Cangul (2004)

Czechoslovak Mathematical Journal

Hecke groups $H (λ_{q})$ are the discrete subgroups of $P S L (2, ℝ)$ generated by $S (z) = - {(z + λ_{q})}^{- 1}$ and $T (z) = - \frac{1}{z}$ . The commutator subgroup of $H$ ( $λ_{q})$ , denoted by $H^{'} (λ_{q})$ , is studied in [2]. It was shown that $H^{'} (λ_{q})$ is a free group of rank $q - 1$ . Here the extended Hecke groups $\bar{H} (λ_{q})$ , obtained by adjoining $R_{1} (z) = 1 / \bar{z}$ to the generators of $H (λ_{q})$ , are considered. The commutator subgroup of $\bar{H} (λ_{q})$ is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the $H (λ_{q})$ case, the index of $H^{'} (λ_{q})$ is changed by $q$ , in the case of $\bar{H} (λ_{q})$ , this number is either 4 for...

Computing fundamental domains for Fuchsian groups

John Voight (2009)

Journal de Théorie des Nombres de Bordeaux

We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group $Γ$ with cofinite area. As a consequence, we compute the invariants of $Γ$ , including an explicit finite presentation for $Γ$ .

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A Note on Link Complements and Arithmetic Groups.

A spectral proof of Rademacher's conjecture for congruence subgroups of the modular group.

A Symbolic Dynamics for Geodesics on Punctured Riemann Surfaces.

Addendum : Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups

An isomorphism between polynomial eigenfunctions of the transfer operator and the Eichler cohomology for modular groups.

Analogs of Δ(z) for triangular Shimura curves