Hyperbolic Triangle Groups Research Articles

Complex hyperbolic triangle groups of type [m,m,0;n1,n2,2]

In this paper we study discreteness of complex hyperbolic triangle groups of type [m,m,0;n1,n2,2], i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders n1,n2,2 in complex geodesics with pairwise distances m, m, 0. For fixed m, the parameter space of such groups is of real dimension one. We determine the possible orders for n1 and n2 and also intervals in the parameter space that correspond to discrete and non-discrete triangle groups.

Soluble quotients of triangle groups

This paper helps explain the prevalence of soluble groups among the automorphism groups of regular maps (at least for ‘small’ genus), by showing that every non-perfect hyperbolic ordinary triangle group Δ+(p,q,r)=〈x,y|xp=yq=(xy)r=1〉 has a smooth finite soluble quotient of derived length c for some c≤3, and infinitely many such quotients of derived length d for every d>c.

Mirror stabilizers for lattice complex hyperbolic triangle groups

Mirror stabilizers for lattice complex hyperbolic triangle groups

Spectral Analysis of the Adjacency Matrices for Alternating Quotients of Hyperbolic Triangle Group ▵*(3,q,r) for q < r Primes

Hyperbolic triangle groups are found within the category of finitely generated groups. These are topological groups formed by the reflections along the sides of a hyperbolic triangle and acting properly discontinuously on the hyperbolic plane. Higman raised a question about the simplicity of finitely generated groups. The best known example of a simple group is the alternating group An, where n≥5. This article establishes a relation between the hyperbolic triangle group denoted as ▵*(3,7,r) and the alternating group. The approach involves employing coset diagrams to establish this connection. The construction of adjacency matrices for these coset diagrams is performed, followed by a detailed examination of their spectral characteristics.

Nielsen equivalence in triangle groups

We show that any non-standard generating pair of a hyperbolic triangle group is represented by a special almost orbifold covering with a good marking.

On Subgroups Finite Index in Complex Hyperbolic Lattice Triangle Groups

We study several explicit finite index subgroups in the known complex hyperbolic lattice triangle groups, and show some of them are neat, some of them have positive first Betti number, some of them have a homomorphisms onto a non-Abelian free group. For some lattice triangle groups, we determine the minimal index of a neat subgroup. Finally, we answer a question raised by Stover and describe an infinite tower of neat ball quotients all with a single cusp.

Triangular modular curves of small genus

Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy text {PGL}_2(mathbb {F}_q) or text {PSL}_2(mathbb {F}_q). We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus.

Systolic length of triangular modular curves

We present a method for computing upper bounds on the systolic length of certain Riemann surfaces uniformized by congruence subgroups of hyperbolic triangle groups, admitting congruence Hurwitz curves as a special case. The uniformizing group is realized as a Fuchsian group and a convenient finite generating set is computed. The upper bound is derived from the traces of the generators. Some explicit computations, including ones for non-arithmetic surfaces, are given. We apply a result of Cosac and Dória to show that the systolic length grows logarithmically with respect to the genus.

Redundancy of Triangle Groups in Spherical CR Representations

Falbel, Koseleff and Rouillier computed a large number of boundary unipotent CR representations of fundamental groups of non compact three-manifolds. Those representations are not always discrete. By experimentally computing their limit set, one can determine that those with fractal limit sets are discrete. Many of those discrete representations can be related to complex hyperbolic triangle groups. By exact computations, we verify the existence of those triangle representations, which have boundary unipotent holonomy. We also show that many representations are redundant: for n fixed, all the representations encountered are conjugate and only one among them is uniformizable.

Rigidity of diagonally embedded triangle groups

AbstractWe show local rigidity of hyperbolic triangle groups generated by reflections in pairs of n-dimensional subspaces of $\mathbb {R}^{2n}$ obtained by composition of the geometric representation in $\mathsf {PGL}(2,\mathbb {R})$ with the diagonal embeddings into $\mathsf {PGL}(2n,\mathbb {R})$ and $\mathsf {PSp}^\pm (2n,\mathbb {R})$ .

Complex hyperbolic (3,n,∞) triangle groups

Let n≥5 be a positive integer. A complex hyperbolic (3,n,∞) triangle group is a representation from the hyperbolic (3,n,∞) triangle group into the holomorphic isometry group of complex hyperbolic plane, which maps the generators to complex reflections fixing complex lines. In this paper, we show that a complex hyperbolic (3,n,∞) triangle group 〈I1,I2,I3〉 is discrete and faithful if and only if I1I3I2I3 is not elliptic. Our result answers a conjecture of Schwartz for complex hyperbolic (3,n,∞) triangle groups.

New Nonarithmetic Complex Hyperbolic Lattices II

We describe a general procedure to produce fundamental domains for complex hyperbolic triangle groups. This allows us to produce new nonarithmetic lattices, bringing the number of known nonarithmetic commensurability classes in PU(2,1) to 22.

Complex hyperbolic triangle groups of type [𝑚,𝑚,0;3,3,2

In this paper we study discreteness of complex hyperbolic triangle groups of type[m,m,0;3,3,2][m,m,0;3,3,2], i.e., groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders3,3,23,3,2in complex geodesics with pairwise distancesm,m,0m,m,0. For fixed mm, the parameter space of such groups is of real dimension one. We determine intervals in this parameter space that correspond to discrete and to non-discrete triangle groups.

Discreteness of ultra‐parallel complex hyperbolic triangle groups of type [m1,m2,0

In this paper we consider ultra-parallel complex hyperbolic triangle groups of type $[m_1,m_2,0]$, i.e. groups of isometries of the complex hyperbolic plane, generated by complex reflections in three ultra-parallel complex geodesics two of which intersect on the boundary. We prove some discreteness and non-discreteness results for these groups and discuss the connection between the discreteness results and ellipticity of certain group elements.

Algebraic curves uniformized by congruence subgroups of triangle groups

We construct certain subgroups of hyperbolic triangle groups which we call \congruence subgroups. These groups include the classical congruence subgroups of SL2(Z), Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimura curves. We determine the eld of moduli of the curves associated to these groups and thereby realize the groups PSL2(Fq) and PGL2(Fq) regularly as Galois groups.

Complex hyperbolic triangle groups with 2-fold symmetry

In this paper we will consider the 2-fold symmetric complex hy­perbolic triangle groups generated by three complex reflections through angle 2Π/p with p ≥ 2. We will mainly concentrate on the groups where some ele­ments are elliptic of finite order. Then we will classify all such groups which are candidates for being discrete. There are only 4 types.

Note on non-discrete complex hyperbolic triangle groups of type $(n,n,\infty;k)$ II

A complex hyperbolic triangle group is a group generated by three complex involutions fixing complex lines in complex hyperbolic space. In our previous papers~[4,5,6,7,8] we discussed complex hyperbolic triangle groups. In particular, in~[5,8] we considered complex hyperbolic triangle groups of type $(n,n,\infty;k)$ and proved that for $n \geq 22$ these groups are not discrete. In this paper we show that if $n \geq 14$, then complex hyperbolic triangle groups of type $(n,n,\infty;k)$ are not discrete and give a new list of non-discrete groups of type $(n,n,\infty;k)$.

Notes on complex hyperbolic triangle groups of type (m, n, ∞)

Abstract In this paper we investigate the discreteness of complex hyperbolic triangle groups of type (m, n, ∞). For the special case n = ∞, more explicit conclusions are given.

On the arithmetic dimension of triangle groups

Let $\Delta=\Delta(a,b,c)$ be a hyperbolic triangle group, a Fuchsian group obtained from reflections in the sides of a triangle with angles $\pi/a,\pi/b,\pi/c$ drawn on the hyperbolic plane. We define the arithmetic dimension of $\Delta$ to be the number of split real places of the quaternion algebra generated by $\Delta$ over its (totally real) invariant trace field. Takeuchi has determined explicitly all triples $(a,b,c)$ with arithmetic dimension $1$, corresponding to the arithmetic triangle groups. We show more generally that the number of triples with fixed arithmetic dimension is finite, and we present an efficient algorithm to completely enumerate the list of triples of bounded arithmetic dimension.

A Note on Quaternionic Hyperbolic Ideal Triangle Groups

AbstractIn this paper, the quaternionic hyperbolic ideal triangle groups are parametrized by a real one-parameter family {ϕ: s ∊ ℝ}. The indexing parameter s is the tangent of the quaternionic angular invariant of a triple of points in forming this ideal triangle. We show that if , then ϕs is not a discrete embedding, and if s , then ϕs is a discrete embedding.