Complex Hyperbolic Triangle Groups Research Articles

Mirror stabilizers for lattice complex hyperbolic triangle groups

Mirror stabilizers for lattice complex hyperbolic triangle groups

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.

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.

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 orders 3 , 3 , 2 3,3,2 in complex geodesics with pairwise distances m , m , 0 m,m,0 . For fixed  m m , 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.

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 $(n,n,\infty;k)$.

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

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 $(n,n,\infty;k)$.

Arithmeticity of complex hyperbolic triangle groups

Complex hyperbolic triangle groups, originally studied by Mostow in building the rst nonarithmetic lattices in PU(2 ; 1), are a natural generalization of the classical triangle groups. A theorem of Takeuchi states that there are only nitely many Fuchsian triangle groups that determine an arithmetic lattice in PSL2(R), so triangle groups are generically nonarithmetic. We prove similar niteness theorems for complex hyperbolic triangle groups that determine an arithmetic lattice in PU(2; 1).

Non-Discrete Complex Hyperbolic Triangle Groups of Type (n,n, ∞;k)

AbstractA complex hyperbolic triangle group is a group generated by three involutions fixing complex lines in complex hyperbolic space. Our purpose in this paper is to improve a previous result and to discuss discreteness of complex hyperbolic triangle groups of type (n,n, ∞;k).

Census of the Complex Hyperbolic Sporadic Triangle Groups

The goal of this paper is to give a conjectural census of complex hyperbolic sporadic triangle groups. We prove that only finitely many of these sporadic groups are lattices. We also give a conjectural list of all lattices among sporadic groups, and for each group in the list we give a conjectural group presentation, as well as a list of cusps and generators for their stabilizers. We describe strong evidence for these conjectural statements, showing that their validity depends on the solution of reasonably small systems of quadratic inequalities in four variables.

Non-discrete complex hyperbolic triangle groups of type (m , m , ∞)

In this note we prove that a complex hyperbolic triangle group of type (m,m,infinity), i.e. a group of isometries of the complex hyperbolic plane, generated by complex reflections in three complex geodesics meeting at angles Pi/m, Pi/m and 0, is not discrete if the product of the three generators is regular elliptic.

Notes on complex hyperbolic triangle groups

We first demonstrate a family of isomorphisms between complex hyperbolic triangle groups and outline a systematic approach classifying the groups. Then we describe conditions that determine the discreteness of certain groups, in particular we prove a slightly weaker version of a conjecture given by Schwartz. Finally we collect together a list of known discrete triangle groups and propose some good candidates for discrete groups.

Unfaithful complex hyperbolic triangle groups, III: Arithmeticity and commensurability

Unfaithful complex hyperbolic triangle groups, III: Arithmeticity and commensurability

Unfaithful complex hyperbolic triangle groups, II: Higher order reflections

We consider symmetric complex hyperbolic triangle groups generated by three complex reflections through angle 2p / p, with p = 3. We restrict our attention to those groups where certain words are elliptic. Our goal is to find necessary conditions for such a group to be discrete. The main application we have in mind is that such groups are candidates for nonarithmetic lattices in SU(2,1).

Unfaithful complex hyperbolic triangle groups, I: Involutions

A complex hyperbolic triangle group is the group of complex hyperbolic isometries generated by complex involutions fixing three complex lines in complex hyperbolic space. Such a group is called equilateral if there is an isometry of order three that cyclically permutes the three complex lines. We consider equilateral triangle groups for which the product of each pair of involutions and the product of all three involutions are all nonloxodromic. We classify all such groups that are discrete.

On the type of triangle groups

We prove a conjecture of R. Schwartz about the type of some complex hyperbolic triangle groups.

A better proof of the Goldman–Parker conjecture

The Goldman-Parker Conjecture classifies the complex hyperbolic C-reflection ideal triangle groups up to discreteness. We proved the Goldman-Parker Conjecture in [Ann. of Math. 153 (2001) 533--598] using a rigorous computer-assisted proof. In this paper we give a new and improved proof of the Goldman-Parker Conjecture. While the proof relies on the computer for extensive guidance, the proof itself is traditional.