[Paper Review] The Geometry and Arithmetic of Kleinian Groups

Colin M. MacLachlan|arXiv (Cornell University)|Nov 11, 2013

Geometric and Algebraic Topology77 references3 citations

TL;DR

This paper investigates the geometry and arithmetic of Kleinian groups—discrete isometry groups of hyperbolic 3-space—focusing on two-generator groups and arithmetic generalised triangle groups. It provides detailed analysis of parameter spaces, discreteness conditions, and covolume minimisation, culminating in classification results for arithmetic lattices and bounds on the Margulis constant, with key contributions to Siegel’s minimal covolume problem and Jørgensen’s inequality in 3D hyperbolic geometry.

ABSTRACT

In this article we survey and describe various aspects of the geometry and arithmetic of Kleinian groups - discrete nonelementary groups of isometries of hyperbolic $3$-space. In particular we make a detailed study of two-generator groups and discuss the classification of the arithmetic generalised triangle groups (and their near relatives). This work is mainly based around my collaborations over the last two decades with Fred Gehring and Colin Maclachlan, both of whom passed away in 2012. There are many others involved as well. Over the last few decades the theory of Kleinian groups has flourished because of its intimate connections with low dimensional topology and geometry. We give little of the general theory and its connections with $3$-manifold theory here, but focus on two main problems: Siegel's problem of identifying the minimal covolume hyperbolic lattice and the Margulis constant problem. These are both "universal constraints" on Kleinian groups -- a feature of discrete isometry groups in negative curvature and include results such as Jørgensen's inequality, the higher dimensional version of Hurwitz's $84g-84$ theorem and a number of other things. We will see that big part of the work necessary to obtain these results is in getting concrete descriptions of various analytic spaces of two-generator Kleinian groups, somewhat akin to the Riley slice.

Motivation & Objective

• To classify arithmetic generalised triangle groups and their near relatives in hyperbolic 3-space, particularly those with high-order torsion.
• To address Siegel’s problem of identifying the minimal covolume hyperbolic lattice in 3-dimensional hyperbolic space.
• To investigate the Margulis constant for discrete Kleinian groups, especially in the context of two-generator and triangle-generated groups.
• To provide concrete analytic descriptions of parameter spaces for two-generator Kleinian groups, analogous to the Riley slice.
• To establish universal geometric constraints on discrete groups in negative curvature, including Jørgensen’s inequality and higher-dimensional analogues of Hurwitz’s 84(g−1) theorem.

Proposed method

• Uses complex parameter spaces to describe two-generator Kleinian groups, particularly projecting to parameters (γ, β, −4) to analyze their geometry.
• Applies trace identities and inequalities—especially Jørgensen’s inequality—modified for higher-order torsion and non-elementary groups.
• Employs geometric decomposition techniques, including thick-thin decompositions and collaring theorems, to study limit sets and group discreteness.
• Analyzes reflection groups and dihedral angles, distinguishing cases where angles are submultiples of π versus not, to classify discrete groups.
• Leverages arithmetic conditions on traces and holonomy to identify arithmetic Kleinian groups, particularly those generated by two parabolics.
• Uses computational enumeration and tabulation of tetrahedral reflection groups with specific dihedral angles to bound covolumes and identify extremal cases.

Experimental results

Research questions

• RQ1 What is the minimal possible covolume of a co-compact hyperbolic lattice in 3-dimensional hyperbolic space, and which group achieves it?
• RQ2 What are the necessary and sufficient conditions for a two-generator Kleinian group to be discrete, particularly in terms of trace parameters and geometric constraints?
• RQ3 How does the Margulis constant behave for groups generated by admissible triangles with non-submultiple dihedral angles in hyperbolic 3-space?
• RQ4 Which arithmetic generalised triangle groups with p, q ≥ 6 are discrete, and what are their covolumes and geometric invariants?
• RQ5 What is the complete classification of arithmetic Kleinian groups generated by two parabolic isometries, and what degree bounds apply to their trace fields?

Key findings

• The paper identifies the (2,3,7) triangle group as the unique minimal coarea lattice in the hyperbolic plane, confirming Siegel’s conjecture in 2D and providing a foundation for 3D analogues.
• For 3-dimensional hyperbolic lattices, the minimal covolume is bounded below by positive values, with explicit candidates such as the (2,3,7) commutator plane groups and tetrahedral reflection groups.
• The Margulis constant for groups generated by an admissible triangle is bounded, with explicit values computed for (2,2,n) and (p,q,r) configurations, particularly for n ≥ 7.
• A complete list of 32 classical tetrahedral reflection groups with dihedral angles submultiples of π is tabulated, with covolumes ranging from 0.035885 to 1.014941.
• Arithmetic generalised triangle groups with p, q ≥ 6 are classified via trace identities and holonomy conditions, with degree bounds and possible γ values explicitly listed.
• The paper provides a detailed classification of arithmetic Kleinian groups generated by two parabolics, including a list of possible γ values and associated orbifold invariants.