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Isometries of length $1$ in purely loxodromic free Kleinian groups and trace inequalities

A. Nedim Narman, İlker S. Yüce

Abstract

In this paper, we prove a generalization of a discreteness criteria for a large class of subgroups of PSL$_2(\mathbb{C})$. In particular, we show that for a given finitely generated, purely loxodromic, free Kleinian group $Γ=\langleξ_1,ξ_2,\dots,ξ_n\rangle$ for $n\geq 2$, the inequality $$\left|\text{trace}^2(ξ_i)-4\right|+\left|\text{trace}(ξ_iξ_jξ_i^{-1}ξ_j^{-1})-2\right|\geq 2\sinh^2\left(\frac{1}{4}\logα_n\right)$$ holds for some $ξ_i$ and $ξ_j$ for $i\neq j$ in $Γ$ provided that certain conditions on the hyperbolic displacements given by $ξ_i$, $ξ_j$ and their length $3$ conjugates formed by the generators are satisfied. Above, the constant $α_n$ turns out to be the real root strictly larger than $(2n-1)^2$ of a fourth degree, integer coefficient polynomial obtained by solving a family of optimization problems via Karush-Kuhn-Tucker theory. The use of this theory in the context of hyperbolic geometry is another novelty of this work.

Isometries of length $1$ in purely loxodromic free Kleinian groups and trace inequalities

Abstract

In this paper, we prove a generalization of a discreteness criteria for a large class of subgroups of PSL. In particular, we show that for a given finitely generated, purely loxodromic, free Kleinian group for , the inequality holds for some and for in provided that certain conditions on the hyperbolic displacements given by , and their length conjugates formed by the generators are satisfied. Above, the constant turns out to be the real root strictly larger than of a fourth degree, integer coefficient polynomial obtained by solving a family of optimization problems via Karush-Kuhn-Tucker theory. The use of this theory in the context of hyperbolic geometry is another novelty of this work.
Paper Structure (6 sections, 20 theorems, 77 equations, 6 tables)

This paper contains 6 sections, 20 theorems, 77 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Basic Definitions and Notations
  3. The Culler-Shalen Machinery
  4. The Max Functions and Classification of Relations
  5. Optimizing the Infimum of the Max Function
  6. Proof of the Main Theorem

Key Result

Theorem 1.1

Let $\Gamma=\langle\xi_1,\xi_2\rangle$ be a purely loxodromic free Kleinian group. If the following inequalitıes hold then, we have $|\textnormal{trace}^2(\xi_1)-4|+|\textnormal{trace}(\xi_1\xi_2\xi_1^{-1}\xi_2^{-1})-2| \geq 1.5937....$

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • Lemma 3.1
  • Proposition 3.1
  • proof
  • Lemma 5.1
  • proof
  • ...and 24 more