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Long relators in groups generated by two parabolic elements

Rotem Yaari

TL;DR

The paper investigates when groups generated by two parabolic elements in $\mathrm{SL}_2(\mathbb{C})$ are free, focusing on $G_{\lambda,\mu}$ with $A_{\lambda}$ and $B_{\mu}$ and the specialization $G_{\lambda}=G_{\lambda,\lambda}$. It introduces syllables and the relator length $\sigma(G)$, proving that as $\lambda\to 2$, relators in $G_{\lambda,\mu}$ must contain long subwords of fixed forms, which forces $\sigma(G_{\lambda,\mu})\to\infty$; this is quantified with effectively computable lower bounds and extended to sequences with $\lambda_n\mu_n\to\ell\in\{\pm4,\pm2i\}$. Two variants of the ping-pong lemma (one for a single group and one for a continuous family) are developed to isolate the obstructions to the classical ping-pong argument and to obtain robust lower bounds on subword lengths, thereby handling non-free cases. The results have implications for the analysis of rational non-freeness and conjectures about parabolic generators, and the discussion outlines extensions to other normalizations, cusp groups, and connections to deformation spaces such as the Riley slice.

Abstract

We find a family of groups generated by a pair of parabolic elements in which every relator must admit a long subword of a specific form. In particular, this collection contains groups in which the number of syllables of any relator is arbitrarily large. This suggests that the existing methods for finding non-free groups with rational parabolic generators may be inadequate in this case, as they depend on the presence of relators with few syllables. Our results rely on two variants of the ping-pong lemma that we develop, applicable to groups that are possibly non-free. These variants aim to isolate the group elements responsible for the failure of the classical ping-pong lemma.

Long relators in groups generated by two parabolic elements

TL;DR

The paper investigates when groups generated by two parabolic elements in are free, focusing on with and and the specialization . It introduces syllables and the relator length , proving that as , relators in must contain long subwords of fixed forms, which forces ; this is quantified with effectively computable lower bounds and extended to sequences with . Two variants of the ping-pong lemma (one for a single group and one for a continuous family) are developed to isolate the obstructions to the classical ping-pong argument and to obtain robust lower bounds on subword lengths, thereby handling non-free cases. The results have implications for the analysis of rational non-freeness and conjectures about parabolic generators, and the discussion outlines extensions to other normalizations, cusp groups, and connections to deformation spaces such as the Riley slice.

Abstract

We find a family of groups generated by a pair of parabolic elements in which every relator must admit a long subword of a specific form. In particular, this collection contains groups in which the number of syllables of any relator is arbitrarily large. This suggests that the existing methods for finding non-free groups with rational parabolic generators may be inadequate in this case, as they depend on the presence of relators with few syllables. Our results rely on two variants of the ping-pong lemma that we develop, applicable to groups that are possibly non-free. These variants aim to isolate the group elements responsible for the failure of the classical ping-pong lemma.
Paper Structure (5 sections, 12 theorems, 37 equations, 2 figures)

This paper contains 5 sections, 12 theorems, 37 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background and main results
  3. The ping-pong lemma
  4. Proofs of the main results
  5. Discussion and questions

Key Result

Theorem 1.2

Let $\mu \in \{2, i\}$. Then for every $N\in \mathbb{N}$ there exists a neighborhood $U\subseteq \mathbb{C}$ of $2$ such that for every $\lambda\in U$, every relator of $G_{\lambda,\mu}$ has a subword $w$ of the form: In particular, $\sigma(G_{\lambda,\mu})\to \infty$ as $\lambda\to 2$. Furthermore, a lower bound on the length of $w$ can be computed effectively. More precisely, let $\lambda_n\to

Figures (2)

  • Figure 1: The angle between the hyperbolic geodesics $[z_\theta, 1]$ and $[z_\theta,-1]$, denoted $\beta$, is greater than $\pi$.
  • Figure 2: The angle between $[z_\theta,\frac{5}{4}]$ and $[z_\theta,1]$, denoted $\alpha$, is greater than $1$.

Theorems & Definitions (31)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Example 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • ...and 21 more