Long relators in groups generated by two parabolic elements

TL;DR

This paper introduces a family of groups generated by two parabolic elements, revealing long relators with specific subwords, and develops new ping-pong lemma variants to analyze their structure.

Contribution

The authors identify long relators in groups generated by parabolic elements and develop novel ping-pong lemma variants for non-free groups, advancing understanding of their algebraic properties.

Findings

Groups contain relators with arbitrarily many syllables

Existing methods may be insufficient for these groups

New ping-pong lemma variants effectively analyze group structure

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.