Context-free rewriting systems and word-hyperbolic structures with uniqueness

TL;DR

This paper demonstrates that monoids presented by confluent context-free monadic rewriting systems are word-hyperbolic and provides an example of a word-hyperbolic monoid lacking a unique representative structure.

Contribution

It establishes a connection between confluent context-free monadic rewriting systems and word-hyperbolicity, and answers an open question about the existence of non-unique word-hyperbolic structures.

Findings

Monoids from confluent context-free monadic rewriting systems are word-hyperbolic

Existence of a word-hyperbolic monoid without a unique structure

Addresses an open question by Duncan & Gilman

Abstract

This paper proves that any monoid presented by a confluent context-free monadic rewriting system is word-hyperbolic. This result then applied to answer a question asked by Duncan & Gilman by exhibiting an example of a word-hyperbolic monoid that does not admit a word-hyperbolic structure with uniqueness (that is, in which the language of representatives maps bijectively onto the monoid).