Context-free rewriting systems and word-hyperbolic structures with   uniqueness
  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). 
Alan J. Cain, Victor Maltcev
math.GR