# Modify a Dehn presentation

Suppose you have a Dehn presentation $\langle X \mid R \rangle$ of (say not the free group) a hyperbolic group.

Has there been some work done on changing this presentation, e.g. adding a relation ("well chosen one") or else some Tietze transformations such that the obtained presentation is still a Dehn presentation?

Edit: It should be emphasised that I do not want to change the group in question. Hence I am looking for a way to change the presentation for this particular group but not the property of being a Dehn presentation.

Since a priori there should be many different ways to write down a Dehn presentation for the same group but how to switch between these? Maybe finding some "optimal" one (where optimal is of course quite vague at this stage).

• After a search, I think perhaps that a Dehn presentation means Definition 3.3 here: math.uchicago.edu/~may/REU2013/REUPapers/Hyun.pdf. Can you verify? – Lee Mosher Feb 19 '16 at 13:34
• Yes exactly. It's a theorem that a fintely generated group is hyperbolic iff it admits a finite dehn presentation. – M.U. Feb 19 '16 at 13:54
• Anyway, there are quantitative/effective versions of that theorem, as you might be able to see from the proof in the link I provided. However, I am not aware of any work along the lines of your question. – Lee Mosher Feb 19 '16 at 16:55