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).

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    $\begingroup$ 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? $\endgroup$ – Lee Mosher Feb 19 '16 at 13:34
  • $\begingroup$ Yes exactly. It's a theorem that a fintely generated group is hyperbolic iff it admits a finite dehn presentation. $\endgroup$ – M.U. Feb 19 '16 at 13:54
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    $\begingroup$ 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. $\endgroup$ – Lee Mosher Feb 19 '16 at 16:55

The answer to your question is positive. The relevant references are:

A.Yu. Olshanski, SQ-universality of hyperbolic groups, Sb. Math. 186 (1995), no. 8, 1199–1211.

T. Delzant, Sous-groupes distingues et quotients des groupes hyperboliques, Duke Math. J. 83 (1996), no. 3, 661–682.

If you cannot access these papers, read the version for relatively hyperbolic groups in

G. Arzhantseva, A. Minasyan, D. Osin, The SQ-universality and residual properties of relatively hyperbolic groups.

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    $\begingroup$ I think the question is not about adding new relations which might change the group, but instead about expanding the presentation without changing the group being presented, by adding to the presentation another already existing relation in the group. That's what "Tietze transformation" seems to suggest. But perhaps the OP can clarify. $\endgroup$ – Lee Mosher Feb 20 '16 at 14:31
  • $\begingroup$ @LeeMosher ... you're absolutely right. I don't want to change the group in question. However I first wanted to go through the paper before commenting on this post. $\endgroup$ – M.U. Feb 20 '16 at 15:44
  • $\begingroup$ In any case I might clarify this in the question. $\endgroup$ – M.U. Feb 20 '16 at 15:44

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