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).
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.