# Maximum relator and hyperbolicity

It is a well-known theorem that a finitely generated group is hyperbolic if and only if it admits a finite Dehn presentation. To prove the "only if" direction one proceeds roughly as follows:

Suppose our group $G$ is hyperbolic with constant $\delta$. Then take the set $R$ to be the set of words of the form $u v^{-1}$, where $u$ varies over all words of length at most $k$, where $k > 8 \delta$ some fixed constant, and $v$ a word of minimal length that is equal to $u$ in $G$. One obtains a Dehn presentation.

In other words in this particular presentation we have that the maximum length of a relator bounds the hyperbolicity constant from above. Clearly if a geodesic metric space is $\delta$-hyperbolic (in the sense that any side of a geodesic triangle is contained in the $\delta$-neighbourhood of the union of the other two sides) then it is $\delta'$-hyperbolic for $\delta' \geq \delta$, i.e. we are interested in a constant which is "as small as possible".

I'm wondering if also some kind of converse of this is true. More precisely suppose we have a Dehn presentation. Is there some relation between the maximum length of a relator of this given presentation, say $m$, and a minimal hyperbolic constant $\delta$?

It would be terrific if something like $\frac{m}{8} \geq \delta$ would hold. However I don't think that something "that good" can be true in general. Nonetheless I'm asking for any "similar" relationship.

My first attempt was to look at the "if"-direction of the above theorem. There one proves (as in Bridson's and Haefliger's book Metric spaces of non-positive curvature) that having a Dehn presentation gives us that the Cayley graph of the group admits a linear isoperimetric bound which is equivalent to being hyperbolic. So I looked at the theorem "linear isoperimetric bound implies hyperbolicity" but the hyperbolic constant derived (in the general setting of a geodesic metric space) gave me no success.

• A useless comment: there is some relation, since there are only finitely many groups of bounded rank with a fixed hyperbolicity constant, and finitely many groups with a maximum length relator. So in principle, one is a function of the other. – Ian Agol Sep 18 '15 at 20:54
• Part of the trouble is that the proofs that linear isoperimetric inequality implies hyperbolic in the literature are either over-complicated or non-constructive. The cleanest proof, due to Drutu, uses the asymptotic cone, so doesn't give a computable constant. – HJRW Sep 19 '15 at 10:46