Do there exist finitely presented $C'(1/6)$ small cancellation groups with arbitrarily high asymptotic dimension?


To offer a little more motivation, Roe proves that all hyperbolic groups have finite asymptotic dimension, a result that was particularly interesting at the time as it ensured that such groups satisfy the Novikov conjecture due to the work of Yu.


Small cancellation groups are a rich and interesting subclass of hyperbolic groups - surface groups (genus $\geq 2$) and random groups in the few relator or low density models are almost surely examples. Calculations and computations are often easier for these groups - for instance they have cohomological dimension at most 2. Unfortunately I cannot find an argument which says either that such groups have uniformly bounded asymptotic dimension, or proving that this is unbounded.



Asymptotic dimension of $C'(1/6)$ small cancellation groups
The asymptotic dimension is bounded by 2. I don't know the original proof of this, but I found some references on this. Torsion-free $C'(\frac16)$ small-cancellation groups have [cohomological dimension 2](http://www.math.uiuc.edu/~kapovich/PAPERS/bry1.pdf) (see Theorem 6.5 (5) due to Bestvina and Mess and the following discussion of $C'(\frac16)$ groups, and any such group is virtually torsion-free), so their boundaries have dimension 1. Thus, they have [asymptotic dimension bounded by 2.](http://dx.doi.org/10.1090/S1061-0022-07-00985-5#sthash.vNHRF186.dpuf)


