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.


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 (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.

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    $\begingroup$ Just in case the links becomes obsolete: the first link points towards a survey by Ilya Kapovich and Nadia Benakli (Boundaries of hyperbolic groups. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), 296, 39-93, 2002.) The second link points towards S. Buyalo and N. Lebedeva, St. Petersburg Math. J. 19 (2008), 45-65, whose abstract includes the result that for a Gromov-hyperbolic group we have $asdim(G)=topdim(\partial G)+1$. $\endgroup$ – YCor Feb 2 '15 at 19:56
  • $\begingroup$ Here's the link for the survey: dx.doi.org/10.1090/conm/296 $\endgroup$ – Ian Agol Feb 2 '15 at 21:39
  • $\begingroup$ Great thanks. It seems unlikely that the asymptotic dimension should behave any differently in the case where torsion is allowed. $\endgroup$ – DavidHume Feb 3 '15 at 9:10
  • $\begingroup$ @DavidHume: Ian's answer covers the torsion case. $\endgroup$ – YCor Feb 3 '15 at 9:22
  • $\begingroup$ @DavidHume: $C'(\frac16)$ groups are virtually torsion-free, by a result of Wise that they are cubulated, and a result of mine that cubulated hyperbolic groups are residually finite, hence virtually torsion-free. This is most certainly overkill - I suspect that the Bestvina-Mess result shows directly that $C'(1/6)$ groups have 1-dimensional boundary, and that torsion is not an issue. But I haven't done a literature search to see if this appears somewhere. It might just follow directly from the fact that there is an aspherical 2-complex on which the group acts. $\endgroup$ – Ian Agol Feb 3 '15 at 19:47

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