Low Dimensional Topology

December 5, 2007

Conformal dimensions and Cantor sets

Filed under: Metric geometry — Jesse Johnson @ 7:08 pm

I don’t know if metric geometry qualifies as low dimensional topology, but today’s subject happens to be low dimensional (usually somewhere between one and two) and in some sense has more to do with topology than geometry. I don’t normally look at the posts over in the .MG section of the arXiv, but I went seeking it out after hearing about it from the author (a grad. student here at Yale). Since I’m new to metric geometry, there’s a reasonable chance some of the details of my description will be off, but here goes:

Given a metric space, one can define its conformal dimension by considering all metric spaces that are quasi-conformal to the original space and taking the infimum of their Hausdorff dimensions. This is useful, for example, when looking at limit sets of discrete groups of hyperbolic isometries, since the metric on such a set is only defined up to conformal maps of the boundary sphere. As one might guess, the conformal dimension is rather hard to calculate. It also seems to have more to do with the topology of the set than its geometry.

John Mackay’s recent preprint [1] demonstrates that the conformal dimension will be strictly greater than one whenever two (essentially topological) conditions are satisfied: First, the space must be N-doubling, for some N, which means that every metric ball in the set is covered by N metric balls of half the original radius. Second, it must be annulus linearly connected, which is a metric analogue to being locally connected with no local cut points. (A cut point is a point such that in a small neighborhood, removing the point makes the neighborhood disconnected.) For the case when the set is the limit set of discrete group of hyperbolic isometries, the local cut point appears to correspond to the group virtually splitting over an elementary group. As a corollary of the main theorem, the author proves that if such a group does not virtually split over an elementary group then its limit set has conformal dimension strictly greater than one.

The proof begins with a result of P. Tukia that allows one to straighten an arc in a doubling metric space. The “straightened” arc has the property that the diameter of every small sub arc is not too much larger than the distance between its endpoints. Because the set has no local cut points, one can then find a second arc that follows parallel to the first one, then straighten it using the theorem. Then one can find two more parallel arcs, then four, etc. These sets limit to a set that looks like a Cantor set cross an interval.  Such a set is known to have conformal dimension bounded away from one and is embedded in the original set, so the conformal dimension of the original set is also bounded below.

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