Conformal dimension and canonical splittings of hyperbolic groups
Can one characterize hyperbolic groups whose conformal dimension equals one ? For the definition of conformal dimension, see Haissinsky’s talk.
1. The Kleinian dimension
For Kleinian groups, an other dimension-like invariant arises, the infimum of Hausdorf dimensions of limit sets of fundamental groups of Kleinian manifolds homeomorphic to .
By definition, . Groups with have a nice characterization.
Theorem 1 (Canary-Minsky-Taylor) if and only if is a generalized book of -bundles, i.e. there exists a collection of essential annuli in such that all closures of connected components of their complement are either
- Solid tori.
In particular, there is no rigid vertex in the JSJ decomposition of .
We shall see that the answer is no.
In general, one understands ideal boundaries of groups which admit quasiconvex splittings: contains one copy of the ideal boundary of each vertex group , a vertex of the splitting tree , whose size decays as tends to infinity, and is compactified by adjoining . We shall first split along finite groups, and then along virtually cyclic groups.
Theorem 2 (Stability under Dunwoody-Stalling splitting) Recall that a Dunwoody-Stalling splitting is a splitting over finite groups with one-ended vertex groups. In this case,
Theorem 3 (Well spread local cut points condition) Let be one-ended. Assume
Recall Bowditch’s JSJ decomposition for hyperbolic groups (see Haissinsky’s talk).
Corollary 4 If all vertex groups in the JSJ decomposition are virtually free, then satisfies condition (WS). Therefore .
Example 1 (of a group with conformal dimension 1 and ) Remove a disk to a 2-torus, take a product with an interval. Then glue to it a solid torus and two -bundles. This satisfies (WS) but it is not a generalized -bundle.
Remark 1 If there exists a rigid vertex groups whose ideal boundary is a circle, then does not satisfy (WS).
It would desirable to fully understand the following family of examples.
Example 2 Glue two surfaces along filling geodesics.
3. A global picture
Above results should be compared to
Theorem 5 (McKay) If the JSJ decomposition of is trivial (no local cut points), then .
McKay’s theorem rules out Sierpinsky curves or Menger curves from the list of possible 1-conformal dimensional ideal boundaries of groups.
Start from a group with . Assume that has no 2-torsion. DS split it, then JSJ split vertex groups, then DS vertex groups again… In the end, only Fuchsian cocompact groups and finite groups may arise. This is why Example 2 is so important.
4. Proof of Theorem 3
The argument is inspired by J. Tyson and J.-M. Wu’s treatment of the Sierpinski gasket.
Thanks to (WS), and by self-similarity, there is a uniform upper bound on the number of points needed to disconnect an -ball into sets of diameter , for all . Fix a large . Split a ball of radius into concentric annuli of width . Each of them contains a bounded number of cut points sufficient to disconnect it from the next layers. Consider the family of curves joining the center to the complement of the ball. Any curve of must pass through at least such cut-points. So make a conformal change of metric where the weight is at cut-points and 0 elsewhere. This will give length to all curves in . On the other hand, the -volume is
which tends to as tends to infinity. This shows that the -modulus of vanishes for all , and thus, that conformal dimension equals 1.