Notes of Matias Carrasco’s lecture

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 {D(G)} of Hausdorf dimensions of limit sets of fundamental groups of Kleinian manifolds homeomorphic to {M_G}.

By definition, {\mathrm{dim}_{AR}(G)\leq D(G)}. Groups with {D(G)=1} have a nice characterization.

Theorem 1 (Canary-Minsky-Taylor) {D(G)=1} if and only if {M} is a generalized book of {I}-bundles, i.e. there exists a collection {A} of essential annuli in {M} such that all closures of connected components of their complement are either

  1. Solid tori.
  2. {I}-bundles.

In particular, there is no rigid vertex in the JSJ decomposition of {G}.

Question. {\mathrm{dim}_{AR}(G)=D(G)} ?

We shall see that the answer is no.

2. Splittings

In general, one understands ideal boundaries of groups which admit quasiconvex splittings: {\partial G} contains one copy of the ideal boundary of each vertex group {\partial G_v}, {v} a vertex of the splitting tree {T}, whose size decays as {v} tends to infinity, and is compactified by adjoining {\partial T}. 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,

\displaystyle  \begin{array}{rcl}  \mathrm{dim}_{AR}(G)=\max\{\mathrm{dim}_{AR}(G_v)\,;\,|G_v|=\infty\}. \end{array}

Theorem 3 (Well spread local cut points condition) Let {G} be one-ended. Assume

\displaystyle  \begin{array}{rcl}  (WS)\quad &&\forall \delta>0,~\exists P_{\delta}\subset\partial G \textrm{ finite subset such that }\\ &&\sup\{\mathrm{diam}(A)\,;\,A\textrm{ connected component of }\partial G\setminus P_{\delta}\}\leq\delta. \end{array}

Then

\displaystyle  \begin{array}{rcl}  \mathrm{dim}_{AR}(G)=1. \end{array}

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 {\partial G} satisfies condition (WS). Therefore {\mathrm{dim}_{AR}(G)=1}.

Example 1 (of a group with conformal dimension 1 and {D(G)>1}) Remove a disk to a 2-torus, take a product with an interval. Then glue to it a solid torus and two {I}-bundles. This satisfies (WS) but it is not a generalized {I}-bundle.

Remark 1 If there exists a rigid vertex groups whose ideal boundary is a circle, then {\partial G} 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 {G} is trivial (no local cut points), then {\mathrm{dim}_{AR}(\partial G)>1}.

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 {G} with {\mathrm{dim}_{AR}(\partial G)=1}. Assume that {G} 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 {R}-ball into sets of diameter {\leq R/2}, for all {R}. Fix a large {N}. Split a ball of radius {1} into concentric annuli of width {2^{-1},\ldots,2^{-N}}. Each of them contains a bounded number of cut points sufficient to disconnect it from the next layers. Consider the family {\Gamma} of curves joining the center to the complement of the ball. Any curve of {\Gamma} must pass through at least {N} such cut-points. So make a conformal change of metric where the weight is {1/N} at cut-points and 0 elsewhere. This will give length {\geq 1} to all curves in {\Gamma}. On the other hand, the {p}-volume is

\displaystyle  \begin{array}{rcl}  \frac{1}{N^p}O(N) \end{array}

which tends to {0} as {N} tends to infinity. This shows that the {p}-modulus of {\Gamma} vanishes for all {p>1}, and thus, that conformal dimension equals 1.

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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