
2. Analytic questions

1. PI spaces

Problem 2.1.

[Bonk] The standard Sierpinski carpet is obtained by removing middle thirds from the unit square. This space is not PI with the standard metric. Is there a metric in the quasi-symmetric guage of the standard metric which makes it Ahlfors $p$-regular and a $p$-PI space?
• Problem 2.2.

[Bonk] What is the conformal dimension $q$ of the standard Sierpinski carpet? It is known $1+\frac{\log 2}{\log 3}\le q<\frac{\log8}{\log3}$ The upper bound above is the Hausdorff dimension (Kigami). For an easier hard problem, prove that $q$ is strictly greater than the lower bound above.
1. Remark. This sort of question is interesting for many other fractals and group boundaries.
• Remark. The standard Sierpinski carpet above is not the boundary of any group because its quasi-symmetry group is finite (quasi-symmetries are symmetries in this case).
• Problem 2.3.

[Haissinsky] Find a criterion that implies a metric space has a quasi-symmetric metric that is a PI space. (Is it the combinatorial Loewner property?)
1. Remark. The standard Sierpinski carpet does have the combinatorial Loewner property.
• Problem 2.4.

Let $G$ be a random hyperbolic group. Is there a metric in the quasi-symmetry gauge of $\partial G$ so that $\partial G$ is a PI space?
• Conformal dimension

Problem 2.5.

[Haissinsky] What can one say about a space $B$ that attains its Ahlfors regular conformal dimension?
1. Remark. If $B=\partial G$ for $G$ hyperbolic, then this condition implies that $\partial G$ is a PI space.
• Problem 2.6.

[Haissinsky] Let $G$ be hyperbolic with $\partial G\subsetneq S^2$. Is its (Ahlfors regular) conformal dimension less than 2?
1. Remark. This implies the Kapovich–Kleiner conjecture (see below).
• Problem 2.7.

If $G=A*_{\mathbb{Z}} B$ what is the relationship between the conformal dimension of $A$, $B$, and $G$?
• Problem 2.8.

[Lafont] Fix $n\ge4$. Does there exist a closed, negatively curved $n$-manifolds $M_i$ such that the conformal dimension of $\partial\widetilde M_i$ goes to infinity?
• Other

Problem 2.9.

[Lafont] Let $M_0,M_1$ be 3-manifolds with curvature $K\le -1$ and some point/plane with curvature $-1$. Suppose the induced metrics on $S^2$ have the same Hausdorff dimension. Must they be bilipschitz?
1. Remark. The statement is true for 2-manifolds. It can be reformulated: does there exist a quasi-isometry between $\widetilde M_0,\widetilde M_1$ with multiplicative constant 1? Jean guesses the answer to the question is No.

Cite this as: AimPL: Boundaries of groups, available at http://aimpl.org/groupbdy.