2. Analytic questions

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 quasisymmetric 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.
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 quasisymmetry group is finite (quasisymmetries are symmetries in this case).


Problem 2.3.
[Haissinsky] Find a criterion that implies a metric space has a quasisymmetric metric that is a PI space. (Is it the combinatorial Loewner property?)
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 quasisymmetry 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?
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?
Remark. This implies the Kapovichâ€“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 3manifolds 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?
Remark. The statement is true for 2manifolds. It can be reformulated: does there exist a quasiisometry 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.