JultikaUniversity of Oulu repository

Abstract

Conformal dimension is an important quasisymmetric invariant. It is used to measure how much the Hausdorff dimension of a space can be reduced by a quasisymmetric mapping. Thus, it provides information on the structure of the collection of metric spaces which are quasisymmetrically equivalent to some given space.

Studying conformal dimension is crucial for solving many important open questions such as Cannon’s conjecture. Moreover, although conformal dimension was first introduced only in 1989, it has already proven to be a powerful tool, for instance, in geometric function theory, geometric group theory, analysis on metric spaces and conformal dynamics.

In this thesis we will survey some fundamental ideas, results and techniques in the theory of conformal dimension. In particular, we will focus on results for obtaining lower bounds for conformal dimension. Main heuristic for this kind of research is to show the existence of a sufficiently rich curve family on the metric measure space, which then implies lower bounds for conformal dimension. This heuristic is made precise by the results discussed in chapter 2. Additionally, we will briefly discuss some relations between tangent spaces and conformal dimension.

In chapter 3 we will apply the obtained results to study the Sierpinski carpet. Particularly, we will obtain lower and upper bounds for the conformal dimension of the Sierpinski carpet. The exact value of the dimension is an open problem. Furthermore, we will apply our results to show that the Sierpinski carpet is not minimal for conformal dimension.