Abstract: I will discuss recent results on rigidity for quasisymmetric maps of Sierpinski carpets, a joint work with Mario Bonk. Quasisymmetric maps are global analogues of quasiconformal maps in the setting of arbitrary metric spaces. We prove, in particular, that every quasisymmetric self-map of the Sierpinski carpet fractal is an isometry, i.e. a rotation or a reflection.
These rigidity questions originate from the study of the boundary at infinity of Gromov hyperbolic groups. One consequense of the above result is that the Sierpinski carpet fractal is not quasisymmetric to the boundary at infinity of such a group.
The main tool that was used, and that I will describe in my talk, is a new invariant for quasisymmetric maps of Sierpinski carpets, a modulus of a curve family with respect to a carpet.