Fractafolds based on the Sierpinski gasket and their spectra

We introduce the notion of ``fractafold'', which is to a fractal what a manifold is to a Euclidean half-space. We specialize to the case when the fractal is the Sierpinski gasket SG. We show that each such compact fractafold can be given by a cellular construction based on a finite cell graph $G$, which is $3$-regular in the case that the fractafold has no boundary. We show explicitly how to obtain the spectrum of the fractafold from the spectrum of the graph, using the spectral decimation method of Fukushima and Shima. This enables us to obtain isospectral pairs of nonhomeomorphic fractafolds. We also show that although SG is topologically rigid, there are fractafolds based on SG that are not topologically rigid.