Gerhard Dorfer: On the fundamental group of the Sierpiński gasket

This is a joint work with Shigeki Akiyama, Jörg Thuswaldner and Reinhard Winkler.

We give a description of the fundamental group $ \pi(\bigtriangleup)$ of the Sierpiński gasket $ \bigtriangleup$ as a subgroup of the inverse limit $ \lim\limits_{\longleftarrow} G_n$, where $ G_n$ is the fundamental group of the $ n$-th approximation to $ \bigtriangleup$ that emerges from a triangle by gradually removing open middle triangles. For this purpose we assign to a loop $ \omega$ in $ \bigtriangleup$ at each approximation level $ n$ a word $ \omega_n$ consisting of the (finite) sequence of ``transition points'' of order $ \le n$ which the loop $ \omega$ passes in a sense. This way homotopy of loops can be handled by an appropriate reduction process on the corresponding words, and a topological condition characterizes those elements in the inverse limit that effectively represent a loop.

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