On the fundamental group of the Sierpiński-gasket
The authors give a precise combinatorial description of the fundamental group of the Sierpiński-gasket. The Sierpiński-gasket is formed from the equilateral triangle \(T\) by subdividing \(T\) into four smaller equilateral triangles in the natural way, removing the interior of the innermost triangle, then infinitely iterating the process with the remaining triangles. The boundary of each finite stage of this infinite construction is a finite graph with each vertex being a local cut point of the graph. The authors approximate the elements of the fundamental group \(\pi_1\) by formal words in these cut points, adjacent letters representing adjacent cut points. A sequence of words that represents an element of \(\pi_1\) is characterized as a sequence in which unbounded oscillation between any two of these cut points is forbidden. The authors describe the group operation in terms of their word sequences.
scientific article; zbMATH DE number 5558956
scientific article