On the fundamental group of the Sierpiński-gasket
type: Item
instance of: scholarly article
title: On the fundamental group of the Sierpiński-gasket
zbMATH Open document ID: 1182.57001
author: Shigeki Akiyama
author: Gerhard Dorfer
author: Jörg M. Thuswaldner
author: Reinhard Winkler
published in: Topology and its Applications
publication date: 2009-05-28T00:00:00Z
review text: 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.
reviewed by: James W. Cannon
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 14F35
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 54F50
Mathematics Subject Classification ID: 55Q99
zbMATH DE Number: 5558956
zbMATH Keywords: Sierpiński-gasket
zbMATH Keywords: fundamental group
zbMATH Keywords: fractal
MaRDI profile type: Publication
full work available at URL: j.topol.2009.01.012
OpenAlex ID: W2095253972
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DOI: 10.1016/J.TOPOL.2009.01.012
description: scientific article; zbMATH DE number 5558956