On the fundamental group of the Sierpiński-gasket

Akiyama, Shigeki; Dorfer, Gerhard; Thuswaldner, Jörg Μ.; Winkler, R.

doi:10.1016/j.topol.2009.01.012

Cited by 20 publications

(37 citation statements)

References 28 publications

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“…Especially the description of such fundamental groups in terms of words turned out to be useful. Cannon and Conner gave such a description for the fundamental group of the Hawaiian Earring (see Figure 1 left side) and in Akiyama et al [1] we gave a representation of the fundamental group π(△) of the Sierpiński gasket △ (see Figure 1 right side) in terms of words. Since △ is a one-dimensional subset of R 2 it is known from Eda and Kawamura [12] that π(△) can be embedded in theČech homotopy groupπ(△) which is known to be a projective limit of free groups.…”

Section: Introductionmentioning

confidence: 92%

“…Throughout this paper let X be a metrizable one-dimensional continuum 1 . Then (see Hurewicz and Wallman [16] or Cannon and Conner [3]) X can be embedded in the three dimensional Euclidean space and represented as the intersection of handle bodies H n , n ∈ N, such that…”

Section: Definition Of the Handlesmentioning

confidence: 99%

“…Meanwhile, properties of fundamental groups of one-dimensional (cf. for instance [1,3,10,11]) and planar (see [5,13]) spaces were derived. Especially the description of such fundamental groups in terms of words turned out to be useful.…”

Section: Introductionmentioning

confidence: 99%

“…Since △ is a one-dimensional subset of R 2 it is known from Eda and Kawamura [12] that π(△) can be embedded in theČech homotopy groupπ(△) which is known to be a projective limit of free groups. In [1] we were able to endow the projective limit definingπ(△) with a word structure. Moreover, we could characterize the elements of the subgroup π(△) by a simple stabilizing condition.…”

Section: Introductionmentioning

confidence: 99%

“…While in the construction for the Sierpiński gasket △ the letters correspond to (local) cut points of △, in our setting letters represent (local) cut sets. This generalization turns out appropriate to extend the approach in [1] for the special case of the Sierpiński gasket to the class of all metrizable one-dimensional continua.…”

Section: Introductionmentioning

confidence: 99%

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Fundamental groups of one-dimensional spaces

2013

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Abstract. Let X be a metrizable one-dimensional continuum. In the present paper we describe the fundamental group of X as a subgroup of itsČech homotopy group. In particular, the elements of theČech homotopy group are represented by sequences of words. Among these sequences the elements of the fundamental group are characterized by a simple stabilization condition. This description of the fundamental group is used to give a new algebro-combinatorial proof of a result due to Eda on continuity properties of homomorphisms from the fundamental group of the Hawaiian earring to that of X.

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Section: Introductionmentioning

confidence: 92%

Section: Definition Of the Handlesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fundamental groups of one-dimensional spaces

2013

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Ends of iterated function systems

Conner

Hojka

2014

Math. Z.

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A. Guided by classical concepts, we de ne the notion of ends of an iterated function system and prove that the number of ends is an upper bound for the number of nondegenerate components of its attractor. e remaining isolated points are then linked to idempotent maps. A commutative diagram illustrates the natural relationships between the innite walks in a semigroup and components of an attractor in more detail. We show in particular that, if an iterated function system is one-ended, the associated attractor is connected, and ask whether every connected attractor (fractal) conversely admits a one-ended system.. I e main aim of the current article is to o er a conceptually new treatment to the study of a general iterated function system (IFS) in which one has no a priori information concerning the attractor. We describe how purely algebraic properties of iterated function systems can anticipate the topological structure of the corresponding attractor. Our method convolves four natural viewpoints: algebraic, geometric, asymptotic, and topological. Algebraically, the semigroup of an IFS, which consists of all countably many compositions of the functions in the system (most notably the idempotent elements), carries a surprising amount of information about the attractor. e arguments are basically geometric in nature. In nite walks, which reside in the intersection of algebra and geometry, play an important role. e asymptotic point of view is embodied by the classical concept of ends. Topologically, we relate components of the attractor and especially the isolated points (the degenerate components) to the algebraic and asymptotic properties of the system. Previous literature has described the attractor of an IFS by detailed analysis of the involved maps. For example [Hat ], [KL ], and more recently, [AT ], or [LT ] require a minute understanding of the action of the IFS on the attractor. In consonance with related elds including geometric group theory and dynamical systems our ambition is to investigate the attractor by examining the relations in the system only on a formal, algebraic level.Accordingly, we obtain an asymptotic decomposition of the Cayley graph in eorem . is decomposition yields a bound on the number of components of the attractor by the number of ends of the IFS, and identi es isolated

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