Gerhard Dorfer A Digital Description of the Fundamental Group of - PDF document

Gerhard Dorfer A Digital Description of the Fundamental Group of Fractals I (joint work with S. Akiyama, J. Thuswaldner and R. Winkler) Project: Metric and Topological Aspects of Number Theoretical Problems Principal Investigator: Reinhard Winkler Analytic Combinatorics and Probabilistic Number Theory National Research Network of the Austrian Science Foundation FWF

Some references [1] J. Cannon and G. Conner. The combinatorial struc- ture of the Hawaiian earring group . Topology Ap- pl. 106 (2000), 225–271 [2] J. Cannon and G. Conner. The big fundamen- tal group, big Hawaiian earrings, and the big free groups . Topology Appl. 106 (2000), 273–291 [3] J. Cannon and G. Conner. On the fundamental group of one dimensional spases . Preprint [4] G. Conner and K. Eda. Fundamental groups ha- ving the whole information of spaces . Topology Appl. 146/147 (2005), 317–328 [5] K. Eda and K. Kawamura. The fundamental groups of one dimensional spaces . Topology Appl. 87 (1998), 163–172 [6] J. Luo and J. Thuswaldner. On the fundamental group of self affine plane tiles . Ann. Inst. Fourier (Grenoble), to appear

Digital representation of the Sierpi´ nski gasket △ by sequences with digits { 0 , 1 , 2 } dyadic points: belong to 2 subtriangles in △ n , the smallest such n is the order of the dyadic point dyadic points P � = (0) , (1) , (2) have 2 repre- sentations as sequences in { 0 , 1 , 2 } N e.g. P = (0 , 1 , 2 , 2 , . . . ) = (0 , 2 , 1 , 1 , . . . ) =: (0 , 1 | 2) dyadic points correspond to sequences which are eventually constant D n : dyadic points of order ≤ n generic points have a unique representation

Symbolic representation of loops in △ ω : [0 , 1] → △ , ω (0) = ω (1) = (0) fixed approximation level n : { ω − 1 ( P ) | P ∈ D n } is a finite family of disjoint closed set ⊆ [0 , 1] → separated family of sets ω �→ σ n ( ω ): contains the (finite!) sequence of dyadic points of order ≤ n that ω “passes” σ n ( ω ) is a finite word over the alphabet D n

Frame for ( σ n ( ω )) n ∈ N S n : the set of all “admissible” words ω n over the alphabet D n , i.e. 1. ω n starts and ends with (0) 2. consecutive letters in ω n are neighboring dyadic points in △ n ( S n , · ): semigroup where · is concatenation of words and one intermediate (0) is cancelled γ n : S n → S n − 1 : γ n deletes all points of order n and cancels out repetitions of points γ n is a semigroup homomorphism ↓ lim − S n inverse limit of semigroups S n ← Proposition. Let ω : [0 , 1] → △ be a loop in △ . Then ( σ n ( ω )) n ∈ N ∈ lim − S n . ←

Reduction process reflecting homotopy reduced words in S n : do not contain subwords of the form PQP , or PQR , where P, Q, R be- long to the same subtriangle of △ n G n : the set of all reduced words over the alphabet D n Red n : S n → G n : reduces subwords � and PQP → P, ( P, Q, R in the same subtriangle) PQR → PR until word is reduced • Red n well defined • Red n ( ω n ) canonical representative of the homotopy class of the elementary path cor- resp. to ω n in △ n

multiplication ∗ in G n : ω n ∗ ¯ ω n := Red n ( ω n · ¯ ω n ) Proposition. ( G n , ∗ ) is isomorphic to the fun- damental group of △ n . � G n → G n − 1 is a group δ n : ω n �→ Red n − 1 ( γ n ( ω n )) homomorphism ↓ lim − G n inverse limit of groups ← Proposition. The ˇ Cech homotopy group of △ is isomorphic to lim − G n . ← the following diagram commutes: γ n S n − → S n − 1 ↓ Red n Red n − 1 ↓ δ n G n − → G n − 1

σ S ( △ ) − → lim − S n ← ↓ Red ↓ [ . ] ϕ π ( △ ) − → lim − G n ← ( S ( △ ) , · ): groupoid of loops in △ with concatenation · [ ω ] homotopy class of ω σ ( ω ) := ( σ n ( ω )) n ∈ N Red(( ω n ) n ∈ N ) := (Red n ( ω n )) n ∈ N ϕ ([ ω ]) := (Red n ( σ n ( ω ))) n ∈ N • ϕ is injective (Eda/Kawamura 1998), i.e. π ( △ ) is a subgroup of lim − G n ←

• ϕ is not surjective: ω 1 = (0) Example 1. ω 2 = C 0 C 1 C − 1 0 ω 3 = C 0 C 1 C − 1 0 C 2 ω 4 = C 0 C 1 C − 1 0 C 2 C 0 C 3 C − 1 0 . . . ( ω n ) n ∈ N ∈ lim − G n , but ( ω n ) n ∈ N / ∈ range( ϕ ): ← a loop ω in △ with ϕ ([ ω ]) = ( ω n ) n ∈ N has to pass the cycle C 0 infinitely often • range( ϕ )= range( ϕ ◦ [ . ]) = range(Red ◦ σ )

• σ is not surjective: Example 2. ω 1 = (0)(0 | 1)(0) ω 2 = (0)(0 , 0 | 1)(0 | 1)(1 , 0 | 1)(0 | 1)(0 , 0 | 1)(0) ω 3 = (0)(0 , 0 , 0 | 1)(0 , 0 | 1) . . . (1 , 1 , 0 | 1) . . . (0) . . . • graph associated to ( ω n ) n ∈ N ∈ lim − S n : ← – every branch corresponds to a dyadic point – there is total order on the branches – this order is dense – every Dedekind cut in the set of branches converges to a point in the Sierpi´ nski gasket

Range and kernel of σ Theorem. ( ω n ) n ∈ N ∈ lim − S n is in the range of ← σ if and only if every irrational Dedekind cut in the set of branches of the graph associated to ( ω n ) n ∈ N converges to a generic point in △ . Theorem. For ω and ¯ ω in S ( △ ) we have σ ( ω ) = σ (¯ ω ) if and only if ω and ¯ ω have a common re-parametrization, i.e. there exist α, β : [0 , 1] → [0 , 1] monotonously increasing and surjective such that ω ◦ α = ¯ ω ◦ β .

Main Theorem. An element ( ω n ) n ≥ 0 of lim − G n ← is in ϕ ( π ( △ )) if and only if for all k ≥ 0 the sequence ( γ nk ( ω n )) n ≥ k stabilizes, where γ nk = γ k +1 ◦ . . . ◦ γ n .