Motivation Approach Word Sequences Generalized Cayley graphs for fundamental groups of one-dimensional spaces Hanspeter Fischer (Ball State University, USA) joint work with Andreas Zastrow (University of Gda´ nsk, Poland) Dubrovnik VII – Geometric Topology July 1, 2011 H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Fundamental Groups of 1-Dimensional Spaces Approach Question Word Sequences The Tame Case Fundamental groups of general 1-dimensional Peano continua are notoriously difficult to analyze: l 1 l 2 l 3 Hawaiian Earring Sierpi´ nski carpet Menger curve H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Fundamental Groups of 1-Dimensional Spaces Approach Question Word Sequences The Tame Case Theorems [Eda 2002-2010] Let X and Y be 1-dimensional Peano continua. A. π 1 ( Sierpi´ nski carpet ) / ↪ π 1 ( Hawaiian Earring ) , π 1 ( Menger curve ) ↪ π 1 ( Hawaiian Earring ) . / B. If X and Y are not locally simply-connected at any point and if π 1 ( X ) ≅ π 1 ( Y ) , then X and Y are homeomorphic. C. If π 1 ( X ) ≅ π 1 ( Y ) , then X and Y are homotopy equivalent. H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Fundamental Groups of 1-Dimensional Spaces Approach Question Word Sequences The Tame Case Theorems [Curtis-Fort 1959] A. Suppose X is a 1-dimensional Peano continuum. Then π 1 ( X ) is free ⇔ X is (semi)locally simply-connected ⇔ π 1 ( X ) is finitely presented ⇔ π 1 ( X ) is countable B. Suppose X is a 1-dimensional separable metric space. Then every finitely generated subgroup of π 1 ( X ) is free. C. The homotopy class of every loop in a 1-dimensional separable metric space has an essentially unique shortest representative. H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Fundamental Groups of 1-Dimensional Spaces Approach Question Word Sequences The Tame Case Question [Cannon-Conner 2006] Given a 1-dimensional path-connected compact metric space X , is there a tree-like object that might be considered the topological Cayley graph for π 1 ( X ) ? Solution A combinatorial description of an R -tree (i.e. a uniquely arcwise connected geodesic space), along with a combinatorial description of π 1 ( X ) , which to the extend possible, functions like a Cayley graph for π 1 ( X ) H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Fundamental Groups of 1-Dimensional Spaces Approach Question Word Sequences The Tame Case Functionality of a classical Cayley graph: G = ⟨ a , b ∣ a 5 = e , b 2 = e , ab = ba − 1 ⟩ g g − 1 h = aba − 1 b h g − 1 h = aa g − 1 h = baaab There is a natural distance based on word length: d ( g , h ) = 2 G acts on the Cayley graph by graph automorphism G acts freely and transitively on the vertex set H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Fundamental Groups of 1-Dimensional Spaces Approach Question Word Sequences The Tame Case The tame case: b e a X = 1-dimensional simplicial complex p Collapsing all translates of a maximal tree in the universal covering space yields a Cayley graph for the free fundamental group π 1 ( X ) H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Fundamental Groups of 1-Dimensional Spaces Approach Question Word Sequences The Tame Case Obstacles In general, we are facing the following obstacles: π 1 ( X ) might be uncountable There might not be a universal covering space Collapsing contractible subsets of X might change π 1 ( X ) π 1 ( ) / ≅ π 1 ( ) H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Injection into the Shape Group Approach Generalized Universal Covering Spaces Word Sequences Theorem [Curtis-Fort ‘59, Eda-Kawamura ‘98] Let X be a 1-dimensional separable metric space, or let X be a 1-dimensional compact Hausdorff space. Then the natural homomorphism ϕ ∶ π 1 ( X ) ↪ ˇ π 1 ( X ) is injective. Suppose X = lim � ( X 1 ← X 2 ← X 3 ← ⋯) with finite graphs X n . f 1 f 2 f 3 ← (Example: If X is the boundary of a CAT(0) 2-complex, we can take metric spheres for X n and geodesic retraction for f n .) π 1 ( X ) = lim � ( π 1 ( X 1 ) ← π 1 ( X 2 ) ← π 1 ( X 3 ) ← ⋯) . f 1# f 2# f 3# Then ˇ ← π 1 ( X ) = coherent sequences of reduced words in free groups. ˇ Problem: How do we identify the image of π 1 ( X ) in ˇ π 1 ( X ) ? H. Fischer, A. Zastrow Generalized Cayley graphs
� � Motivation Injection into the Shape Group Approach Generalized Universal Covering Spaces Word Sequences Theorem [F-Zastrow 2007] Suppose X is a path-connected topological space. If the natural homomorphism ϕ ∶ π 1 ( X ) ↪ ˇ π 1 ( X ) is injective, then there is a generalized universal covering p ∶ ̃ X → X , that is, a continuous surjection characterized by the usual lifting criterion: ̃ X = path-conn, (̃ X , ̃ x ) loc path-conn, ⇐ ⇒ f # ( π 1 ( Y , y )) = 1 ∃ ! g simply conn. p Y = path-conn, � ( X , x ) ( Y , y ) ∀ f loc path-conn. π 1 ( X ) ≅ Aut (̃ → X ) acts freely and transitively on p − 1 ( x ) ; p X If X is 1-dimensional separable metric, then ̃ X is an R -tree. (There is no R -tree metric for which π 1 ( X ) acts by isometry.) Problem: How do we combinatorially describe ̃ X ? H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Combinatorial Description Approach Dynamic Word Length Word Sequences Results General Assumption Let X be a 1-dimensional path-connected compact metric space. Express X = lim � ( X 1 ← X 2 ← X 3 ← ⋯) with finite graphs X n . f 1 f 2 f 3 ← Arrange that f n ∶ X n + 1 → X ∗ n maps each edge linearly onto an edge n of X n and fix a base point ( x n ) n ∈ X . of a regular subdivision X ∗ Let W n = { all words v 1 v 2 ⋯ v k over the vertex alphabet of X n which describe paths starting at the base vertex x n } Set of word sequences : W = lim � (W 1 ← � W 2 ← � W 3 ← � ⋯ ) φ 1 φ 2 φ 3 ← where φ n ∶ W n + 1 → W n is the natural combinatorial projection. H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Combinatorial Description Approach Dynamic Word Length Word Sequences Results Example: F H I A B E J K G L N M Q R O P T C U S V W D Y Z ω 2 = E F H I M Q T V U R L J ω 1 = φ 2 ( ω 2 ) = ABCB A B C C B H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Combinatorial Description Approach Dynamic Word Length Word Sequences Results Example: F H I A B E J K G L N M Q R O P T C U S V W D Y Z ω 2 = E F H I M Q T V U R L J H G ω 1 = φ 2 ( ω 2 ) = ABCB/A A B C C B Formally, we allow for words of the form “ v 1 v 2 ⋯ v k / v k + 1 ” in W n , unless this can eventually be avoided. (“0 . 999 ... = 1 . 000 ... ”) H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Combinatorial Description Approach Dynamic Word Length Word Sequences Results Combinatorial Reduction: Given a word ω n , repeatedly apply the following replacements ↝ ... uvu ... ... u ... ... uv / u ↝ ... u / v until this is no longer possible. Denote the resulting word by ω ′ n . Example: ω n = ABEDCDADEA A B C D E H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Combinatorial Description Approach Dynamic Word Length Word Sequences Results Combinatorial Reduction: Given a word ω n , repeatedly apply the following replacements ↝ ... uvu ... ... u ... ... uv / u ↝ ... u / v until this is no longer possible. Denote the resulting word by ω ′ n . Example: ω n = ABEDCDADEA A B ABED ADEA C D E H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Combinatorial Description Approach Dynamic Word Length Word Sequences Results Combinatorial Reduction: Given a word ω n , repeatedly apply the following replacements ↝ ... uvu ... ... u ... ... uv / u ↝ ... u / v until this is no longer possible. Denote the resulting word by ω ′ n . Example: ω n = ABEDCDADEA A B ABED ADEA ABED EA C D E H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Combinatorial Description Approach Dynamic Word Length Word Sequences Results Combinatorial Reduction: Given a word ω n , repeatedly apply the following replacements ↝ ... uvu ... ... u ... ... uv / u ↝ ... u / v until this is no longer possible. Denote the resulting word by ω ′ n . Example: ω n = ABEDCDADEA A B ABED ADEA ABED EA ABE A C D E H. Fischer, A. Zastrow Generalized Cayley graphs
Motivation Combinatorial Description Approach Dynamic Word Length Word Sequences Results Combinatorial Reduction: Given a word ω n , repeatedly apply the following replacements ↝ ... uvu ... ... u ... ... uv / u ↝ ... u / v until this is no longer possible. Denote the resulting word by ω ′ n . Example: ω n = ABEDCDADEA A B ABED ADEA ABED EA ABE A ω ′ n = ABEA C D E H. Fischer, A. Zastrow Generalized Cayley graphs
Recommend
More recommend
