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Combinatorial Group Theory (Pdf) Combinatorial Group Theory George D. Torres Spring 2021 1 Introduction 2 2 Free Products 3 2.1 Free Groups . .3 2.2 Residual Finiteness . .6 2.3 Free Products . .8 2.4 Free Products with Amalgamation and HNN Extensions . 14 These are lecture notes from the Spring 2021 course M392C Combinatorial Group Theory at the University of Texas at Austin taught by Prof. Cameron Gordon. The recommended prerequisites for following these notes are a first course in algebraic topology and an undergraduate level knowledge of group theory. Not strictly re- quired, but beneficial, is a familiarity with low-dimensional topology. Note: the theorem/definition/lemma etc. numbers in these notes don’t necessarily coincide with those in Cameron’s notes. Please email me at [email protected] with any typos and suggestions. Any portions marked TODO are on my to-do list to finish time permitting. Last updated March 4, 2021 1 1. Introduction v (from the syllabus) We will discuss some topics in combinatorial group theory, withan emphasis on topological methods and applications. The topics covered will include:free groups, free products, free products with amalgamation, HNN extensions, Grushko’sTheorem, group presentations, Tietze transformations, Dehn’s decision problems: the word, conjugacy and isomorphism problems, residual finiteness, coherence, 3-manifoldgroups, 1-relator groups, the Andrews-Curtis Conjecture, recursive presentations, the Higman Embedding Theorem, the Adjan- Rabin Theorem, the unsolvability of the homeomorphism problem for manifolds of dimension ≥ 4. The association X π1(X) defines a useful relationship between topology and group theory. For example, covering spaces of a space X correspond to subgroups of π1(X). In the case of low dimensional (≤ 3 dimen- sions) topology, topological spaces allow us to study combinatorial group theory. For example, 1-complexes correspond to free groups and finite 2-complexes correspond to finitely presented groups. For dimension 3, the story is more complicated and more interesting. Here is a brief description of some results in combinatorial group theory that we will discuss in this course. • A closed orientable 3-manifold M is a nontrivial connected sum M = M1#M2 if and only if π1(M) is a nontrivial free product, i.e. π1(M) = A1 ∗ A2. 3 • Given an incompressible surface F ⊂ M , Dehn’s Lemma can show that π1(M) is one of two things. If F is 0 00 0 00 separating (i.e. M n F is not connected), then π1(M) = π1(M ) ∗π1(F ) π1(M ) (where M n F = M t M ). 0 If F is not separating, then π1(M) is what is known as an HNN extension π1(M )∗π1(F ). The converse for this characterization holds as well. 3 3 ∼ • In 1910, Dehn showed (modulo his famous lemma) that a knot K ⊂ S is trivial if and only if π1(S nK) = Z. This led to the study of Dehn’s decision problems. These are group decidability problems like the Word Problem. The Word Problem asks, given an arbitrary finitely generated group G, if there is an algorithm to decide if any word in the generators of G is the trivial element. There is also the Conjugacy Problem and the Isomorphism Problem, all of which have been shown to be undecidable. ∼ • If F1;F2 are closed surfaces, then it is a standard exercise to show that F1 = F2 (homeomorphic) if and only ∼ if H1(F1) = H1(F2). In the case where we have closed orientable 3-manifolds M1;M2; there is an analogous ∼ ∼ 1 result for fundamental groups. Namely, M1 = M2 if and only if π1(M1) = π1(M2) . A generalization of this to higher dimensions exists as well (only for closed manifolds with trivial higher homotopy) and is known as Borel’s Conjecture. • The Homoeomorphism problem for manifolds can be formulated as a group theory problem. This prob- lem asks if you can decide whether or not two closed n-manifolds are homoemorphic. The answer is yes for n ≤ 3 and no for n ≥ 4. Similarly, the sphere recognition problem, which asks if you can determine a closed n-manifold is homeomorphic to Sn, has a positive answer for n ≤ 3 and a negative answer for n ≥ 5. The case of n = 4 is unknown. 1The Poincaré conjecture in dimension 3 is a special case of this result. 2 2. Free Products v Lecture 1/19 Definition 2.1. Let fAλgλ2Λ be a collection of groups. A free product of fAλg is a group P and a homomorphism iλ : Aλ ! P for every λ such that for any group G and maps αλ : Aλ ! G, there exists a unique map φ : P ! G such that the diagram below commutes. αλ Aλ G φ iλ P In other words, it is the coproduct in the category of groups. Standard categorical arguments show that any free product, if it exists, is unique and that the maps iλ are ` injective. Hence, we refer to “the” free product P as λ2Λ Aλ and regard each Aλ as a subgroup of the free product. The notation for a free product of two groups is A ∗ B. 2.1 Free Groups v A special case of this is where Aλ = hxλi for xλ 2 S, i.e. each group is isomorphic to Z. Then the free product ` λ2Λhxλi is denoted F (S) and is called the free group on S. Note that F (S) has a certain universal property. That is, there exists i : S ! F (S) such that for any group G and function f : S ! G, there exists a unique homomorphism φ : F (S) ! G such that the diagram below commutes. f S G φ i F (S) Lecture 1/21 We can prove that free groups actually exist using Van Kampen’s Theorem from topology. Theorem 2.2. Free groups exist. Proof: 1 W 1 We know that π1(S ) = Z. Let S = fxλgλ2Λ and let X = λ2Λ Sλ and set b to be the wedge point. Let 1 Xλ = Sλ [ X0, where X0 is an open neighborhood of b in X. Note that Xλ strong deformation retracts 1 ∼ ∼ ` to Sλ and X0 strong deformation retracts to b. Since π1(Xλ; b) = Z, we have π1(X; b) = hxλi = F (S) by Van Kampen’s theorem. One can show that two sets S and S0 have the same cardinality if and only if F (S) =∼ F (S0). One direction is clear; the other direction can be shown by passing to the topological category, looking at the abelianization of π1 (the first homology) and tensoring with a field. This will show that the two basis sets have the same cardinality. Definition 2.3. If H < G is a subgroup, then a retraction of G onto H is a homomorphism ρ : G ! H such that ρjH = idH . In this case, we say H is a retract of G. 3 2 Free Products Proposition 2.4. Let S1;S2 be disjoint sets. Then ∼ 1. F (S1 [ S2) = F (S1) ∗ F (S2), and 2. F (S1) is a retract of F (S1 [ S2). These both follow from the standard topological notions. The wedge of two boquets of circles is another bouquet of circles, and a retraction of spaces induces a retraction of fundamental groups. 2.1.1 The Nelson-Schreier Theorem Recall the following facts about trees and graphs (seen as cell complexes): • Every tree is contractible. • Every path connected graph Γ has a maximal tree T (one that contains all the vertices of Γ). • The pair (Γ;T ) has the homotopy extension property. In particular, the quotient map Γ ! Γ=T is a homo- W 1 topy equivalence. Moreover Γ=T = S is a wedge of circles. ∼ Therefore π1(Γ) = π1(Γ=T ) is the free group on the set of edges in Γ n T . Theorem 2.5 (Nelson-Schreier). A subgroup of a free group is free. Proof: Let F be a free group and write F = π1(X) for X a wedge of circles. If H < F is a subgroup, then there ~ ~ ∼ ~ exists a connected covering space X ! X such that π1(X) = H. However, X is a connected graph and ~ so therefore π1(X) is free. ∼ Wm 1 If Γ is a finite graph, then Γ = X := i=1 S . Then χ(Γ) = χ(X) = 1 − m. In other words, the Euler characteristic of a finite graph is one minus the rank of π1(Γ). Proposition 2.6. Let F be a free group of rank n and let H < F have finite index k. Then H is a (free) group of rank kn − k + 1. Proof: ~ Write F = π1(X) for X a wedge of n circles. There is a k-sheeted covering space X ! X whose ~ ∼ fundamental group is π1(X) = H. The Euler characteristic is multiplicative on covering spaces, so χ(X~) = kχ(X) = k(1 − n). Therefore Rank(H) = 1 − χ(X~) = kn − k + 1. Let Fn denote the free group of rank n (or on n generators). As a Corollary to the above, any free group of countable rank is a subgroup of F2 (see figure 2.1). Theorem 2.7. Let F = F (fxλ : λ 2 Λg) be a free group. Then: x 2 F fx g Qn xki k 2 n f0g 1. Every can be expressed as a reduced word in λ , i.e. a product of of the form i=1 λi with i Z and λi 6= λi+1. 2. If n ≥ 1, then every nontrivial word x is not 1 2 F . 4 2 Free Products Figure 2.1: A k-covering of a wedge of two circles, whose fundamental group is F2.
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