Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Quasi-isometry Limit groups Free actions Lecture 4, Geometric and asymptotic group theory Olga Kharlampovich NYC, Sep 16 1 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions The universal property of free groups allows one to describe arbitrary groups in terms of generators and relators. Let G be a group with a generating set S. By the universal property of free groups there exists a homomorphism ': F (S) ! G such that '(s) = s for s 2 S. It follows that ' is onto, so by the first isomorphism theorem G ' F (S)=ker('): In this event ker(') is viewed as the set of relators of G, and a group word w 2 ker(') is called a relator of G in generators S. If a subset R ⊂ ker(') generates ker(') as a normal subgroup of F (S) then it is termed a set of defining relations of G relative to S. 2 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions The pair hS j Ri is called a presentation of G, it determines G uniquely up to isomorphism. The presentation hS j Ri is finite if both sets S and R are finite. A group is finitely presented if it has at least one finite presentation. Presentations provide a universal method to describe groups. Example of finite presentations 1 G = hs1;:::; sn j [si ; sj ]; 81 ≤ i < j ≤ ni is the free abelian group of rank n. n 2 Cn = hs j s = 1i is the cyclic group of order n. 3 Both presentations ha; b j ba2b−1a−3i and ha; b j ba2b−1a−3; [bab−1; a3]i define the Baumslag-Solitar group BS(2; 3) (HWP7 Prove that these presentations define isomorphic groups). 3 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions If a group G is defined by a presentation, then one can try to find homomorphisms from G into other groups. Lemma Let G = hS j Ri be a group defined by a (finite) presentation with the set of relators R = fr = y (j) ::: y (j) j y (j) 2 S±1; 1 ≤ j ≤ mg, j i1 ij i and let H be an arbitrary group. A map : S±1 ! H extends to a homomorphism ~: G ! H, if and only if (r ) = (y (j)) : : : (y (j)) = 1 in H for all r 2 R. j i1 ij j Proof Define the map ~: G ! H by ~ (yn1 ::: ynt ) = (yn1 ) : : : (ynt ); ±1 ~ whenever yni 2 S . If is a homomorphism, then obviously ~(rj ) = 1 for all rj 2 R. 4 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions The converse follows from Lemma (Mapping property of quotient groups) Let N be a normal subgroup of G, let G = G=N, and let π : G ! G be the canonical map, π(g) =g ¯ = gN. Let φ : G ! G 0 be a homomorphism such that N ≤ Ker(φ). Then there is a unique homomorphism φ : G ! G 0 such that φ ◦ π = φ. This map is defined by the rule φ(¯g) = φ(g): π G ! G φ &# φ G 0 5 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions Let G be a group, by the commutant (or derived subgroup) G 0 of G we mean the subgroup generated by all the commutators [g; b] = gbg −1b−1 in G. Since a[g; b]a−1 = [aga−1; aba−1], the commutant is a normal subgroup of G. The quotient G=G 0 is called the abelianization of G. G=G 0 is an abelian group. For example, the abelianization of a free group Fn is the free abelian group of rank n. In general, if G = hs1;:::; sn j r1;:::; rmi; then 0 G=G = hs1;:::; sn j r1;:::; rm; [si ; sj ](1 ≤ i < j ≤ n)i 6 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions As the following corollary shows, the abelianization G=G 0 is the largest abelian quotient of G, in a sense. Corollary Let H be an abelian quotient of G, and let ν : G ! G=G 0 and : G ! H be the natural homomorphisms. Then there is a homomorphism ': G=G 0 ! H so that the following diagram commutes: G ! G=G 0 &# ' H 7 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions Proof. 0 Let G be generated by S = fs1;:::; sng, then G=G is generated by ν(S) = fν(s1); : : : ; ν(sn)g. We still denote ν(si ) by si , since we want to fix the alphabet S±1 for both G and G=G 0. Hence, G=G 0 has the presentation above. Define a map '0 : ν(S) ! H by 0 0 ' (si ) = (si ) for all i. Observe that ' (rj ) = (rj ) = 1 in H, since is a homomorphism and rj = 1 in G. Also, 0 ' ([si ; sj ]) = ([si ; sj ]) = [ (si ); (sj )] = 1, since H is abelian. It follows now from the previous lemma that the map '0 extends to a homomorphism from G=G 0 to H. 8 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions More groups given by generators and relations The free Burnside group of exponent n with two generators is given by the presentation ha; b j un = 1i for all words u in the alphabet a; b. The fundamental group of the orientable surface of genus n is given by the presentation ha1; b1; :::; an; bn j [a1; b1]:::[an; bn] = 1i: 9 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions More groups given by generators and relations The free Burnside group of exponent n with two generators is given by the presentation ha; b j un = 1i for all words u in the alphabet a; b. The fundamental group of the orientable surface of genus n is given by the presentation ha1; b1; :::; an; bn j [a1; b1]:::[an; bn] = 1i: 9 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions More groups given by generators and relations The free Burnside group of exponent n with two generators is given by the presentation ha; b j un = 1i for all words u in the alphabet a; b. The fundamental group of the orientable surface of genus n is given by the presentation ha1; b1; :::; an; bn j [a1; b1]:::[an; bn] = 1i: 9 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions More groups given by generators and relations The free Burnside group of exponent n with two generators is given by the presentation ha; b j un = 1i for all words u in the alphabet a; b. The fundamental group of the orientable surface of genus n is given by the presentation ha1; b1; :::; an; bn j [a1; b1]:::[an; bn] = 1i: 9 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions More groups given by generators and relations The free Burnside group of exponent n with two generators is given by the presentation ha; b j un = 1i for all words u in the alphabet a; b. The fundamental group of the orientable surface of genus n is given by the presentation ha1; b1; :::; an; bn j [a1; b1]:::[an; bn] = 1i: 9 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions More groups given by generators and relations The free Burnside group of exponent n with two generators is given by the presentation ha; b j un = 1i for all words u in the alphabet a; b. The fundamental group of the orientable surface of genus n is given by the presentation ha1; b1; :::; an; bn j [a1; b1]:::[an; bn] = 1i: 9 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions More groups given by generators and relations The free Burnside group of exponent n with two generators is given by the presentation ha; b j un = 1i for all words u in the alphabet a; b. The fundamental group of the orientable surface of genus n is given by the presentation ha1; b1; :::; an; bn j [a1; b1]:::[an; bn] = 1i: 9 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions More groups given by generators and relations The free Burnside group of exponent n with two generators is given by the presentation ha; b j un = 1i for all words u in the alphabet a; b. The fundamental group of the orientable surface of genus n is given by the presentation ha1; b1; :::; an; bn j [a1; b1]:::[an; bn] = 1i: 9 / 32 Preliminaries Presentations of groups Finitely generated groups viewed as metric spaces Homomorphisms of groups Quasi-isometry Limit groups Free actions Group presentations The object of study in Geometric Group Theory- finitely generated groups given by presentations ha1; :::; an j r1; r2; :::i, where ri is a word in a1; :::; an.