Lecture 4, Geometric and Asymptotic Group Theory

Home , Finitely generated group, Real tree

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 ϕ: F (S) → G such that ϕ(s) = s for s ∈ S. It follows that ϕ is onto, so by the ﬁrst isomorphism theorem

G ' F (S)/ker(ϕ).

In this event ker(ϕ) is viewed as the set of relators of G, and a group word w ∈ ker(ϕ) is called a relator of G in generators S. If a subset R ⊂ ker(ϕ) generates ker(ϕ) 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 | Ri is called a presentation of G, it determines G uniquely up to isomorphism. The presentation hS | 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 | [si , sj ], ∀1 ≤ i < j ≤ ni is the free abelian group of rank n. n 2 Cn = hs | s = 1i is the cyclic group of order n. 3 Both presentations ha, b | ba2b−1a−3i and ha, b | 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 | Ri be a group defined by a (finite) presentation with the set of relators R = {r = y (j) ... y (j) | y (j) ∈ S±1, 1 ≤ j ≤ m}, j i1 ij i and let H be an arbitrary group. A map ψ : S±1 → H extends to a homomorphism ψ˜: G → H, if and only if ψ(r ) = ψ(y (j)) . . . ψ(y (j)) = 1 in H for all r ∈ R. j i1 ij j

Proof Deﬁne the map ψ˜: G → H by ˜ ψ(yn1 ... ynt ) = ψ(yn1 ) . . . ψ(ynt ), ±1 ˜ whenever yni ∈ S . If ψ is a homomorphism, then obviously ψ˜(rj ) = 1 for all rj ∈ 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 | r1,..., rmi, then 0 G/G = hs1,..., sn | 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 ψ : G → H be the natural homomorphisms. Then there is a homomorphism ϕ: G/G 0 → H so that the following diagram commutes:

G → G/G 0 ψ & ↓ ϕ 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 = {s1,..., sn}, then G/G is generated by ν(S) = {ν(s1), . . . , ν(sn)}. 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 ϕ0 : ν(S) → H by 0 0 ϕ (si ) = ψ(si ) for all i. Observe that ϕ (rj ) = ψ(rj ) = 1 in H, since ψ is a homomorphism and rj = 1 in G. Also, 0 ϕ ([si , sj ]) = ψ([si , sj ]) = [ψ(si ), ψ(sj )] = 1, since H is abelian. It follows now from the previous lemma that the map ϕ0 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 | un = 1i