On the hyperbolicity of strict Pride groups - Workshop ... · then G is called a strict Pride group. 21.05.2013 Matthias Neumann-Brosig On the hyperbolicity of strict Pride groups

On the hyperbolicity of strict Pride groupsWorkshop “Questions, Algorithms, and Computationsin Abstract Group Theory” in BraunschweigMay 20-24, 2013

Matthias Neumann-Brosig, 21.05.2013

Outline of the talk

Strict Pride groups

Hyperbolic groups

Small Cancellation Theory

The hyperbolicity of finitely presented groups with high-powered relators

Examples: generalized triangle and tetrahedron groups

21.05.2013 Matthias Neumann-Brosig On the hyperbolicity of strict Pride groups

Strict Pride groups

Definition

Let G be a group. If G admits a presentation 〈S|R〉 such that

1. |S|+ |R| <∞,

2. every element of R involves at most two elements of S and

3. for every (unordered) pair (s1, s2), there is at most one element of r ∈ Rsuch that r involves exactly the elements s1 and s2,

then G is called a strict Pride group.


Examples of strict Pride groups

The following groups are strict Pride groups:

1. Finitely generated, free groups or free abelian groups

2. Braid groups

3. Coxeter groups

4. Generalized triangle and tetrahedron groups


Hyperbolic groups

DefinitionLet Γ be a group. A set S ⊂ Γ of generators of Γ with 1 /∈ S is calledsymmetric if x ∈ S ⇒ x−1 ∈ S holds for all x ∈ S.

DefinitionA group Γ having a symmetric, finite set of generators S is called δ-hyperbolicif the Cayley-graph for (Γ ,S) is δ-hyperbolic (as a metric space). Γ is calledhyperbolic (or word-hyperbolic) if there is a real number δ ∈ R+

0 and a set ofgenerators S of Γ such that (Γ ,S) is δ−hyperbolic as a metric space.


Hyperbolic groups

DefinitionLet Γ be a group. A set S ⊂ Γ of generators of Γ with 1 /∈ S is calledsymmetric if x ∈ S ⇒ x−1 ∈ S holds for all x ∈ S.

DefinitionA group Γ having a symmetric, finite set of generators S is called δ-hyperbolicif the Cayley-graph for (Γ ,S) is δ-hyperbolic (as a metric space). Γ is calledhyperbolic (or word-hyperbolic) if there is a real number δ ∈ R+

0 and a set ofgenerators S of Γ such that (Γ ,S) is δ−hyperbolic as a metric space.


Examples of hyperbolic groups

The following groups are hyperbolic:

1. Finite groups

2. Finitely generated free groups

3. Fundamental groups of surfaces with negative Euler characteristics

4. Groups that act cocompactly and properly discontinously on a properCAT (k)-space with k < 0.



DefinitionLet X be a set, F(X) the free group on X and R ⊂ F(X) such that eachelement is cyclically reduced and the following condition holds:

r ∈ R ⇒ r−1 ∈ R

and every cyclically reduced conjugate r ′ of r is also in R. In this case, wesay that R is symmetric.

DefinitionLet G = 〈S|R〉 be a group, with symmetric set R of relators. We say that thecondition C ′(λ) is fulfilled if for all r ∈ R and r ≡ bc, where b is a piece (amaximal subword of r that is completely cancelled in a product r ′r), theinequality |b| < λ|r | holds.



DefinitionLet X be a set, F(X) the free group on X and R ⊂ F(X) such that eachelement is cyclically reduced and the following condition holds:

r ∈ R ⇒ r−1 ∈ R

and every cyclically reduced conjugate r ′ of r is also in R. In this case, wesay that R is symmetric.

DefinitionLet G = 〈S|R〉 be a group, with symmetric set R of relators. We say that thecondition C ′(λ) is fulfilled if for all r ∈ R and r ≡ bc, where b is a piece (amaximal subword of r that is completely cancelled in a product r ′r), theinequality |b| < λ|r | holds.


C ′(16) and hyperbolic groups

The following theorem is well-known:

Theorem

Let G ∼= 〈S|R〉 be a group, s.th. R fulfills the condition C ′(λ) for λ 6 16 . Then

G is hyperbolic.

DefinitionLet G ∼= 〈S | R〉 be a finitely presented group. Then the pair (S,R) is calledadmissible if S and R are symmetric, and if r1, r2 are two elements of R thatare not inverse to each other, then neither r1 nor r2 is completely cancelled inthe product r1r2.


C ′(16) and hyperbolic groups

The following theorem is well-known:

Theorem

Let G ∼= 〈S|R〉 be a group, s.th. R fulfills the condition C ′(λ) for λ 6 16 . Then

G is hyperbolic.

DefinitionLet G ∼= 〈S | R〉 be a finitely presented group. Then the pair (S,R) is calledadmissible if S and R are symmetric, and if r1, r2 are two elements of R thatare not inverse to each other, then neither r1 nor r2 is completely cancelled inthe product r1r2.


The existence of admissible presentations

LemmaLet G ∼= 〈S | R〉 such that |S|+ |R| <∞.Then there is a set R ′ of relatorssuch that G ∼= 〈S | R ′〉 and the pair (S,R ′) is admissible.

Sketch of proof: We assume that R is a minimal set of relators (relativeto ⊆) such that 〈S | R〉 ∼= G holds.

1. If there are words r1, r2 ∈ R such that w1 and w2 are cyclically reducedconjugates of rε1

1 and rε22 , respectively, and w1 is completely cancelled

in the product w1w2, exchange r1 by w1 and r2 by w1w2 in R.2. Repeat step 1 until no further reduction is possible. Each time, the

total length of the presentation is reduced, so this process terminatesafter a �nite number of steps.

3. Take the smallest symmetric set of generators R ′ such that R ′ containsR. The pair (S,R ′) is admissible by construction, and - due to Tietze -〈S | R ′〉 ∼= G holds.

�


The existence of admissible presentations

LemmaLet G ∼= 〈S | R〉 such that |S|+ |R| <∞.Then there is a set R ′ of relatorssuch that G ∼= 〈S | R ′〉 and the pair (S,R ′) is admissible.

Sketch of proof: We assume that R is a minimal set of relators (relativeto ⊆) such that 〈S | R〉 ∼= G holds.

1. If there are words r1, r2 ∈ R such that w1 and w2 are cyclically reducedconjugates of rε1

1 and rε22 , respectively, and w1 is completely cancelled

in the product w1w2, exchange r1 by w1 and r2 by w1w2 in R.2. Repeat step 1 until no further reduction is possible. Each time, the

total length of the presentation is reduced, so this process terminatesafter a �nite number of steps.

3. Take the smallest symmetric set of generators R ′ such that R ′ containsR. The pair (S,R ′) is admissible by construction, and - due to Tietze -〈S | R ′〉 ∼= G holds.

�


Finitely presented groups with high-powered relators

TheoremLet G ∼= 〈S | R〉 such that the pair (S,R) is admissible. Then there is anatural number n ∈ N such that 〈S | {rn|r ∈ R} 〉 is hyperbolic.

Sketch of proof: Idea: show that for n large enough, C ′( 16 ) holds.

1. Look at products rm1 rm

2 , where r1, r2 ∈ R.

2. (S,R) admissible, and R is symmetric ⇒ no copy of r1 or r2 is cancelledout completely in the product r1r2.

3. Looking at the presentation 〈S | {rm | r ∈ R}〉, we see that the length ofa piece of rm

1 cannot be longer then a piece of r1 in 〈S | R〉.4. Choose n > 6 and apply the well-known theorem.

�


Finitely presented groups with high-powered relators

TheoremLet G ∼= 〈S | R〉 such that the pair (S,R) is admissible. Then there is anatural number n ∈ N such that 〈S | {rn|r ∈ R} 〉 is hyperbolic.

Sketch of proof: Idea: show that for n large enough, C ′( 16 ) holds.

1. Look at products rm1 rm

2 , where r1, r2 ∈ R.

2. (S,R) admissible, and R is symmetric ⇒ no copy of r1 or r2 is cancelledout completely in the product r1r2.

3. Looking at the presentation 〈S | {rm | r ∈ R}〉, we see that the length ofa piece of rm

1 cannot be longer then a piece of r1 in 〈S | R〉.4. Choose n > 6 and apply the well-known theorem.

�


Application: generalized triangle groups

CorollaryLet G = 〈x, y |xp = yq = R(x, y)r = 1〉, where R(x, y) = xn1 ym1 ...xnk ymk ,k > 0, 0 < ni < p, 0 < mi < q for all i , 1 < p, q, r , be a generalized trianglegroup. If p > 6 max{ ni | i = 1, ..., k}, q > 6 max{ mi | i = 1, ..., k} and r > 6holds, G is hyperbolic.

If we choose the ni in the intervall − p2 , ...,

p2 and the exponents mi in − q

2 , ...,q2 ,

we can extend this result a bit. The same is true for the next corollary.


Application: generalized tetrahedron groups

CorollaryLet

G = 〈x, y, z | xp = yq = zr =

= R1(x, y)n = R2(y, z)

m = R3(z, x)s = 1〉

be a generelized tetrahedron group, where

R1(x, y) = xn1 ym1 ...xnk ymk ,R2(y, z) = yp1 zq1 ...ypk zqk ,

R3(z, x) = zr1 xs1 ...zrk xsk .

If n,m, s > 6 and

p > 6 max{ ni | i = 1, ..., k} ∪ { si | i = 1, ..., k},

q > 6 max{ mi | i = 1, ..., k} ∪ { pi | i = 1, ..., k},

r > 6 max{ qi | i = 1, ..., k} ∪ { ri | i = 1, ..., k},

then G is hyperbolic.


Documents

On the hyperbolicity of strict Pride groups - Workshop ... · then G is called a strict Pride group. 21.05.2013 Matthias Neumann-Brosig On the hyperbolicity of strict Pride groups