Uncountably many quasi-isometry classes of groups of type FP Ignat - - PowerPoint PPT Presentation

Uncountably many quasi-isometry classes of groups of type FP

Ignat Soroko

University of Oklahoma ignat.soroko@ou.edu Joint work with Robert Kropholler, Tufts University and Ian J. Leary, University of Southampton

Bielefeld U., April 3–6, 2018

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 1 / 10

TOPOLOGY ALGEBRA Space X π1(X), Hn(X), πn(X), etc.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 2 / 10

TOPOLOGY ALGEBRA Space X π1(X), Hn(X), πn(X), etc. ALGEBRA TOPOLOGY Group G Eilenberg–Mac Lane space X = K(G, 1) :

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 2 / 10

TOPOLOGY ALGEBRA Space X π1(X), Hn(X), πn(X), etc. ALGEBRA TOPOLOGY Group G Eilenberg–Mac Lane space X = K(G, 1) : X is a CW-complex, π1(X) = G,

• X is contractible.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 2 / 10

TOPOLOGY ALGEBRA Space X π1(X), Hn(X), πn(X), etc. ALGEBRA TOPOLOGY Group G Eilenberg–Mac Lane space X = K(G, 1) : X is a CW-complex, π1(X) = G,

• X is contractible.

We build X = K(G, 1) as follows: X has a single 0–cell, 1–cells of X correspond to generators of G, 2–cells of X correspond to relations of G, 3–cells of X are added to kill π2(X), 4–cells of X are added to kill π3(X),

• etc. . .

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 2 / 10

If the n–skeleton of K(G, 1) has finitely many cells, group G is of type Fn:

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 3 / 10

If the n–skeleton of K(G, 1) has finitely many cells, group G is of type Fn: F1 = finitely generated groups,

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 3 / 10

If the n–skeleton of K(G, 1) has finitely many cells, group G is of type Fn: F1 = finitely generated groups, F2 = finitely presented groups.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 3 / 10

If the n–skeleton of K(G, 1) has finitely many cells, group G is of type Fn: F1 = finitely generated groups, F2 = finitely presented groups. If K(G, 1) has finitely many cells, group G is of type F.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 3 / 10

If the n–skeleton of K(G, 1) has finitely many cells, group G is of type Fn: F1 = finitely generated groups, F2 = finitely presented groups. If K(G, 1) has finitely many cells, group G is of type F. If X = K(G, 1), G acts cellularly on X and we have a long exact sequence · · · − → Ci( X) − → · · · − → C1( X) − → C0( X) − → Z − → 0 consisting of free ZG–modules. This leads to a definition:

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 3 / 10

If the n–skeleton of K(G, 1) has finitely many cells, group G is of type Fn: F1 = finitely generated groups, F2 = finitely presented groups. If K(G, 1) has finitely many cells, group G is of type F. If X = K(G, 1), G acts cellularly on X and we have a long exact sequence · · · − → Ci( X) − → · · · − → C1( X) − → C0( X) − → Z − → 0 consisting of free ZG–modules. This leads to a definition: A group G is of type FPn if the trivial ZG–module Z has a projective resolution which is finitely generated in dimensions 0 to n: · · · − → Pn − → · · · − → P1 − → P0 − → Z − → 0

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 3 / 10

If the n–skeleton of K(G, 1) has finitely many cells, group G is of type Fn: F1 = finitely generated groups, F2 = finitely presented groups. If K(G, 1) has finitely many cells, group G is of type F. If X = K(G, 1), G acts cellularly on X and we have a long exact sequence · · · − → Ci( X) − → · · · − → C1( X) − → C0( X) − → Z − → 0 consisting of free ZG–modules. This leads to a definition: A group G is of type FPn if the trivial ZG–module Z has a projective resolution which is finitely generated in dimensions 0 to n: · · · − → Pn − → · · · − → P1 − → P0 − → Z − → 0 If, in addition, all Pi = 0 for i > N, for some N, group G is of type FP. Clearly,

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 3 / 10

If the n–skeleton of K(G, 1) has finitely many cells, group G is of type Fn: F1 = finitely generated groups, F2 = finitely presented groups. If K(G, 1) has finitely many cells, group G is of type F. If X = K(G, 1), G acts cellularly on X and we have a long exact sequence · · · − → Ci( X) − → · · · − → C1( X) − → C0( X) − → Z − → 0 consisting of free ZG–modules. This leads to a definition: A group G is of type FPn if the trivial ZG–module Z has a projective resolution which is finitely generated in dimensions 0 to n: · · · − → Pn − → · · · − → P1 − → P0 − → Z − → 0 If, in addition, all Pi = 0 for i > N, for some N, group G is of type FP. Clearly, FPn ⊃ FPn+1 and Fn ⊃ Fn+1. FPn ⊃ Fn, and FP ⊃ F.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 3 / 10

Question 1: Are these inclusions strict? Answer: Yes.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 4 / 10

Question 1: Are these inclusions strict? Answer: Yes. Stallings’63: example of F2 \ F3,

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 4 / 10

Question 1: Are these inclusions strict? Answer: Yes. Stallings’63: example of F2 \ F3, Bieri’76: Fn \ Fn+1

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 4 / 10

Question 1: Are these inclusions strict? Answer: Yes. Stallings’63: example of F2 \ F3, Bieri’76: Fn \ Fn+1 Bestvina–Brady’97: FP2 \ F2.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 4 / 10

Question 1: Are these inclusions strict? Answer: Yes. Stallings’63: example of F2 \ F3, Bieri’76: Fn \ Fn+1 Bestvina–Brady’97: FP2 \ F2.

Bestvina–Brady machine:

Input: A flag simplicial complex L. Output: A group BBL with nice properties: L is (n − 1)–connected ⇐ ⇒ BBL is of type Fn, L is (n − 1)–acyclic ⇐ ⇒ BBL is of type FPn.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 4 / 10

Question 1: Are these inclusions strict? Answer: Yes. Stallings’63: example of F2 \ F3, Bieri’76: Fn \ Fn+1 Bestvina–Brady’97: FP2 \ F2.

Bestvina–Brady machine:

Input: A flag simplicial complex L. Output: A group BBL with nice properties: L is (n − 1)–connected ⇐ ⇒ BBL is of type Fn, L is (n − 1)–acyclic ⇐ ⇒ BBL is of type FPn. L is octahedron: π1(L) = 1, π2(L) = 0, = ⇒ Stallings’s example. L is n–dimensional octahedron (orthoplex) = ⇒ Bieri’s example. L has π1(L) = 1, but H1(L) = 0 = ⇒ BBL of type FP2 \ F2.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 4 / 10

Question 1: Are these inclusions strict? Answer: Yes. Stallings’63: example of F2 \ F3, Bieri’76: Fn \ Fn+1 Bestvina–Brady’97: FP2 \ F2.

Bestvina–Brady machine:

Input: A flag simplicial complex L. Output: A group BBL with nice properties: L is (n − 1)–connected ⇐ ⇒ BBL is of type Fn, L is (n − 1)–acyclic ⇐ ⇒ BBL is of type FPn. L is octahedron: π1(L) = 1, π2(L) = 0, = ⇒ Stallings’s example. L is n–dimensional octahedron (orthoplex) = ⇒ Bieri’s example. L has π1(L) = 1, but H1(L) = 0 = ⇒ BBL of type FP2 \ F2. Question 2: How many groups are there of type FP2? Answer 1: Up to isomorphism: 2ℵ0 (I.Leary’15) Answer 2: Up to quasi-isometry: 2ℵ0 (R.Kropholler–I.Leary–S.’17)

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 4 / 10

I.J.Leary’s groups GL(S)

Input: A flag simplicial complex L, a finite collection Γ of directed edge loops in L that normally generates π1(L), a subset S ⊂ Z.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 5 / 10

I.J.Leary’s groups GL(S)

Input: A flag simplicial complex L, a finite collection Γ of directed edge loops in L that normally generates π1(L), a subset S ⊂ Z. Output: Group GL(S) defined as: Generators: directed edges of L, the opposite edge to a being a−1. (Triangle relations) For each directed triangle (a, b, c) in L, two relations: abc = 1 and a−1b−1c−1 = 1. (Long cycle relations) For each n ∈ S \ 0 and each (a1, . . . , aℓ) ∈ Γ, a relation: an

1an 2 . . . an ℓ = 1.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 5 / 10

I.J.Leary’s groups GL(S)

Input: A flag simplicial complex L, a finite collection Γ of directed edge loops in L that normally generates π1(L), a subset S ⊂ Z. Output: Group GL(S) defined as: Generators: directed edges of L, the opposite edge to a being a−1. (Triangle relations) For each directed triangle (a, b, c) in L, two relations: abc = 1 and a−1b−1c−1 = 1. (Long cycle relations) For each n ∈ S \ 0 and each (a1, . . . , aℓ) ∈ Γ, a relation: an

1an 2 . . . an ℓ = 1.

Theorem (I.J.Leary)

If L is a flag complex with π1(L) = 1, then groups GL(S) form 2ℵ0 isomorphism classes. If, in addition, L is aspherical and acyclic, then groups GL(S) are all of type FP.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 5 / 10

I.J.Leary’s groups GL(S)

Input: A flag simplicial complex L, a finite collection Γ of directed edge loops in L that normally generates π1(L), a subset S ⊂ Z. Output: Group GL(S) defined as: Generators: directed edges of L, the opposite edge to a being a−1. (Triangle relations) For each directed triangle (a, b, c) in L, two relations: abc = 1 and a−1b−1c−1 = 1. (Long cycle relations) For each n ∈ S \ 0 and each (a1, . . . , aℓ) ∈ Γ, a relation: an

1an 2 . . . an ℓ = 1.

Theorem (I.J.Leary)

If L is a flag complex with π1(L) = 1, then groups GL(S) form 2ℵ0 isomorphism classes. If, in addition, L is aspherical and acyclic, then groups GL(S) are all of type FP. What is a possible example of an aspherical and acyclic flag simplicial complex L?

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 5 / 10

Take the famous Higman’s group: H = a, b, c, d | ab = a2, bc = b2, cd = c2, da = d2. Let K be its presentation complex. It is aspherical and acyclic. Take L to be the 2nd barycentric subdivision of K. Then L is a flag simplicial complex with 97 vertices, 336 edges and 240 triangles. Thus, GL(S) = 336 gen’s | 240 × 2 triangle relators, 1 long relator ∀n ∈ S.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 6 / 10

Take the famous Higman’s group: H = a, b, c, d | ab = a2, bc = b2, cd = c2, da = d2. Let K be its presentation complex. It is aspherical and acyclic. Take L to be the 2nd barycentric subdivision of K. Then L is a flag simplicial complex with 97 vertices, 336 edges and 240 triangles. Thus, GL(S) = 336 gen’s | 240 × 2 triangle relators, 1 long relator ∀n ∈ S.

Theorem (R.Kropholler–Leary–S.)

Groups GL(S) form 2ℵ0 classes up to quasi-isometry.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 6 / 10

Take the famous Higman’s group: H = a, b, c, d | ab = a2, bc = b2, cd = c2, da = d2. Let K be its presentation complex. It is aspherical and acyclic. Take L to be the 2nd barycentric subdivision of K. Then L is a flag simplicial complex with 97 vertices, 336 edges and 240 triangles. Thus, GL(S) = 336 gen’s | 240 × 2 triangle relators, 1 long relator ∀n ∈ S.

Theorem (R.Kropholler–Leary–S.)

Groups GL(S) form 2ℵ0 classes up to quasi-isometry. Recall that groups G1, G2 are quasi-isometric (qi), if their Cayley graphs are qi as metric spaces, i.e. there exists f : Cay(G1, d1) → Cay(G2, d2), and A ≥ 1, B ≥ 0, C ≥ 0 such that for all x, y ∈ Cay(G1): 1 Ad1(x, y) − B ≤ d2(f (x), f (y)) ≤ Ad1(x, y) + B, and for all z ∈ Cay(G2) there exists x ∈ Cay(G1) such that d2(z, f (x)) ≤ C.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 6 / 10

How to distinguish groups up to qi?

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 7 / 10

How to distinguish groups up to qi?

Grigorchuk’84: growth functions of groups.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 7 / 10

How to distinguish groups up to qi?

Grigorchuk’84: growth functions of groups. Bowditch’98: a concept of taut loops in Cayley graphs. These are the loops which are not consequences of shorter loops. More formally:

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 7 / 10

How to distinguish groups up to qi?

Grigorchuk’84: growth functions of groups. Bowditch’98: a concept of taut loops in Cayley graphs. These are the loops which are not consequences of shorter loops. More formally: If Γ is the Cayley graph of G, we can form a sequence of 2–complexes Γ ⊂ Γ1 ⊂ Γ2 ⊂ Γ3 ⊂ . . . , where Γℓ = Γℓ−1 ∪

• |γ|≤ℓ

Cone(γ).

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 7 / 10

How to distinguish groups up to qi?

Grigorchuk’84: growth functions of groups. Bowditch’98: a concept of taut loops in Cayley graphs. These are the loops which are not consequences of shorter loops. More formally: If Γ is the Cayley graph of G, we can form a sequence of 2–complexes Γ ⊂ Γ1 ⊂ Γ2 ⊂ Γ3 ⊂ . . . , where Γℓ = Γℓ−1 ∪

• |γ|≤ℓ

Cone(γ). We get π1(Γ) → π1(Γ1) → π1(Γ2) → . . . . A loop γ ⊂ Γ of length ℓ is taut if it lies in the kernel ker

• π1(Γℓ) → π1(Γℓ+1)
• .

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 7 / 10

How to distinguish groups up to qi?

Grigorchuk’84: growth functions of groups. Bowditch’98: a concept of taut loops in Cayley graphs. These are the loops which are not consequences of shorter loops. More formally: If Γ is the Cayley graph of G, we can form a sequence of 2–complexes Γ ⊂ Γ1 ⊂ Γ2 ⊂ Γ3 ⊂ . . . , where Γℓ = Γℓ−1 ∪

• |γ|≤ℓ

Cone(γ). We get π1(Γ) → π1(Γ1) → π1(Γ2) → . . . . A loop γ ⊂ Γ of length ℓ is taut if it lies in the kernel ker

• π1(Γℓ) → π1(Γℓ+1)
• .

Let TL(G) ⊂ N be the spectrum of lengths of taut loops in the Cayley graph of a group G.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 7 / 10

How to distinguish groups up to qi?

Grigorchuk’84: growth functions of groups. Bowditch’98: a concept of taut loops in Cayley graphs. These are the loops which are not consequences of shorter loops. More formally: If Γ is the Cayley graph of G, we can form a sequence of 2–complexes Γ ⊂ Γ1 ⊂ Γ2 ⊂ Γ3 ⊂ . . . , where Γℓ = Γℓ−1 ∪

• |γ|≤ℓ

Cone(γ). We get π1(Γ) → π1(Γ1) → π1(Γ2) → . . . . A loop γ ⊂ Γ of length ℓ is taut if it lies in the kernel ker

• π1(Γℓ) → π1(Γℓ+1)
• .

Let TL(G) ⊂ N be the spectrum of lengths of taut loops in the Cayley graph of a group G. Bowditch: Groups G1 and G2 quasi-isometric = ⇒ TL(G1) and TL(G2) quasi-isometric in R.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 7 / 10

How to distinguish groups up to qi?

Grigorchuk’84: growth functions of groups. Bowditch’98: a concept of taut loops in Cayley graphs. These are the loops which are not consequences of shorter loops. More formally: If Γ is the Cayley graph of G, we can form a sequence of 2–complexes Γ ⊂ Γ1 ⊂ Γ2 ⊂ Γ3 ⊂ . . . , where Γℓ = Γℓ−1 ∪

• |γ|≤ℓ

Cone(γ). We get π1(Γ) → π1(Γ1) → π1(Γ2) → . . . . A loop γ ⊂ Γ of length ℓ is taut if it lies in the kernel ker

• π1(Γℓ) → π1(Γℓ+1)
• .

Let TL(G) ⊂ N be the spectrum of lengths of taut loops in the Cayley graph of a group G. Bowditch: Groups G1 and G2 quasi-isometric = ⇒ TL(G1) and TL(G2) quasi-isometric in R. I.e. there exist constants A, B, N > 0 such that for every l1 ∈ TL(G1), l1 > N, there exist an l2 ∈ TL(G2) such that l1 ∈ [Al2, Bl2] and vice versa.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 7 / 10

Goal: to engineer groups with taut loops spectra “wildly interspersed” in N, this will make the linear relation above impossible.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 8 / 10

Goal: to engineer groups with taut loops spectra “wildly interspersed” in N, this will make the linear relation above impossible. Bowditch does this for small cancellation groups: he proves that there exist continuously many qi classes of 2–generator small cancellation groups.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 8 / 10

Goal: to engineer groups with taut loops spectra “wildly interspersed” in N, this will make the linear relation above impossible. Bowditch does this for small cancellation groups: he proves that there exist continuously many qi classes of 2–generator small cancellation groups. Recall: GL(S) has: (Triangle relations) For each directed triangle (a, b, c) in L, two relations: abc = 1 and a−1b−1c−1 = 1. (Long cycle relations) For each n ∈ S \ 0 and each (a1, . . . , aℓ) ∈ Γ, a relation: an

1an 2 . . . an ℓ = 1.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 8 / 10

Goal: to engineer groups with taut loops spectra “wildly interspersed” in N, this will make the linear relation above impossible. Bowditch does this for small cancellation groups: he proves that there exist continuously many qi classes of 2–generator small cancellation groups. Recall: GL(S) has: (Triangle relations) For each directed triangle (a, b, c) in L, two relations: abc = 1 and a−1b−1c−1 = 1. (Long cycle relations) For each n ∈ S \ 0 and each (a1, . . . , aℓ) ∈ Γ, a relation: an

1an 2 . . . an ℓ = 1.

Intuitively, we expect TL(GL(S)) ≈ ℓ · S.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 8 / 10

Goal: to engineer groups with taut loops spectra “wildly interspersed” in N, this will make the linear relation above impossible. Bowditch does this for small cancellation groups: he proves that there exist continuously many qi classes of 2–generator small cancellation groups. Recall: GL(S) has: (Triangle relations) For each directed triangle (a, b, c) in L, two relations: abc = 1 and a−1b−1c−1 = 1. (Long cycle relations) For each n ∈ S \ 0 and each (a1, . . . , aℓ) ∈ Γ, a relation: an

1an 2 . . . an ℓ = 1.

Intuitively, we expect TL(GL(S)) ≈ ℓ · S. Many triangles = ⇒ no small cancellation.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 8 / 10

Goal: to engineer groups with taut loops spectra “wildly interspersed” in N, this will make the linear relation above impossible. Bowditch does this for small cancellation groups: he proves that there exist continuously many qi classes of 2–generator small cancellation groups. Recall: GL(S) has: (Triangle relations) For each directed triangle (a, b, c) in L, two relations: abc = 1 and a−1b−1c−1 = 1. (Long cycle relations) For each n ∈ S \ 0 and each (a1, . . . , aℓ) ∈ Γ, a relation: an

1an 2 . . . an ℓ = 1.

Intuitively, we expect TL(GL(S)) ≈ ℓ · S. Many triangles = ⇒ no small cancellation. Use CAT(0) geometry of branched covers of cubical complexes to get estimates for the taut loops spectra.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 8 / 10

Goal: to engineer groups with taut loops spectra “wildly interspersed” in N, this will make the linear relation above impossible. Bowditch does this for small cancellation groups: he proves that there exist continuously many qi classes of 2–generator small cancellation groups. Recall: GL(S) has: (Triangle relations) For each directed triangle (a, b, c) in L, two relations: abc = 1 and a−1b−1c−1 = 1. (Long cycle relations) For each n ∈ S \ 0 and each (a1, . . . , aℓ) ∈ Γ, a relation: an

1an 2 . . . an ℓ = 1.

Intuitively, we expect TL(GL(S)) ≈ ℓ · S. Many triangles = ⇒ no small cancellation. Use CAT(0) geometry of branched covers of cubical complexes to get estimates for the taut loops

• spectra. We proved:

If S ⊂ {C 2n | n ∈ N}, for some C > 7, then TL(GL(S)) lies in some multiplicative [A, B] neighborhood of S.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 8 / 10

Goal: to engineer groups with taut loops spectra “wildly interspersed” in N, this will make the linear relation above impossible. Bowditch does this for small cancellation groups: he proves that there exist continuously many qi classes of 2–generator small cancellation groups. Recall: GL(S) has: (Triangle relations) For each directed triangle (a, b, c) in L, two relations: abc = 1 and a−1b−1c−1 = 1. (Long cycle relations) For each n ∈ S \ 0 and each (a1, . . . , aℓ) ∈ Γ, a relation: an

1an 2 . . . an ℓ = 1.

Intuitively, we expect TL(GL(S)) ≈ ℓ · S. Many triangles = ⇒ no small cancellation. Use CAT(0) geometry of branched covers of cubical complexes to get estimates for the taut loops

• spectra. We proved:

If S ⊂ {C 2n | n ∈ N}, for some C > 7, then TL(GL(S)) lies in some multiplicative [A, B] neighborhood of S. Now there are uncountably many subsets S in the above set, and these give 2ℵ0 quasi-isometry classes of groups GL(S).

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 8 / 10

Connection to the Relation Gap problem

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 9 / 10

Connection to the Relation Gap problem

If G is arbitrary group, G = a1, . . . , am | r1, . . . , rn = F/R, where F = F(a1, . . . , am) and R = r1, . . . , rn .

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 9 / 10

Connection to the Relation Gap problem

If G is arbitrary group, G = a1, . . . , am | r1, . . . , rn = F/R, where F = F(a1, . . . , am) and R = r1, . . . , rn . F acts on R by conjugation, so it induces an action of G on Rab = R/[R, R], the relation module.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 9 / 10

Connection to the Relation Gap problem

If G is arbitrary group, G = a1, . . . , am | r1, . . . , rn = F/R, where F = F(a1, . . . , am) and R = r1, . . . , rn . F acts on R by conjugation, so it induces an action of G on Rab = R/[R, R], the relation module. Rank(Rab) as a ZG–module ≤ min number of normal generators of R. The difference of the two is the relation gap.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 9 / 10

Connection to the Relation Gap problem

If G is arbitrary group, G = a1, . . . , am | r1, . . . , rn = F/R, where F = F(a1, . . . , am) and R = r1, . . . , rn . F acts on R by conjugation, so it induces an action of G on Rab = R/[R, R], the relation module. Rank(Rab) as a ZG–module ≤ min number of normal generators of R. The difference of the two is the relation gap. Bestvina–Brady kernels BBL have infinite relation gap, and so do GL(S).

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 9 / 10

Connection to the Relation Gap problem

If G is arbitrary group, G = a1, . . . , am | r1, . . . , rn = F/R, where F = F(a1, . . . , am) and R = r1, . . . , rn . F acts on R by conjugation, so it induces an action of G on Rab = R/[R, R], the relation module. Rank(Rab) as a ZG–module ≤ min number of normal generators of R. The difference of the two is the relation gap. Bestvina–Brady kernels BBL have infinite relation gap, and so do GL(S). Open Question: Are there groups with nonzero finite relation gap?

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 9 / 10

Connection to the Relation Gap problem

If G is arbitrary group, G = a1, . . . , am | r1, . . . , rn = F/R, where F = F(a1, . . . , am) and R = r1, . . . , rn . F acts on R by conjugation, so it induces an action of G on Rab = R/[R, R], the relation module. Rank(Rab) as a ZG–module ≤ min number of normal generators of R. The difference of the two is the relation gap. Bestvina–Brady kernels BBL have infinite relation gap, and so do GL(S). Open Question: Are there groups with nonzero finite relation gap? Take our group G = GL(S) with infinite S.

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 9 / 10

Connection to the Relation Gap problem

If G is arbitrary group, G = a1, . . . , am | r1, . . . , rn = F/R, where F = F(a1, . . . , am) and R = r1, . . . , rn . F acts on R by conjugation, so it induces an action of G on Rab = R/[R, R], the relation module. Rank(Rab) as a ZG–module ≤ min number of normal generators of R. The difference of the two is the relation gap. Bestvina–Brady kernels BBL have infinite relation gap, and so do GL(S). Open Question: Are there groups with nonzero finite relation gap? Take our group G = GL(S) with infinite S. Exhaust S by finite sets: ∅ ⊂ S1 ⊂ S2 ⊂ S3 ⊂ · · · ⊂ S

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 9 / 10

Connection to the Relation Gap problem

If G is arbitrary group, G = a1, . . . , am | r1, . . . , rn = F/R, where F = F(a1, . . . , am) and R = r1, . . . , rn . F acts on R by conjugation, so it induces an action of G on Rab = R/[R, R], the relation module. Rank(Rab) as a ZG–module ≤ min number of normal generators of R. The difference of the two is the relation gap. Bestvina–Brady kernels BBL have infinite relation gap, and so do GL(S). Open Question: Are there groups with nonzero finite relation gap? Take our group G = GL(S) with infinite S. Exhaust S by finite sets: ∅ ⊂ S1 ⊂ S2 ⊂ S3 ⊂ · · · ⊂ S GL(∅) → GL(S1) → GL(S2) → GL(S3) → · · · → GL(S)

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 9 / 10

Connection to the Relation Gap problem

If G is arbitrary group, G = a1, . . . , am | r1, . . . , rn = F/R, where F = F(a1, . . . , am) and R = r1, . . . , rn . F acts on R by conjugation, so it induces an action of G on Rab = R/[R, R], the relation module. Rank(Rab) as a ZG–module ≤ min number of normal generators of R. The difference of the two is the relation gap. Bestvina–Brady kernels BBL have infinite relation gap, and so do GL(S). Open Question: Are there groups with nonzero finite relation gap? Take our group G = GL(S) with infinite S. Exhaust S by finite sets: ∅ ⊂ S1 ⊂ S2 ⊂ S3 ⊂ · · · ⊂ S GL(∅) → GL(S1) → GL(S2) → GL(S3) → · · · → GL(S) Fact: they all have the same relation module!

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 9 / 10

Connection to the Relation Gap problem

If G is arbitrary group, G = a1, . . . , am | r1, . . . , rn = F/R, where F = F(a1, . . . , am) and R = r1, . . . , rn . F acts on R by conjugation, so it induces an action of G on Rab = R/[R, R], the relation module. Rank(Rab) as a ZG–module ≤ min number of normal generators of R. The difference of the two is the relation gap. Bestvina–Brady kernels BBL have infinite relation gap, and so do GL(S). Open Question: Are there groups with nonzero finite relation gap? Take our group G = GL(S) with infinite S. Exhaust S by finite sets: ∅ ⊂ S1 ⊂ S2 ⊂ S3 ⊂ · · · ⊂ S GL(∅) → GL(S1) → GL(S2) → GL(S3) → · · · → GL(S) Fact: they all have the same relation module! Their relation gaps are:

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 9 / 10

Connection to the Relation Gap problem

If G is arbitrary group, G = a1, . . . , am | r1, . . . , rn = F/R, where F = F(a1, . . . , am) and R = r1, . . . , rn . F acts on R by conjugation, so it induces an action of G on Rab = R/[R, R], the relation module. Rank(Rab) as a ZG–module ≤ min number of normal generators of R. The difference of the two is the relation gap. Bestvina–Brady kernels BBL have infinite relation gap, and so do GL(S). Open Question: Are there groups with nonzero finite relation gap? Take our group G = GL(S) with infinite S. Exhaust S by finite sets: ∅ ⊂ S1 ⊂ S2 ⊂ S3 ⊂ · · · ⊂ S GL(∅) → GL(S1) → GL(S2) → GL(S3) → · · · → GL(S) Fact: they all have the same relation module! Their relation gaps are: ? ? ? ? . . . ∞

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 9 / 10

Connection to the Relation Gap problem

If G is arbitrary group, G = a1, . . . , am | r1, . . . , rn = F/R, where F = F(a1, . . . , am) and R = r1, . . . , rn . F acts on R by conjugation, so it induces an action of G on Rab = R/[R, R], the relation module. Rank(Rab) as a ZG–module ≤ min number of normal generators of R. The difference of the two is the relation gap. Bestvina–Brady kernels BBL have infinite relation gap, and so do GL(S). Open Question: Are there groups with nonzero finite relation gap? Take our group G = GL(S) with infinite S. Exhaust S by finite sets: ∅ ⊂ S1 ⊂ S2 ⊂ S3 ⊂ · · · ⊂ S GL(∅) → GL(S1) → GL(S2) → GL(S3) → · · · → GL(S) Fact: they all have the same relation module! Their relation gaps are: ? ? ? ? . . . ∞ So groups GL(Si) for finite Si are candidates to have finite relation gap!

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 9 / 10

Bibliography

[1] M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), 445–470. [2] R. Bieri, Normal subgroups in duality groups and in groups of cohomological dimension 2, Jour. Pure App. Algebra 7, 35–52 (1976). [3] B. H. Bowditch, Continuously many quasi-isometry classes of 2-generator groups, Comm. Math. Helv. 73 (1998), 232–236. [4] R. Kropholler, I. J. Leary, I. Soroko, Uncountably many quasi-isometry classes of groups of type FP, arXiv:1712.05826. [5] I. J. Leary, Uncountably many groups of type FP, arXiv:1512.06609v2. [6] J.R. Stallings, A finitely presented group whose 3–dimensional integral homology is not finitely generated, Am. J. Math. 85, 541–543 (1963). Thank you!

Ignat Soroko (OU) Uncountably many qi classes of FP groups Bielefeld U., April 3–6, 2018 10 / 10