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Rank of intersection of free subgroups in free amalgamated products of groups Alexander Zakharov Moscow State University July 30, 2012 Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products Free group case The

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Rank of intersection of free subgroups in free amalgamated products of groups

Alexander Zakharov

Moscow State University

July 30, 2012

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Free group case

The reduced rank of a free group H: r(H) = max{0, r(H) − 1}

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Free group case

The reduced rank of a free group H: r(H) = max{0, r(H) − 1} Theorem (Hanna Neumann, 1957) Suppose G is a free group, H1 and H2 are finitely generated subgroups in G. Then H1 ∩ H2 is also finitely generated (Howson) and r(H1 ∩ H2) 2 r(H1) r(H2)

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Free group case

The reduced rank of a free group H: r(H) = max{0, r(H) − 1} Theorem (Hanna Neumann, 1957) Suppose G is a free group, H1 and H2 are finitely generated subgroups in G. Then H1 ∩ H2 is also finitely generated (Howson) and r(H1 ∩ H2) 2 r(H1) r(H2) Theorem (Igor Mineyev, 2011) r(H1 ∩ H2) r(H1) r(H2) (Hanna Neumann conjecture)

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Free product case

Factor-free subgroup of a free product (or an amalgamated free product): one that intersects trivially with the conjugates to the factors of the product.

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Free product case

Factor-free subgroup of a free product (or an amalgamated free product): one that intersects trivially with the conjugates to the factors of the product. Factor-free subgroups are free.

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Free product case

Factor-free subgroup of a free product (or an amalgamated free product): one that intersects trivially with the conjugates to the factors of the product. Factor-free subgroups are free. Theorem (S.Ivanov, 2000) Suppose G = A ∗ B, and H1, H2 are factor-free subgroups of G with finite ranks. Then H1 ∩ H2 also has finite rank and r(H1 ∩ H2) 6r(H1)r(H2).

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Free product case

Factor-free subgroup of a free product (or an amalgamated free product): one that intersects trivially with the conjugates to the factors of the product. Factor-free subgroups are free. Theorem (S.Ivanov, 2000) Suppose G = A ∗ B, and H1, H2 are factor-free subgroups of G with finite ranks. Then H1 ∩ H2 also has finite rank and r(H1 ∩ H2) 6r(H1)r(H2). (W.Dicks and S.Ivanov, 2008: more precise estimate).

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Amalgamated free product case

Theorem (A.Z., 2011) Suppose G = A ∗T B, T is finite, and H1, H2 are factor-free subgroups of G with finite ranks. Then H1 ∩ H2 also has finite rank, and r(H1 ∩ H2) 6|T| · r(H1)r(H2). (Recall factor-free subgroups are those which intersect trivially with the conjugates to the factors A, B.)

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Amalgamated free product case

Theorem (A.Z., 2011) Suppose G = A ∗T B, T is finite, and H1, H2 are factor-free subgroups of G with finite ranks. Then H1 ∩ H2 also has finite rank, and r(H1 ∩ H2) 6|T| · r(H1)r(H2). (Recall factor-free subgroups are those which intersect trivially with the conjugates to the factors A, B.) Idea of the proof is given further.

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Graph Ψ(H)

Suppose H is a factor-free subgroup of G = A ∗T B Graph Ψ(H) associated with subgroup H:

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Graph Ψ(H)

Suppose H is a factor-free subgroup of G = A ∗T B Graph Ψ(H) associated with subgroup H: 2 types of vertices of Ψ(H):

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Graph Ψ(H)

Suppose H is a factor-free subgroup of G = A ∗T B Graph Ψ(H) associated with subgroup H: 2 types of vertices of Ψ(H):

1 Primary vertices correspond to the right cosets of H in G; Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Graph Ψ(H)

Suppose H is a factor-free subgroup of G = A ∗T B Graph Ψ(H) associated with subgroup H: 2 types of vertices of Ψ(H):

1 Primary vertices correspond to the right cosets of H in G; 2 Secondary vertices correspond to double cosets HgA and HgB

(g ∈ G).

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Graph Ψ(H)

Suppose H is a factor-free subgroup of G = A ∗T B Graph Ψ(H) associated with subgroup H: 2 types of vertices of Ψ(H):

1 Primary vertices correspond to the right cosets of H in G; 2 Secondary vertices correspond to double cosets HgA and HgB

(g ∈ G). Edges of Ψ(H): each primary vertex Hg is connected by an edge with the secondary vertex HgA and with HgB.

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Graph Γ(H)

Suppose H is a factor-free subgroup of G = A ∗T B Graph Γ(H) associated with subgroup H:

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Graph Γ(H)

Suppose H is a factor-free subgroup of G = A ∗T B Graph Γ(H) associated with subgroup H: 2 types of vertices of Γ(H):

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Graph Γ(H)

Suppose H is a factor-free subgroup of G = A ∗T B Graph Γ(H) associated with subgroup H: 2 types of vertices of Γ(H):

1 Primary vertices correspond to double cosets HgT (g ∈ G); Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Graph Γ(H)

Suppose H is a factor-free subgroup of G = A ∗T B Graph Γ(H) associated with subgroup H: 2 types of vertices of Γ(H):

1 Primary vertices correspond to double cosets HgT (g ∈ G); 2 Secondary vertices correspond to double cosets HgA and HgB

(g ∈ G).

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Graph Γ(H)

Suppose H is a factor-free subgroup of G = A ∗T B Graph Γ(H) associated with subgroup H: 2 types of vertices of Γ(H):

1 Primary vertices correspond to double cosets HgT (g ∈ G); 2 Secondary vertices correspond to double cosets HgA and HgB

(g ∈ G). Edges of Γ(H): each primary vertex HgT is connected by an edge with the secondary vertex HgA and with HgB.

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Core graphs

π : Ψ(H) → Γ(H) – the projection: π(Hg) = HgT, π(HgA) = HgA, π(HgB) = HgB (extended to the edges in a natural way).

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Core graphs

π : Ψ(H) → Γ(H) – the projection: π(Hg) = HgT, π(HgA) = HgA, π(HgB) = HgB (extended to the edges in a natural way). Γ1(H) – the core of Γ(H) (the union of all reduced closed paths ending at HT vertex)

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Core graphs

π : Ψ(H) → Γ(H) – the projection: π(Hg) = HgT, π(HgA) = HgA, π(HgB) = HgB (extended to the edges in a natural way). Γ1(H) – the core of Γ(H) (the union of all reduced closed paths ending at HT vertex) Ψ1(H) – the (full) inverse image of Γ1(H) under π (a subgraph of Ψ(H) obtained from it by deleting all ”unnecessary” edges and vertices)

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Facts about the graphs

If H has finite rank, then Γ1(H) is finite.

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Facts about the graphs

If H has finite rank, then Γ1(H) is finite. H ∼ = π1(Γ1(H)).

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Facts about the graphs

If H has finite rank, then Γ1(H) is finite. H ∼ = π1(Γ1(H)). Therefore, r(H) = −χ(Γ1(H)) = 1 2

(deg v − 2),

where χ is Euler characteristics of a graph and the last sum expands over all secondary vertices of Γ1(H).

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Facts about the graphs

If H has finite rank, then Γ1(H) is finite. H ∼ = π1(Γ1(H)). Therefore, r(H) = −χ(Γ1(H)) = 1 2

(deg v − 2),

where χ is Euler characteristics of a graph and the last sum expands over all secondary vertices of Γ1(H). For any w – secondary vertex of Ψ1(H) deg Ψ1(H) w = |T| · deg Γ1(H) π(w).

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Facts about the graphs

If H has finite rank, then Γ1(H) is finite. H ∼ = π1(Γ1(H)). Therefore, r(H) = −χ(Γ1(H)) = 1 2

(deg v − 2),

where χ is Euler characteristics of a graph and the last sum expands over all secondary vertices of Γ1(H). For any w – secondary vertex of Ψ1(H) deg Ψ1(H) w = |T| · deg Γ1(H) π(w). Therefore, the rank of H can be calculated using the graph Ψ1(H).

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Intersecting subgroups

Now we shall work with graphs Ψ1(H1 ∩ H2), Ψ1(H1) and Ψ1(H2).

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Intersecting subgroups

Now we shall work with graphs Ψ1(H1 ∩ H2), Ψ1(H1) and Ψ1(H2). Note that the map τ : (H1 ∩ H2)g → (H1g, H2g) is injective, while η : (H1 ∩ H2)gA → (H1gA, H2gA) might be not.

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Intersecting subgroups

Now we shall work with graphs Ψ1(H1 ∩ H2), Ψ1(H1) and Ψ1(H2). Note that the map τ : (H1 ∩ H2)g → (H1g, H2g) is injective, while η : (H1 ∩ H2)gA → (H1gA, H2gA) might be not. Suppose that w1, ..., wk are all secondary vertices of Ψ1(H1 ∩ H2) such that η(wi) = (v1, v2), i = 1...k, where v1, v2 are fixed secondary vertices of Ψ1(H1), Ψ1(H2)

respectively. Then

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Intersecting subgroups

Now we shall work with graphs Ψ1(H1 ∩ H2), Ψ1(H1) and Ψ1(H2). Note that the map τ : (H1 ∩ H2)g → (H1g, H2g) is injective, while η : (H1 ∩ H2)gA → (H1gA, H2gA) might be not. Suppose that w1, ..., wk are all secondary vertices of Ψ1(H1 ∩ H2) such that η(wi) = (v1, v2), i = 1...k, where v1, v2 are fixed secondary vertices of Ψ1(H1), Ψ1(H2)

respectively. Then

deg wi ≤ deg v1, deg wi ≤ deg v2, i = 1...k (since subgroups are factor-free)

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Intersecting subgroups

Now we shall work with graphs Ψ1(H1 ∩ H2), Ψ1(H1) and Ψ1(H2). Note that the map τ : (H1 ∩ H2)g → (H1g, H2g) is injective, while η : (H1 ∩ H2)gA → (H1gA, H2gA) might be not. Suppose that w1, ..., wk are all secondary vertices of Ψ1(H1 ∩ H2) such that η(wi) = (v1, v2), i = 1...k, where v1, v2 are fixed secondary vertices of Ψ1(H1), Ψ1(H2)

respectively. Then

deg wi ≤ deg v1, deg wi ≤ deg v2, i = 1...k (since subgroups are factor-free) k

i=1 deg wi ≤ deg v1 · deg v2 (since τ is injective)

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Intersecting subgroups

Now we shall work with graphs Ψ1(H1 ∩ H2), Ψ1(H1) and Ψ1(H2). Note that the map τ : (H1 ∩ H2)g → (H1g, H2g) is injective, while η : (H1 ∩ H2)gA → (H1gA, H2gA) might be not. Suppose that w1, ..., wk are all secondary vertices of Ψ1(H1 ∩ H2) such that η(wi) = (v1, v2), i = 1...k, where v1, v2 are fixed secondary vertices of Ψ1(H1), Ψ1(H2)

respectively. Then

deg wi ≤ deg v1, deg wi ≤ deg v2, i = 1...k (since subgroups are factor-free) k

i=1 deg wi ≤ deg v1 · deg v2 (since τ is injective)

After summing over all pairs (v1, v2) and using the facts above we obtain the desired estimate.

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Further generalizations

The next step perhaps is to generalize this estimate for:

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Further generalizations

The next step perhaps is to generalize this estimate for: amalgamated free products with more than two factors

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Further generalizations

The next step perhaps is to generalize this estimate for: amalgamated free products with more than two factors HNN-extensions

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Further generalizations

The next step perhaps is to generalize this estimate for: amalgamated free products with more than two factors HNN-extensions fundamental groups of graph of groups.

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Further generalizations

The next step perhaps is to generalize this estimate for: amalgamated free products with more than two factors HNN-extensions fundamental groups of graph of groups. Conjecture Suppose G is a fundamental group of a finite graph of groups X with finite edge groups, and H1, H2 are factor-free subgroups of G with finite ranks (a subgroup is factor-free if it intersects trivially with the conjugates to all vertex groups). Then H1 ∩ H2 also has finite rank, and r(H1 ∩ H2) 6n · r(H1)r(H2), where n is the maximum of orders of edge groups of X.

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products

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Thank you!

Alexander Zakharov Rank of intersection of free subgroups in free amalgamated products