Algorithmic problems for free-abelian times free groups Jordi - - PowerPoint PPT Presentation

Zm × Fn generalities Algorithmic problems for Zm × Fn

Algorithmic problems for free-abelian times free groups

Jordi Delgado (joint with Enric Ventura)

Universitat Polit` ecnica de Catalunya

GAGTA-7 May 29th, 2013 CCNY, New York

Zm × Fn generalities Algorithmic problems for Zm × Fn

Index

Zm × Fn generalities Algorithmic problems for Zm × Fn

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters).

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters). Consider the group G = T, X | [ T, T ⊔ X ]

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters). Consider the group G = T, X | [ T, T ⊔ X ] = Z × F.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters). Consider the group G = T, X | [ T, T ⊔ X ] = Z × F. If |T| = m, |X| = n,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters). Consider the group G = T, X | [ T, T ⊔ X ] = Z × F. If |T| = m, |X| = n, then G = Zm × Fn,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters). Consider the group G = T, X | [ T, T ⊔ X ] = Z × F. If |T| = m, |X| = n, then G = Zm × Fn, and for every word w(T, X) we have the normal form:

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters). Consider the group G = T, X | [ T, T ⊔ X ] = Z × F. If |T| = m, |X| = n, then G = Zm × Fn, and for every word w(T, X) we have the normal form: w(T, X) = ta1

1 · · · tam m u(x1, . . . , xn)

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters). Consider the group G = T, X | [ T, T ⊔ X ] = Z × F. If |T| = m, |X| = n, then G = Zm × Fn, and for every word w(T, X) we have the normal form: w(T, X) = ta1

1 · · · tam m u(x1, . . . , xn) =: ta u ,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters). Consider the group G = T, X | [ T, T ⊔ X ] = Z × F. If |T| = m, |X| = n, then G = Zm × Fn, and for every word w(T, X) we have the normal form: w(T, X) = ta1

1 · · · tam m u(x1, . . . , xn) =: ta u ,

where a = (a1, . . . , am) ∈ Zm and u ∈ Fn.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters). Consider the group G = T, X | [ T, T ⊔ X ] = Z × F. If |T| = m, |X| = n, then G = Zm × Fn, and for every word w(T, X) we have the normal form: w(T, X) = ta1

1 · · · tam m u(x1, . . . , xn) =: ta u ,

where a = (a1, . . . , am) ∈ Zm and u ∈ Fn.

Remarks

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters). Consider the group G = T, X | [ T, T ⊔ X ] = Z × F. If |T| = m, |X| = n, then G = Zm × Fn, and for every word w(T, X) we have the normal form: w(T, X) = ta1

1 · · · tam m u(x1, . . . , xn) =: ta u ,

where a = (a1, . . . , am) ∈ Zm and u ∈ Fn.

Remarks

• Zm × F1 ≃ Zm+1 × F0.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters). Consider the group G = T, X | [ T, T ⊔ X ] = Z × F. If |T| = m, |X| = n, then G = Zm × Fn, and for every word w(T, X) we have the normal form: w(T, X) = ta1

1 · · · tam m u(x1, . . . , xn) =: ta u ,

where a = (a1, . . . , am) ∈ Zm and u ∈ Fn.

Remarks

• Zm × F1 ≃ Zm+1 × F0.
• this is the only redundance for Zm × Fn,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters). Consider the group G = T, X | [ T, T ⊔ X ] = Z × F. If |T| = m, |X| = n, then G = Zm × Fn, and for every word w(T, X) we have the normal form: w(T, X) = ta1

1 · · · tam m u(x1, . . . , xn) =: ta u ,

where a = (a1, . . . , am) ∈ Zm and u ∈ Fn.

Remarks

• Zm × F1 ≃ Zm+1 × F0.
• this is the only redundance for Zm × Fn,
• We exclude the case n = 1 (without loss of generality).

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters). Consider the group G = T, X | [ T, T ⊔ X ] = Z × F. If |T| = m, |X| = n, then G = Zm × Fn, and for every word w(T, X) we have the normal form: w(T, X) = ta1

1 · · · tam m u(x1, . . . , xn) =: ta u ,

where a = (a1, . . . , am) ∈ Zm and u ∈ Fn.

Remarks

• Zm × F1 ≃ Zm+1 × F0.
• this is the only redundance for Zm × Fn,
• We exclude the case n = 1 (without loss of generality).
• Then (n = 1),

Zm × Fn generalities Algorithmic problems for Zm × Fn

Free-abelian times free groups

T = {ti} and X = {xj} disjoint sets (of letters). Consider the group G = T, X | [ T, T ⊔ X ] = Z × F. If |T| = m, |X| = n, then G = Zm × Fn, and for every word w(T, X) we have the normal form: w(T, X) = ta1

1 · · · tam m u(x1, . . . , xn) =: ta u ,

where a = (a1, . . . , am) ∈ Zm and u ∈ Fn.

Remarks

• Zm × F1 ≃ Zm+1 × F0.
• this is the only redundance for Zm × Fn,
• We exclude the case n = 1 (without loss of generality).
• Then (n = 1), the split-rank (m, n) of Zm × Fn identifies the

isomorphic class.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroups

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroups

Remark

There are subgroups H Zm × Fn of the form H ≃ Zm′ × Fn′,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroups

Remark

There are subgroups H Zm × Fn of the form H ≃ Zm′ × Fn′, for every m′ ∈ [0, m],

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroups

Remark

There are subgroups H Zm × Fn of the form H ≃ Zm′ × Fn′, for every m′ ∈ [0, m], and every n′ ∈ [0, ∞].

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroups

Remark

There are subgroups H Zm × Fn of the form H ≃ Zm′ × Fn′, for every m′ ∈ [0, m], and every n′ ∈ [0, ∞].

Proposition

Every subgroup H Zm × Fn is of the form H ≃ Zm′ × Fn′, where m′ ∈ [0, m], and n′ ∈ [0, ∞].

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroups

Remark

There are subgroups H Zm × Fn of the form H ≃ Zm′ × Fn′, for every m′ ∈ [0, m], and every n′ ∈ [0, ∞].

Proposition

Every subgroup H Zm × Fn is of the form H ≃ Zm′ × Fn′, where m′ ∈ [0, m], and n′ ∈ [0, ∞]. Concretely H = (H ∩ Zm) × Hπα

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroups

Remark

There are subgroups H Zm × Fn of the form H ≃ Zm′ × Fn′, for every m′ ∈ [0, m], and every n′ ∈ [0, ∞].

Proposition

Every subgroup H Zm × Fn is of the form H ≃ Zm′ × Fn′, where m′ ∈ [0, m], and n′ ∈ [0, ∞]. Concretely H = (H ∩ Zm) × Hπα where π: tau → u is the projection map to the free part,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroups

Remark

There are subgroups H Zm × Fn of the form H ≃ Zm′ × Fn′, for every m′ ∈ [0, m], and every n′ ∈ [0, ∞].

Proposition

Every subgroup H Zm × Fn is of the form H ≃ Zm′ × Fn′, where m′ ∈ [0, m], and n′ ∈ [0, ∞]. Concretely H = (H ∩ Zm) × Hπα where π: tau → u is the projection map to the free part, α: Hπ → H is a splitting of π|H.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroups

Remark

There are subgroups H Zm × Fn of the form H ≃ Zm′ × Fn′, for every m′ ∈ [0, m], and every n′ ∈ [0, ∞].

Proposition

Every subgroup H Zm × Fn is of the form H ≃ Zm′ × Fn′, where m′ ∈ [0, m], and n′ ∈ [0, ∞]. Concretely H = (H ∩ Zm) × Hπα where π: tau → u is the projection map to the free part, α: Hπ → H is a splitting of π|H.

Corollary

H f. g. ⇔ Hπ f. g.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Basis

Zm × Fn generalities Algorithmic problems for Zm × Fn

Basis

Given H Zm × Fn, we have H = (H ∩ Zm) × Hπα

Zm × Fn generalities Algorithmic problems for Zm × Fn

Basis

Given H Zm × Fn, we have H = (H ∩ Zm) × Hπα where H ∩ Zm Zm,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Basis

Given H Zm × Fn, we have H = (H ∩ Zm) × Hπα where H ∩ Zm Zm, and Hπα ≃ Hπ Fn.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Basis

Given H Zm × Fn, we have H = (H ∩ Zm) × Hπα where H ∩ Zm Zm, and Hπα ≃ Hπ Fn.

Definition

We say that E ⊆ G is a basis of H if E = ET ∪ EX, where

Zm × Fn generalities Algorithmic problems for Zm × Fn

Basis

Given H Zm × Fn, we have H = (H ∩ Zm) × Hπα where H ∩ Zm Zm, and Hπα ≃ Hπ Fn.

Definition

We say that E ⊆ G is a basis of H if E = ET ∪ EX, where

• ET is a basis (free-abelian) of H ∩ Zm,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Basis

Given H Zm × Fn, we have H = (H ∩ Zm) × Hπα where H ∩ Zm Zm, and Hπα ≃ Hπ Fn.

Definition

We say that E ⊆ G is a basis of H if E = ET ∪ EX, where

• ET is a basis (free-abelian) of H ∩ Zm,
• EX is a basis (free) of Hπα.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Basis

Given H Zm × Fn, we have H = (H ∩ Zm) × Hπα where H ∩ Zm Zm, and Hπα ≃ Hπ Fn.

Definition

We say that E ⊆ G is a basis of H if E = ET ∪ EX, where

• ET is a basis (free-abelian) of H ∩ Zm,
• EX is a basis (free) of Hπα.

Proposition

If H is given by a finite family of generators, then we can compute a (finite) basis for H.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Proposition

The endomorphisms of G = Zm × Fn are of the form: Ψ : (ta, u) → (taQ+uP, uφa)

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Proposition

The endomorphisms of G = Zm × Fn are of the form: Ψ : (ta, u) → (taQ+uP, uφa) where u = u ab ∈ Zn,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Proposition

The endomorphisms of G = Zm × Fn are of the form: Ψ : (ta, u) → (taQ+uP, uφa) where u = u ab ∈ Zn, Q and P are integer matrices,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Proposition

The endomorphisms of G = Zm × Fn are of the form: Ψ : (ta, u) → (taQ+uP, uφa) where u = u ab ∈ Zn, Q and P are integer matrices, and φa : Fn → Fn is either

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Proposition

The endomorphisms of G = Zm × Fn are of the form: Ψ : (ta, u) → (taQ+uP, uφa) where u = u ab ∈ Zn, Q and P are integer matrices, and φa : Fn → Fn is either (I) an endomorphism φ: Fn → Fn (independent from a),

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Proposition

The endomorphisms of G = Zm × Fn are of the form: Ψ : (ta, u) → (taQ+uP, uφa) where u = u ab ∈ Zn, Q and P are integer matrices, and φa : Fn → Fn is either (I) an endomorphism φ: Fn → Fn (independent from a), (II) a map u → wα(a,u), where 1 = w ∈ Fn is not a proper power.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Proposition

The endomorphisms of G = Zm × Fn are of the form: Ψ : (ta, u) → (taQ+uP, uφa) where u = u ab ∈ Zn, Q and P are integer matrices, and φa : Fn → Fn is either (I) an endomorphism φ: Fn → Fn (independent from a), (II) a map u → wα(a,u), where 1 = w ∈ Fn is not a proper power.

Proposition

Ψ is mono ⇔ Ψ is of type I with φ mono and Q mono.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Proposition

The endomorphisms of G = Zm × Fn are of the form: Ψ : (ta, u) → (taQ+uP, uφa) where u = u ab ∈ Zn, Q and P are integer matrices, and φa : Fn → Fn is either (I) an endomorphism φ: Fn → Fn (independent from a), (II) a map u → wα(a,u), where 1 = w ∈ Fn is not a proper power.

Proposition

Ψ is mono ⇔ Ψ is of type I with φ mono and Q mono.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Proposition

The endomorphisms of G = Zm × Fn are of the form: Ψ : (ta, u) → (taQ+uP, uφa) where u = u ab ∈ Zn, Q and P are integer matrices, and φa : Fn → Fn is either (I) an endomorphism φ: Fn → Fn (independent from a), (II) a map u → wα(a,u), where 1 = w ∈ Fn is not a proper power.

Proposition

Ψ is mono ⇔ Ψ is of type I with φ mono and Q mono. Ψ is epi ⇔ Ψ is of type I with φ epi and Q epi.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Proposition

The endomorphisms of G = Zm × Fn are of the form: Ψ : (ta, u) → (taQ+uP, uφa) where u = u ab ∈ Zn, Q and P are integer matrices, and φa : Fn → Fn is either (I) an endomorphism φ: Fn → Fn (independent from a), (II) a map u → wα(a,u), where 1 = w ∈ Fn is not a proper power.

Proposition

Ψ is mono ⇔ Ψ is of type I with φ mono and Q mono. Ψ is epi ⇔ Ψ is of type I with φ epi and Q epi.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Proposition

The endomorphisms of G = Zm × Fn are of the form: Ψ : (ta, u) → (taQ+uP, uφa) where u = u ab ∈ Zn, Q and P are integer matrices, and φa : Fn → Fn is either (I) an endomorphism φ: Fn → Fn (independent from a), (II) a map u → wα(a,u), where 1 = w ∈ Fn is not a proper power.

Proposition

Ψ is mono ⇔ Ψ is of type I with φ mono and Q mono. Ψ is epi ⇔ Ψ is of type I with φ epi and Q epi.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Proposition

The endomorphisms of G = Zm × Fn are of the form: Ψ : (ta, u) → (taQ+uP, uφa) where u = u ab ∈ Zn, Q and P are integer matrices, and φa : Fn → Fn is either (I) an endomorphism φ: Fn → Fn (independent from a), (II) a map u → wα(a,u), where 1 = w ∈ Fn is not a proper power.

Proposition

Ψ is mono ⇔ Ψ is of type I with φ mono and Q mono. Ψ is epi ⇔ Ψ is of type I with φ epi and Q epi.

Corollary

Fn × Zm is hopfian (and not cohopfian).

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Proposition

The endomorphisms of G = Zm × Fn are of the form: Ψ : (ta, u) → (taQ+uP, uφa) where u = u ab ∈ Zn, Q and P are integer matrices, and φa : Fn → Fn is either (I) an endomorphism φ: Fn → Fn (independent from a), (II) a map u → wα(a,u), where 1 = w ∈ Fn is not a proper power.

Proposition

Ψ is mono ⇔ Ψ is of type I with φ mono and Q mono. Ψ is auto ⇔ Ψ is of type I with φ auto and Q auto.

Corollary

Fn × Zm is hopfian (and not cohopfian).

Zm × Fn generalities Algorithmic problems for Zm × Fn

Endomorphisms

Proposition

The endomorphisms of G = Zm × Fn are of the form: Ψ : (ta, u) → (taQ+uP, uφa) where u = u ab ∈ Zn, Q and P are integer matrices, and φa : Fn → Fn is either (I) an endomorphism φ: Fn → Fn (independent from a), (II) a map u → wα(a,u), where 1 = w ∈ Fn is not a proper power.

Proposition

Ψ is mono ⇔ Ψ is of type I with φ mono and Q mono. Ψ is auto ⇔ Ψ is of type I with φ auto and Q auto.

Corollary

Fn × Zm is hopfian (and not cohopfian).

Zm × Fn generalities Algorithmic problems for Zm × Fn

Index

Zm × Fn generalities Algorithmic problems for Zm × Fn

Zm × Fn generalities Algorithmic problems for Zm × Fn

Dehn and Membership problems

Zm × Fn generalities Algorithmic problems for Zm × Fn

Dehn and Membership problems

Proposition

• The Word Problem WP(Zm × Fn) is solvable.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Dehn and Membership problems

Proposition

• The Word Problem WP(Zm × Fn) is solvable.
• The Conjugacy Problem CP(Zm × Fn) is solvable.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Dehn and Membership problems

Proposition

• The Word Problem WP(Zm × Fn) is solvable.
• The Conjugacy Problem CP(Zm × Fn) is solvable.
• The Isomorphism Problem IP(Zm × Fn) is solvable.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Dehn and Membership problems

Proposition

• The Word Problem WP(Zm × Fn) is solvable.
• The Conjugacy Problem CP(Zm × Fn) is solvable.
• The Isomorphism Problem IP(Zm × Fn) is solvable.

Membership Problem, MP(G)

Given g0, g1, . . . , gk ∈ G, decide whether g0 ∈ g1, . . . , gk; if so, compute g0 = w(g1, . . . , gk).

Zm × Fn generalities Algorithmic problems for Zm × Fn

Dehn and Membership problems

Proposition

• The Word Problem WP(Zm × Fn) is solvable.
• The Conjugacy Problem CP(Zm × Fn) is solvable.
• The Isomorphism Problem IP(Zm × Fn) is solvable.

Membership Problem, MP(G)

Given g0, g1, . . . , gk ∈ G, decide whether g0 ∈ g1, . . . , gk; if so, compute g0 = w(g1, . . . , gk).

Proposition

The Membership Problem MP(Zm × Fn) is solvable.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Whitehead Problems

Zm × Fn generalities Algorithmic problems for Zm × Fn

Whitehead Problems

Given g, h ∈ G, decide whether there exist an endo (mono, auto)

• f G sending g to h.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Whitehead Problems

Given g, h ∈ G, decide whether there exist an endo (mono, auto)

• f G sending g to h.

In general, given F ⊆ End G,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Whitehead Problems

Given g, h ∈ G, decide whether there exist an endo (mono, auto)

• f G sending g to h.

In general, given F ⊆ End G, WhP(G, F) ≡ ¿∃ϕ ∈ F | gϕ = h? (g,h∈G)

Zm × Fn generalities Algorithmic problems for Zm × Fn

Whitehead Problems

Given g, h ∈ G, decide whether there exist an endo (mono, auto)

• f G sending g to h.

In general, given F ⊆ End G, WhP(G, F) ≡ ¿∃ϕ ∈ F | gϕ = h? (g,h∈G)

Theorem

• WhP(Fn, End Fn) is solvable (Makanin, 1982).

Zm × Fn generalities Algorithmic problems for Zm × Fn

Whitehead Problems

Given g, h ∈ G, decide whether there exist an endo (mono, auto)

• f G sending g to h.

In general, given F ⊆ End G, WhP(G, F) ≡ ¿∃ϕ ∈ F | gϕ = h? (g,h∈G)

Theorem

• WhP(Fn, End Fn) is solvable (Makanin, 1982).
• WhP(Fn, Mon Fn) is solvable (Ciobanu and Houcine, 2010).

Zm × Fn generalities Algorithmic problems for Zm × Fn

Whitehead Problems

Given g, h ∈ G, decide whether there exist an endo (mono, auto)

• f G sending g to h.

In general, given F ⊆ End G, WhP(G, F) ≡ ¿∃ϕ ∈ F | gϕ = h? (g,h∈G)

Theorem

• WhP(Fn, End Fn) is solvable (Makanin, 1982).
• WhP(Fn, Mon Fn) is solvable (Ciobanu and Houcine, 2010).
• WhP(Fn, Aut Fn) is solvable (Whitehead, 1936).

Zm × Fn generalities Algorithmic problems for Zm × Fn

Whitehead Problems

Given g, h ∈ G, decide whether there exist an endo (mono, auto)

• f G sending g to h.

In general, given F ⊆ End G, WhP(G, F) ≡ ¿∃ϕ ∈ F | gϕ = h? (g,h∈G)

Theorem

• WhP(Fn, End Fn) is solvable (Makanin, 1982).
• WhP(Fn, Mon Fn) is solvable (Ciobanu and Houcine, 2010).
• WhP(Fn, Aut Fn) is solvable (Whitehead, 1936).

Theorem

Whitehead problems for endos, monos and autos are solvable in Zm × Fn.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroup and Coset Intersection Problems

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroup and Coset Intersection Problems

Remark

Zm and Fn are Howson, but Zm × Fn is not.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroup and Coset Intersection Problems

Remark

Zm and Fn are Howson, but Zm × Fn is not.

Example (Moldavanski)

H1 = a, b, H2 = ta, b Z × F2 = t | × a, b | .

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroup and Coset Intersection Problems

Remark

Zm and Fn are Howson, but Zm × Fn is not.

Example (Moldavanski)

H1 = a, b, H2 = ta, b Z × F2 = t | × a, b | .

Subgroup Intersection Problem, SIP(G)

Given f.g. subgroups H, H′ ≤f.g. G, decide whether H ∩ H′ is f.g.; if so, compute a generating set.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroup and Coset Intersection Problems

Remark

Zm and Fn are Howson, but Zm × Fn is not.

Example (Moldavanski)

H1 = a, b, H2 = ta, b Z × F2 = t | × a, b | .

Subgroup Intersection Problem, SIP(G)

Given f.g. subgroups H, H′ ≤f.g. G, decide whether H ∩ H′ is f.g.; if so, compute a generating set.

Coset Intersection Problem, CIP(G)

Given H, H′ ≤f.g. G, and g, g′ ∈ G, decide whether gH ∩ g′H′ = ∅; if so, compute a coset representative.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Subgroup and Coset Intersection Problems

Remark

Zm and Fn are Howson, but Zm × Fn is not.

Example (Moldavanski)

H1 = a, b, H2 = ta, b Z × F2 = t | × a, b | .

Subgroup Intersection Problem, SIP(G)

Given f.g. subgroups H, H′ ≤f.g. G, decide whether H ∩ H′ is f.g.; if so, compute a generating set.

Coset Intersection Problem, CIP(G)

Given H, H′ ≤f.g. G, and g, g′ ∈ G, decide whether gH ∩ g′H′ = ∅; if so, compute a coset representative.

Theorem

SIP(Zm × Fn) and CIP(Zm × Fn) are solvable.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Finite Index Problem, FIP(G)

Zm × Fn generalities Algorithmic problems for Zm × Fn

Finite Index Problem, FIP(G)

Given a subgroup H f.g. G, decide whether [G : H] < ∞; if so, compute the index and a system of coset representatives.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Finite Index Problem, FIP(G)

Given a subgroup H f.g. G, decide whether [G : H] < ∞; if so, compute the index and a system of coset representatives.

Theorem

FIP(Zm × Fn) is solvable.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Finite Index Problem, FIP(G)

Given a subgroup H f.g. G, decide whether [G : H] < ∞; if so, compute the index and a system of coset representatives.

Theorem

FIP(Zm × Fn) is solvable.

Remark

Given a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} of H Zm × Fn,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Finite Index Problem, FIP(G)

Given a subgroup H f.g. G, decide whether [G : H] < ∞; if so, compute the index and a system of coset representatives.

Theorem

FIP(Zm × Fn) is solvable.

Remark

Given a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} of H Zm × Fn, we have

Zm × Fn generalities Algorithmic problems for Zm × Fn

Finite Index Problem, FIP(G)

Given a subgroup H f.g. G, decide whether [G : H] < ∞; if so, compute the index and a system of coset representatives.

Theorem

FIP(Zm × Fn) is solvable.

Remark

Given a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} of H Zm × Fn, we have

• {tb1, . . . , tbm′} free-abelian basis of H ∩ Zm, and

Zm × Fn generalities Algorithmic problems for Zm × Fn

Finite Index Problem, FIP(G)

Given a subgroup H f.g. G, decide whether [G : H] < ∞; if so, compute the index and a system of coset representatives.

Theorem

FIP(Zm × Fn) is solvable.

Remark

Given a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} of H Zm × Fn, we have

• {tb1, . . . , tbm′} free-abelian basis of H ∩ Zm, and
• {ta1u1, . . . , tan′un′} free basis of Hπα ≃ Hπ,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Finite Index Problem, FIP(G)

Given a subgroup H f.g. G, decide whether [G : H] < ∞; if so, compute the index and a system of coset representatives.

Theorem

FIP(Zm × Fn) is solvable.

Remark

Given a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} of H Zm × Fn, we have

• {tb1, . . . , tbm′} free-abelian basis of H ∩ Zm, and
• {ta1u1, . . . , tan′un′} free basis of Hπα ≃ Hπ,

and we will write

Zm × Fn generalities Algorithmic problems for Zm × Fn

Finite Index Problem, FIP(G)

Given a subgroup H f.g. G, decide whether [G : H] < ∞; if so, compute the index and a system of coset representatives.

Theorem

FIP(Zm × Fn) is solvable.

Remark

Given a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} of H Zm × Fn, we have

• {tb1, . . . , tbm′} free-abelian basis of H ∩ Zm, and
• {ta1u1, . . . , tan′un′} free basis of Hπα ≃ Hπ,

and we will write

• L = b1, . . . , bm′ Zm

Zm × Fn generalities Algorithmic problems for Zm × Fn

Finite Index Problem, FIP(G)

Given a subgroup H f.g. G, decide whether [G : H] < ∞; if so, compute the index and a system of coset representatives.

Theorem

FIP(Zm × Fn) is solvable.

Remark

Given a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} of H Zm × Fn, we have

• {tb1, . . . , tbm′} free-abelian basis of H ∩ Zm, and
• {ta1u1, . . . , tan′un′} free basis of Hπα ≃ Hπ,

and we will write

• L = b1, . . . , bm′ Zm (subgroup of rank m′),

Zm × Fn generalities Algorithmic problems for Zm × Fn

Finite Index Problem, FIP(G)

Given a subgroup H f.g. G, decide whether [G : H] < ∞; if so, compute the index and a system of coset representatives.

Theorem

FIP(Zm × Fn) is solvable.

Remark

Given a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} of H Zm × Fn, we have

• {tb1, . . . , tbm′} free-abelian basis of H ∩ Zm, and
• {ta1u1, . . . , tan′un′} free basis of Hπα ≃ Hπ,

and we will write

• L = b1, . . . , bm′ Zm (subgroup of rank m′),
• A =

a1 . . .

an′

• ∈ Mn′×m(Z).

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn,

Remark

• H f.i. Zm × Fn

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn,

Remark

• H f.i. Zm × Fn ⇔

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn,

Remark

• H f.i. Zm × Fn ⇔

H ∩ Zm f.i. Zm

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn,

Remark

• H f.i. Zm × Fn ⇔

H ∩ Zm f.i. Zm H ∩ Fn f.i. Fn

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn,

Remark

• H f.i. Zm × Fn ⇔

H ∩ Zm f.i. Zm H ∩ Fn f.i. Fn

• H ∩ Fn Hπ Fn,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn,

Remark

• H f.i. Zm × Fn ⇔

H ∩ Zm f.i. Zm H ∩ Fn f.i. Fn

• H ∩ Fn Hπ Fn,

where π: Zm × Fn → Fn is the projection map.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm

Remark

• H f.i. Zm × Fn ⇔

H ∩ Zm f.i. Zm H ∩ Fn f.i. Fn

• H ∩ Fn Hπ Fn,

where π: Zm × Fn → Fn is the projection map.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm 2 Hπ f.i. Fn

Remark

• H f.i. Zm × Fn ⇔

H ∩ Zm f.i. Zm H ∩ Fn f.i. Fn

• H ∩ Fn Hπ Fn,

where π: Zm × Fn → Fn is the projection map.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm 2 Hπ f.i. Fn 3 H ∩ Fn f.i. Hπ

Remark

• H f.i. Zm × Fn ⇔

H ∩ Zm f.i. Zm H ∩ Fn f.i. Fn

• H ∩ Fn Hπ Fn,

where π: Zm × Fn → Fn is the projection map.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm 2 Hπ f.i. Fn 3 H ∩ Fn f.i. Hπ

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm 2 Hπ f.i. Fn 3 H ∩ Fn f.i. Hπ

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn 3 H ∩ Fn f.i. Hπ

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn 3 H ∩ Fn f.i. Hπ

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn 3 H ∩ Fn f.i. Hπ

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L},

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L}, H ∩ Fn = {ω(u1, . . . , un′) ∈ Fn | ω ∈ Fn′ , ωA ∈ L}

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L}, H ∩ Fn = {ω(u1, . . . , un′) ∈ Fn | ω ∈ Fn′ , ωA ∈ L} Hπ H ∩ Fn

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L}, H ∩ Fn = {ω(u1, . . . , un′) ∈ Fn | ω ∈ Fn′ , ωA ∈ L} Hπ H ∩ Fn

• Fn

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L}, H ∩ Fn = {ω(u1, . . . , un′) ∈ Fn | ω ∈ Fn′ , ωA ∈ L} Hπ H ∩ Fn

• ≃
• Fn

Fn′

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L}, H ∩ Fn = {ω(u1, . . . , un′) ∈ Fn | ω ∈ Fn′ , ωA ∈ L} Hπ H ∩ Fn

• ≃
• Fn

Fn′ Zn′

ρ

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L}, H ∩ Fn = {ω(u1, . . . , un′) ∈ Fn | ω ∈ Fn′ , ωA ∈ L} Hπ H ∩ Fn

• ≃
• Fn

Fn′ Zn′ Zm

ρ A

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L}, H ∩ Fn = {ω(u1, . . . , un′) ∈ Fn | ω ∈ Fn′ , ωA ∈ L} Hπ H ∩ Fn

• ≃
• Fn

Fn′ Zn′ Zm L

ρ A

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L}, H ∩ Fn = {ω(u1, . . . , un′) ∈ Fn | ω ∈ Fn′ , ωA ∈ L} Hπ H ∩ Fn

• ≃
• Fn

Fn′ Zn′ Zm L (L)A−1

ρ A

• ✤

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L}, H ∩ Fn = {ω(u1, . . . , un′) ∈ Fn | ω ∈ Fn′ , ωA ∈ L} Hπ H ∩ Fn

• ≃
• Fn

Fn′ Zn′ Zm L (L)A−1 (L)A−1ρ−1

ρ A

• ✤
• ✤

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L}, H ∩ Fn = {ω(u1, . . . , un′) ∈ Fn | ω ∈ Fn′ , ωA ∈ L} = (L)A−1ρ−1. Hπ H ∩ Fn

• ≃

≃

• Fn

Fn′ Zn′ Zm L (L)A−1 (L)A−1ρ−1

ρ A

• ✤
• ✤

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L}, H ∩ Fn = {ω(u1, . . . , un′) ∈ Fn | ω ∈ Fn′ , ωA ∈ L} = (L)A−1ρ−1. Hπ H ∩ Fn

• ≃

≃

• Fn

Fn′ Zn′ Zm L (L)A−1 (L)A−1ρ−1

ρ A

• ✤
• ✤

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L}, H ∩ Fn = {ω(u1, . . . , un′) ∈ Fn | ω ∈ Fn′ , ωA ∈ L} = (L)A−1ρ−1. Hπ H ∩ Fn

• ≃

≃

• Fn

Fn′ Zn′ Zm L (L)A−1 (L)A−1ρ−1

ρ A

• ✤
• ✤

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ (L)A−1 f.i. Zn′ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L}, H ∩ Fn = {ω(u1, . . . , un′) ∈ Fn | ω ∈ Fn′ , ωA ∈ L} = (L)A−1ρ−1. Hπ H ∩ Fn

• ≃

≃

• Fn

Fn′ Zn′ Zm L (L)A−1 (L)A−1ρ−1

ρ A

• ✤
• ✤

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ (L)A−1 f.i. Zn′ rk((L)A−1) = n′ H = {ta ω(u1, . . . , un′) | ω ∈ Fn′, a ∈ ωA + L}, H ∩ Fn = {ω(u1, . . . , un′) ∈ Fn | ω ∈ Fn′ , ωA ∈ L} = (L)A−1ρ−1. Hπ H ∩ Fn

• ≃

≃

• Fn

Fn′ Zn′ Zm L (L)A−1 (L)A−1ρ−1

ρ A

• ✤
• ✤

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ (L)A−1 f.i. Zn′ rk((L)A−1) = n′

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ (L)A−1 f.i. Zn′ rk((L)A−1) = n′

Search Problem

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ (L)A−1 f.i. Zn′ rk((L)A−1) = n′

Search Problem

Suppose H f.i. Zm × Fn,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ (L)A−1 f.i. Zn′ rk((L)A−1) = n′

Search Problem

Suppose H f.i. Zm × Fn, then 1 , 2 and 3 hold,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ (L)A−1 f.i. Zn′ rk((L)A−1) = n′

Search Problem

Suppose H f.i. Zm × Fn, then 1 , 2 and 3 hold, and provide respective transversals {ci}i, {vj}j and {vk}k:

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ (L)A−1 f.i. Zn′ rk((L)A−1) = n′

Search Problem

Suppose H f.i. Zm × Fn, then 1 , 2 and 3 hold, and provide respective transversals {ci}i, {vj}j and {vk}k: Zm =

i ciL ,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ (L)A−1 f.i. Zn′ rk((L)A−1) = n′

Search Problem

Suppose H f.i. Zm × Fn, then 1 , 2 and 3 hold, and provide respective transversals {ci}i, {vj}j and {vk}k: Zm =

i ciL , Fn = j vj(Hπ) ,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ (L)A−1 f.i. Zn′ rk((L)A−1) = n′

Search Problem

Suppose H f.i. Zm × Fn, then 1 , 2 and 3 hold, and provide respective transversals {ci}i, {vj}j and {vk}k: Zm =

i ciL , Fn = j vj(Hπ) , and Hπ = k wk(H ∩ Fn).

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ (L)A−1 f.i. Zn′ rk((L)A−1) = n′

Search Problem

Suppose H f.i. Zm × Fn, then 1 , 2 and 3 hold, and provide respective transversals {ci}i, {vj}j and {vk}k: Zm =

i ciL , Fn = j vj(Hπ) , and Hπ = k wk(H ∩ Fn).

Then,

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ (L)A−1 f.i. Zn′ rk((L)A−1) = n′

Search Problem

Suppose H f.i. Zm × Fn, then 1 , 2 and 3 hold, and provide respective transversals {ci}i, {vj}j and {vk}k: Zm =

i ciL , Fn = j vj(Hπ) , and Hπ = k wk(H ∩ Fn).

Then, L = {tcivjwk}i,j,k is a full s.c.r for H f.i. Zm × Fn.

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ (L)A−1 f.i. Zn′ rk((L)A−1) = n′

Search Problem

Suppose H f.i. Zm × Fn, then 1 , 2 and 3 hold, and provide respective transversals {ci}i, {vj}j and {vk}k: Zm =

i ciL , Fn = j vj(Hπ) , and Hπ = k wk(H ∩ Fn).

Then, L = {tcivjwk}i,j,k is a full s.c.r for H f.i. Zm × Fn. Finally, remove redundancies from L using MP(Zm × Fn).

Zm × Fn generalities Algorithmic problems for Zm × Fn

Sketch of proof for FIP(Zm × Fn)

• Compute a basis {tb1, . . . , tbm′, ta1u1, . . . , tan′un′} for H.

Decision Problem

In order to decide whether H f.i. Zm × Fn, it is enough to decide whether: 1 H ∩ Zm f.i. Zm L = b1, . . . , bm′ f.i. Zm m′ = m 2 Hπ f.i. Fn u1, . . . , un′ f.i. Fn S(H) complete 3 H ∩ Fn f.i. Hπ (L)A−1 f.i. Zn′ rk((L)A−1) = n′

Search Problem

Suppose H f.i. Zm × Fn, then 1 , 2 and 3 hold, and provide respective transversals {ci}i, {vj}j and {vk}k: Zm =

i ciL , Fn = j vj(Hπ) , and Hπ = k wk(H ∩ Fn).

Then, L = {tcivjwk}i,j,k is a full s.c.r for H f.i. Zm × Fn. Finally, remove redundancies from L using MP(Zm × Fn).

Zm × Fn generalities Algorithmic problems for Zm × Fn

Thanks!