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

Z m × F n generalities Algorithmic problems for Z m × F n Basis Given H � Z m × F n , we have H = ( H ∩ Z m ) × H πα

Z m × F n generalities Algorithmic problems for Z m × F n Basis Given H � Z m × F n , we have H = ( H ∩ Z m ) × H πα where H ∩ Z m � Z m ,

Z m × F n generalities Algorithmic problems for Z m × F n Basis Given H � Z m × F n , we have H = ( H ∩ Z m ) × H πα where H ∩ Z m � Z m , and H πα ≃ H π � F n .

Z m × F n generalities Algorithmic problems for Z m × F n Basis Given H � Z m × F n , we have H = ( H ∩ Z m ) × H πα where H ∩ Z m � Z m , and H πα ≃ H π � F n . Definition We say that E ⊆ G is a basis of H if E = E T ∪ E X , where

Z m × F n generalities Algorithmic problems for Z m × F n Basis Given H � Z m × F n , we have H = ( H ∩ Z m ) × H πα where H ∩ Z m � Z m , and H πα ≃ H π � F n . Definition We say that E ⊆ G is a basis of H if E = E T ∪ E X , where • E T is a basis (free-abelian) of H ∩ Z m ,

Z m × F n generalities Algorithmic problems for Z m × F n Basis Given H � Z m × F n , we have H = ( H ∩ Z m ) × H πα where H ∩ Z m � Z m , and H πα ≃ H π � F n . Definition We say that E ⊆ G is a basis of H if E = E T ∪ E X , where • E T is a basis (free-abelian) of H ∩ Z m , • E X is a basis (free) of H πα .

Z m × F n generalities Algorithmic problems for Z m × F n Basis Given H � Z m × F n , we have H = ( H ∩ Z m ) × H πα where H ∩ Z m � Z m , and H πα ≃ H π � F n . Definition We say that E ⊆ G is a basis of H if E = E T ∪ E X , where • E T is a basis (free-abelian) of H ∩ Z m , • E X is a basis (free) of H πα . Proposition If H is given by a finite family of generators, then we can compute a (finite) basis for H.

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms Proposition The endomorphisms of G = Z m × F n are of the form: Ψ : ( t a , u ) �→ ( t aQ + uP , u φ a )

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms Proposition The endomorphisms of G = Z m × F n are of the form: Ψ : ( t a , u ) �→ ( t aQ + uP , u φ a ) where u = u ab ∈ Z n ,

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms Proposition The endomorphisms of G = Z m × F n are of the form: Ψ : ( t a , u ) �→ ( t aQ + uP , u φ a ) where u = u ab ∈ Z n , Q and P are integer matrices,

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms Proposition The endomorphisms of G = Z m × F n are of the form: Ψ : ( t a , u ) �→ ( t aQ + uP , u φ a ) where u = u ab ∈ Z n , Q and P are integer matrices, and φ a : F n → F n is either

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms Proposition The endomorphisms of G = Z m × F n are of the form: Ψ : ( t a , u ) �→ ( t aQ + uP , u φ a ) where u = u ab ∈ Z n , Q and P are integer matrices, and φ a : F n → F n is either (I) an endomorphism φ : F n → F n (independent from a ),

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms Proposition The endomorphisms of G = Z m × F n are of the form: Ψ : ( t a , u ) �→ ( t aQ + uP , u φ a ) where u = u ab ∈ Z n , Q and P are integer matrices, and φ a : F n → F n is either (I) an endomorphism φ : F n → F n (independent from a ), (II) a map u → w α ( a , u ) , where 1 � = w ∈ F n is not a proper power.

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms Proposition The endomorphisms of G = Z m × F n are of the form: Ψ : ( t a , u ) �→ ( t aQ + uP , u φ a ) where u = u ab ∈ Z n , Q and P are integer matrices, and φ a : F n → F n is either (I) an endomorphism φ : F n → F n (independent from a ), (II) a map u → w α ( a , u ) , where 1 � = w ∈ F n is not a proper power. Proposition Ψ is mono ⇔ Ψ is of type I with φ mono and Q mono.

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms Proposition The endomorphisms of G = Z m × F n are of the form: Ψ : ( t a , u ) �→ ( t aQ + uP , u φ a ) where u = u ab ∈ Z n , Q and P are integer matrices, and φ a : F n → F n is either (I) an endomorphism φ : F n → F n (independent from a ), (II) a map u → w α ( a , u ) , where 1 � = w ∈ F n is not a proper power. Proposition Ψ is mono ⇔ Ψ is of type I with φ mono and Q mono.

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms Proposition The endomorphisms of G = Z m × F n are of the form: Ψ : ( t a , u ) �→ ( t aQ + uP , u φ a ) where u = u ab ∈ Z n , Q and P are integer matrices, and φ a : F n → F n is either (I) an endomorphism φ : F n → F n (independent from a ), (II) a map u → w α ( a , u ) , where 1 � = w ∈ F n 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.

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms Proposition The endomorphisms of G = Z m × F n are of the form: Ψ : ( t a , u ) �→ ( t aQ + uP , u φ a ) where u = u ab ∈ Z n , Q and P are integer matrices, and φ a : F n → F n is either (I) an endomorphism φ : F n → F n (independent from a ), (II) a map u → w α ( a , u ) , where 1 � = w ∈ F n 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.

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms Proposition The endomorphisms of G = Z m × F n are of the form: Ψ : ( t a , u ) �→ ( t aQ + uP , u φ a ) where u = u ab ∈ Z n , Q and P are integer matrices, and φ a : F n → F n is either (I) an endomorphism φ : F n → F n (independent from a ), (II) a map u → w α ( a , u ) , where 1 � = w ∈ F n 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.

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms Proposition The endomorphisms of G = Z m × F n are of the form: Ψ : ( t a , u ) �→ ( t aQ + uP , u φ a ) where u = u ab ∈ Z n , Q and P are integer matrices, and φ a : F n → F n is either (I) an endomorphism φ : F n → F n (independent from a ), (II) a map u → w α ( a , u ) , where 1 � = w ∈ F n 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 F n × Z m is hopfian (and not cohopfian).

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms Proposition The endomorphisms of G = Z m × F n are of the form: Ψ : ( t a , u ) �→ ( t aQ + uP , u φ a ) where u = u ab ∈ Z n , Q and P are integer matrices, and φ a : F n → F n is either (I) an endomorphism φ : F n → F n (independent from a ), (II) a map u → w α ( a , u ) , where 1 � = w ∈ F n 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 F n × Z m is hopfian (and not cohopfian).

Z m × F n generalities Algorithmic problems for Z m × F n Endomorphisms Proposition The endomorphisms of G = Z m × F n are of the form: Ψ : ( t a , u ) �→ ( t aQ + uP , u φ a ) where u = u ab ∈ Z n , Q and P are integer matrices, and φ a : F n → F n is either (I) an endomorphism φ : F n → F n (independent from a ), (II) a map u → w α ( a , u ) , where 1 � = w ∈ F n 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 F n × Z m is hopfian (and not cohopfian).

Z m × F n generalities Algorithmic problems for Z m × F n Index Z m × F n generalities Algorithmic problems for Z m × F n

Z m × F n generalities Algorithmic problems for Z m × F n Dehn and Membership problems

Z m × F n generalities Algorithmic problems for Z m × F n Dehn and Membership problems Proposition • The Word Problem WP ( Z m × F n ) is solvable.

Z m × F n generalities Algorithmic problems for Z m × F n Dehn and Membership problems Proposition • The Word Problem WP ( Z m × F n ) is solvable. • The Conjugacy Problem CP ( Z m × F n ) is solvable.

Z m × F n generalities Algorithmic problems for Z m × F n Dehn and Membership problems Proposition • The Word Problem WP ( Z m × F n ) is solvable. • The Conjugacy Problem CP ( Z m × F n ) is solvable. • The Isomorphism Problem IP ( Z m × F n ) is solvable.

Z m × F n generalities Algorithmic problems for Z m × F n Dehn and Membership problems Proposition • The Word Problem WP ( Z m × F n ) is solvable. • The Conjugacy Problem CP ( Z m × F n ) is solvable. • The Isomorphism Problem IP ( Z m × F n ) is solvable. Membership Problem, MP( G ) Given g 0 , g 1 , . . . , g k ∈ G , decide whether g 0 ∈ � g 1 , . . . , g k � ; if so, compute g 0 = w ( g 1 , . . . , g k ).

Z m × F n generalities Algorithmic problems for Z m × F n Dehn and Membership problems Proposition • The Word Problem WP ( Z m × F n ) is solvable. • The Conjugacy Problem CP ( Z m × F n ) is solvable. • The Isomorphism Problem IP ( Z m × F n ) is solvable. Membership Problem, MP( G ) Given g 0 , g 1 , . . . , g k ∈ G , decide whether g 0 ∈ � g 1 , . . . , g k � ; if so, compute g 0 = w ( g 1 , . . . , g k ). Proposition The Membership Problem MP ( Z m × F n ) is solvable.

Z m × F n generalities Algorithmic problems for Z m × F n Whitehead Problems

Z m × F n generalities Algorithmic problems for Z m × F n Whitehead Problems Given g , h ∈ G , decide whether there exist an endo (mono, auto) of G sending g to h .

Z m × F n generalities Algorithmic problems for Z m × F n Whitehead Problems Given g , h ∈ G , decide whether there exist an endo (mono, auto) of G sending g to h . In general, given F ⊆ End G ,

Z m × F n generalities Algorithmic problems for Z m × F n Whitehead Problems Given g , h ∈ G , decide whether there exist an endo (mono, auto) of G sending g to h . In general, given F ⊆ End G , WhP( G , F ) ≡ ¿ ∃ ϕ ∈ F | g ϕ = h ? ( g , h ∈ G )

Z m × F n generalities Algorithmic problems for Z m × F n Whitehead Problems Given g , h ∈ G , decide whether there exist an endo (mono, auto) of G sending g to h . In general, given F ⊆ End G , WhP( G , F ) ≡ ¿ ∃ ϕ ∈ F | g ϕ = h ? ( g , h ∈ G ) Theorem • WhP( F n , End F n ) is solvable (Makanin, 1982) .

Z m × F n generalities Algorithmic problems for Z m × F n Whitehead Problems Given g , h ∈ G , decide whether there exist an endo (mono, auto) of G sending g to h . In general, given F ⊆ End G , WhP( G , F ) ≡ ¿ ∃ ϕ ∈ F | g ϕ = h ? ( g , h ∈ G ) Theorem • WhP( F n , End F n ) is solvable (Makanin, 1982) . • WhP( F n , Mon F n ) is solvable (Ciobanu and Houcine, 2010) .

Z m × F n generalities Algorithmic problems for Z m × F n Whitehead Problems Given g , h ∈ G , decide whether there exist an endo (mono, auto) of G sending g to h . In general, given F ⊆ End G , WhP( G , F ) ≡ ¿ ∃ ϕ ∈ F | g ϕ = h ? ( g , h ∈ G ) Theorem • WhP( F n , End F n ) is solvable (Makanin, 1982) . • WhP( F n , Mon F n ) is solvable (Ciobanu and Houcine, 2010) . • WhP( F n , Aut F n ) is solvable (Whitehead, 1936) .

Z m × F n generalities Algorithmic problems for Z m × F n Whitehead Problems Given g , h ∈ G , decide whether there exist an endo (mono, auto) of G sending g to h . In general, given F ⊆ End G , WhP( G , F ) ≡ ¿ ∃ ϕ ∈ F | g ϕ = h ? ( g , h ∈ G ) Theorem • WhP( F n , End F n ) is solvable (Makanin, 1982) . • WhP( F n , Mon F n ) is solvable (Ciobanu and Houcine, 2010) . • WhP( F n , Aut F n ) is solvable (Whitehead, 1936) . Theorem Whitehead problems for endos, monos and autos are solvable in Z m × F n .

Z m × F n generalities Algorithmic problems for Z m × F n Subgroup and Coset Intersection Problems

Z m × F n generalities Algorithmic problems for Z m × F n Subgroup and Coset Intersection Problems Remark Z m and F n are Howson, but Z m × F n is not.

Z m × F n generalities Algorithmic problems for Z m × F n Subgroup and Coset Intersection Problems Remark Z m and F n are Howson, but Z m × F n is not. Example ( Moldavanski ) H 1 = � a , b � , H 2 = � ta , b � � Z × F 2 = � t | � × � a , b | � .

Z m × F n generalities Algorithmic problems for Z m × F n Subgroup and Coset Intersection Problems Remark Z m and F n are Howson, but Z m × F n is not. Example ( Moldavanski ) H 1 = � a , b � , H 2 = � ta , b � � Z × F 2 = � 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.

Z m × F n generalities Algorithmic problems for Z m × F n Subgroup and Coset Intersection Problems Remark Z m and F n are Howson, but Z m × F n is not. Example ( Moldavanski ) H 1 = � a , b � , H 2 = � ta , b � � Z × F 2 = � 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.

Z m × F n generalities Algorithmic problems for Z m × F n Subgroup and Coset Intersection Problems Remark Z m and F n are Howson, but Z m × F n is not. Example ( Moldavanski ) H 1 = � a , b � , H 2 = � ta , b � � Z × F 2 = � 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( Z m × F n ) and CIP( Z m × F n ) are solvable.

Z m × F n generalities Algorithmic problems for Z m × F n Finite Index Problem, FIP( G )

Z m × F n generalities Algorithmic problems for Z m × F n 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.

Z m × F n generalities Algorithmic problems for Z m × F n 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( Z m × F n ) is solvable.

Z m × F n generalities Algorithmic problems for Z m × F n 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( Z m × F n ) is solvable. Remark Given a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } of H � Z m × F n ,

Z m × F n generalities Algorithmic problems for Z m × F n 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( Z m × F n ) is solvable. Remark Given a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } of H � Z m × F n , we have

Z m × F n generalities Algorithmic problems for Z m × F n 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( Z m × F n ) is solvable. Remark Given a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } of H � Z m × F n , we have • { t b 1 , . . . , t b m ′ } free-abelian basis of H ∩ Z m , and

Z m × F n generalities Algorithmic problems for Z m × F n 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( Z m × F n ) is solvable. Remark Given a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } of H � Z m × F n , we have • { t b 1 , . . . , t b m ′ } free-abelian basis of H ∩ Z m , and • { t a 1 u 1 , . . . , t a n ′ u n ′ } free basis of H πα ≃ H π ,

Z m × F n generalities Algorithmic problems for Z m × F n 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( Z m × F n ) is solvable. Remark Given a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } of H � Z m × F n , we have • { t b 1 , . . . , t b m ′ } free-abelian basis of H ∩ Z m , and • { t a 1 u 1 , . . . , t a n ′ u n ′ } free basis of H πα ≃ H π , and we will write

Z m × F n generalities Algorithmic problems for Z m × F n 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( Z m × F n ) is solvable. Remark Given a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } of H � Z m × F n , we have • { t b 1 , . . . , t b m ′ } free-abelian basis of H ∩ Z m , and • { t a 1 u 1 , . . . , t a n ′ u n ′ } free basis of H πα ≃ H π , and we will write • L = � b 1 , . . . , b m ′ � � Z m

Z m × F n generalities Algorithmic problems for Z m × F n 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( Z m × F n ) is solvable. Remark Given a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } of H � Z m × F n , we have • { t b 1 , . . . , t b m ′ } free-abelian basis of H ∩ Z m , and • { t a 1 u 1 , . . . , t a n ′ u n ′ } free basis of H πα ≃ H π , and we will write • L = � b 1 , . . . , b m ′ � � Z m (subgroup of rank m ′ ),

Z m × F n generalities Algorithmic problems for Z m × F n 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( Z m × F n ) is solvable. Remark Given a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } of H � Z m × F n , we have • { t b 1 , . . . , t b m ′ } free-abelian basis of H ∩ Z m , and • { t a 1 u 1 , . . . , t a n ′ u n ′ } free basis of H πα ≃ H π , and we will write • L = � b 1 , . . . , b m ′ � � Z m (subgroup of rank m ′ ), � a 1 � . . • A = ∈ M n ′ × m ( Z ). . a n ′

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n )

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H .

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n ,

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , Remark • H � f . i . Z m × F n

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , Remark � • H � f . i . Z m × F n ⇔

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , Remark � H ∩ Z m � f . i . Z m • H � f . i . Z m × F n ⇔

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , Remark � H ∩ Z m � f . i . Z m • H � f . i . Z m × F n ⇔ H ∩ F n � f . i . F n

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , Remark � H ∩ Z m � f . i . Z m • H � f . i . Z m × F n ⇔ H ∩ F n � f . i . F n • H ∩ F n � H π � F n ,

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , Remark � H ∩ Z m � f . i . Z m • H � f . i . Z m × F n ⇔ H ∩ F n � f . i . F n • H ∩ F n � H π � F n , where π : Z m × F n → F n is the projection map.

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , it is enough to decide whether: 1 H ∩ Z m � f . i . Z m Remark � H ∩ Z m � f . i . Z m • H � f . i . Z m × F n ⇔ H ∩ F n � f . i . F n • H ∩ F n � H π � F n , where π : Z m × F n → F n is the projection map.

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , it is enough to decide whether: 1 H ∩ Z m � f . i . Z m 2 H π � f . i . F n Remark � H ∩ Z m � f . i . Z m • H � f . i . Z m × F n ⇔ H ∩ F n � f . i . F n • H ∩ F n � H π � F n , where π : Z m × F n → F n is the projection map.

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , it is enough to decide whether: 1 H ∩ Z m � f . i . Z m 2 H π � f . i . F n 3 H ∩ F n � f . i . H π Remark � H ∩ Z m � f . i . Z m • H � f . i . Z m × F n ⇔ H ∩ F n � f . i . F n • H ∩ F n � H π � F n , where π : Z m × F n → F n is the projection map.

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , it is enough to decide whether: 1 H ∩ Z m � f . i . Z m 2 H π � f . i . F n 3 H ∩ F n � f . i . H π

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , it is enough to decide whether: 1 H ∩ Z m � f . i . Z m � L = � b 1 , . . . , b m ′ � � f . i . Z m 2 H π � f . i . F n 3 H ∩ F n � f . i . H π

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , it is enough to decide whether: 1 H ∩ Z m � f . i . Z m � L = � b 1 , . . . , b m ′ � � f . i . Z m � m ′ = m 2 H π � f . i . F n 3 H ∩ F n � f . i . H π

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , it is enough to decide whether: 1 H ∩ Z m � f . i . Z m � L = � b 1 , . . . , b m ′ � � f . i . Z m � m ′ = m � 2 H π � f . i . F n 3 H ∩ F n � f . i . H π

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , it is enough to decide whether: 1 H ∩ Z m � f . i . Z m � L = � b 1 , . . . , b m ′ � � f . i . Z m � m ′ = m � 2 H π � f . i . F n � � u 1 , . . . , u n ′ � � f . i . F n 3 H ∩ F n � f . i . H π

Z m × F n generalities Algorithmic problems for Z m × F n Sketch of proof for FIP( Z m × F n ) • Compute a basis { t b 1 , . . . , t b m ′ , t a 1 u 1 , . . . , t a n ′ u n ′ } for H . Decision Problem In order to decide whether H � f . i . Z m × F n , it is enough to decide whether: 1 H ∩ Z m � f . i . Z m � L = � b 1 , . . . , b m ′ � � f . i . Z m � m ′ = m � 2 H π � f . i . F n � � u 1 , . . . , u n ′ � � f . i . F n � S ( H ) complete 3 H ∩ F n � f . i . H π