Fixed subgroups of automorphisms and auto-closures in free-abelian - PowerPoint PPT Presentation

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Fixed subgroups of automorphisms and auto-closures in free-abelian times free groups Mallika Roy Universitat Polit` ecnica de Catalunya Barcelona Graduate School of Mathematics Joint work with Enric Ventura GRAS - June 14, 2019

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Outline Motivations 1 Preliminaries 2 Periodic subgroup, stabilizer and auto-fixed closure Previous results about fixed subgroups of automorphisms in free groups Fixed point subgroups: F n and Z m × F n 3 Finite ordered automorphisms of Z m × F n 4 Finite ordered automorphisms and periodic points of endomorphisms of Z m × F n 5 Algorithmic computation of auto-fixed closure of a subgroup of Z m × F n Our goal and complications Main theorem and Questions

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Outline Motivations 1 Preliminaries 2 Periodic subgroup, stabilizer and auto-fixed closure Previous results about fixed subgroups of automorphisms in free groups Fixed point subgroups: F n and Z m × F n 3 Finite ordered automorphisms of Z m × F n 4 Finite ordered automorphisms and periodic points of endomorphisms of Z m × F n 5 Algorithmic computation of auto-fixed closure of a subgroup of Z m × F n Our goal and complications Main theorem and Questions

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Motivations How the fixed point subgroups of automorphisms look like in Z m × F n , while for Q ∈ GL m ( Z ) , it is E 1 ( Q ) (the eigenspace with respect to the eigenvalue 1) and for free groups, F n , it is very well studied in the literature. The dual problem of the above, i.e., given a subgroup, decide whether it can be realized as the fixed subgroup of a family of automorphisms and in the affirmative case, compute such a family of automorphisms.

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Motivations How the fixed point subgroups of automorphisms look like in Z m × F n , while for Q ∈ GL m ( Z ) , it is E 1 ( Q ) (the eigenspace with respect to the eigenvalue 1) and for free groups, F n , it is very well studied in the literature. The dual problem of the above, i.e., given a subgroup, decide whether it can be realized as the fixed subgroup of a family of automorphisms and in the affirmative case, compute such a family of automorphisms.

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Outline Motivations 1 Preliminaries 2 Periodic subgroup, stabilizer and auto-fixed closure Previous results about fixed subgroups of automorphisms in free groups Fixed point subgroups: F n and Z m × F n 3 Finite ordered automorphisms of Z m × F n 4 Finite ordered automorphisms and periodic points of endomorphisms of Z m × F n 5 Algorithmic computation of auto-fixed closure of a subgroup of Z m × F n Our goal and complications Main theorem and Questions

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Periodic subgroup and Stabilizer Periodic subgroup For an endomorphism of a group G , Ψ ∈ End( G ) , the periodic p = 1 Fix Ψ p . subgroup of Ψ is denoted as Per Ψ and Per Ψ = ∪ ∞ Stabilizer Let H � G . We denote by Aut H ( G ) the subgroup of Aut( G ) consisting of all automorphisms of G which fix H pointwise, Aut H ( G ) = { φ ∈ Aut( G ) | H � Fix φ } , usually called the (pointwise) stabilizer of H . Similar definition for End H ( G ) . Clearly, Aut H ( G ) � End H ( G ) .

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Periodic subgroup and Stabilizer Periodic subgroup For an endomorphism of a group G , Ψ ∈ End( G ) , the periodic p = 1 Fix Ψ p . subgroup of Ψ is denoted as Per Ψ and Per Ψ = ∪ ∞ Stabilizer Let H � G . We denote by Aut H ( G ) the subgroup of Aut( G ) consisting of all automorphisms of G which fix H pointwise, Aut H ( G ) = { φ ∈ Aut( G ) | H � Fix φ } , usually called the (pointwise) stabilizer of H . Similar definition for End H ( G ) . Clearly, Aut H ( G ) � End H ( G ) .

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Auto-fixed closure Auto-fixed closure The auto-closure of H in G , denoted a-Cl G ( H ) , is the group � a-Cl G ( H ) = Fix(Aut H ( G )) = Fix φ, φ ∈ Aut( G ) H � Fix φ i.e., the samallest auto-fixed subgroup of G containing H . Similarly, the endo-closure of H in G , is e-Cl G ( H ) = Fix(End H ( G )) . H is called auto-fixed if a-Cl G ( H ) = H . Example In F n = � a 1 , . . . , a n � , a-Cl F n ( � a i 2 � ) = � a i � . Note that, as Aut H ( G ) � End H ( G ) , it is obvious that e-Cl G ( H ) � a-Cl G ( H ) . A subgroup H � Z m is endo-fixed ⇔ H is auto-fixed ⇔ H is 1-endo-fixed ⇔ H is 1-auto-fixed ⇔ H � ⊕ Z m .

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Auto-fixed closure Auto-fixed closure The auto-closure of H in G , denoted a-Cl G ( H ) , is the group � a-Cl G ( H ) = Fix(Aut H ( G )) = Fix φ, φ ∈ Aut( G ) H � Fix φ i.e., the samallest auto-fixed subgroup of G containing H . Similarly, the endo-closure of H in G , is e-Cl G ( H ) = Fix(End H ( G )) . H is called auto-fixed if a-Cl G ( H ) = H . Example In F n = � a 1 , . . . , a n � , a-Cl F n ( � a i 2 � ) = � a i � . Note that, as Aut H ( G ) � End H ( G ) , it is obvious that e-Cl G ( H ) � a-Cl G ( H ) . A subgroup H � Z m is endo-fixed ⇔ H is auto-fixed ⇔ H is 1-endo-fixed ⇔ H is 1-auto-fixed ⇔ H � ⊕ Z m .

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Auto-fixed closure Auto-fixed closure The auto-closure of H in G , denoted a-Cl G ( H ) , is the group � a-Cl G ( H ) = Fix(Aut H ( G )) = Fix φ, φ ∈ Aut( G ) H � Fix φ i.e., the samallest auto-fixed subgroup of G containing H . Similarly, the endo-closure of H in G , is e-Cl G ( H ) = Fix(End H ( G )) . H is called auto-fixed if a-Cl G ( H ) = H . Example In F n = � a 1 , . . . , a n � , a-Cl F n ( � a i 2 � ) = � a i � . Note that, as Aut H ( G ) � End H ( G ) , it is obvious that e-Cl G ( H ) � a-Cl G ( H ) . A subgroup H � Z m is endo-fixed ⇔ H is auto-fixed ⇔ H is 1-endo-fixed ⇔ H is 1-auto-fixed ⇔ H � ⊕ Z m .

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Previous results about fixed subgroups of automorphisms in free groups Theorem (Bogopolski–Maslakova, ’16 ; Feingh–Handel, ’18) Let φ : F n → F n be an automorphism. Then, a free-basis for Fix φ is computable. Theorem (Culler–Imrich–Turner–Stallings, 1984, 1989, 1991) For every φ ∈ End( F n ) , we have Per φ = Fix φ ( 6 n − 6 )! . Theorem (McCool, 1975) Let H � fg F n , given by a finite set of generators. Then the stabilizer, Aut H ( F n ) , of H is also finitely generated (in fact, finitely presented), and a finite set of generators (and relations) is algorithmically computable.

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Previous results about fixed subgroups of automorphisms in free groups Theorem (Bogopolski–Maslakova, ’16 ; Feingh–Handel, ’18) Let φ : F n → F n be an automorphism. Then, a free-basis for Fix φ is computable. Theorem (Culler–Imrich–Turner–Stallings, 1984, 1989, 1991) For every φ ∈ End( F n ) , we have Per φ = Fix φ ( 6 n − 6 )! . Theorem (McCool, 1975) Let H � fg F n , given by a finite set of generators. Then the stabilizer, Aut H ( F n ) , of H is also finitely generated (in fact, finitely presented), and a finite set of generators (and relations) is algorithmically computable.

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Previous results about fixed subgroups of automorphisms in free groups Theorem (Bogopolski–Maslakova, ’16 ; Feingh–Handel, ’18) Let φ : F n → F n be an automorphism. Then, a free-basis for Fix φ is computable. Theorem (Culler–Imrich–Turner–Stallings, 1984, 1989, 1991) For every φ ∈ End( F n ) , we have Per φ = Fix φ ( 6 n − 6 )! . Theorem (McCool, 1975) Let H � fg F n , given by a finite set of generators. Then the stabilizer, Aut H ( F n ) , of H is also finitely generated (in fact, finitely presented), and a finite set of generators (and relations) is algorithmically computable.

5. Automorphisms of Z m × Fn 1. Motivations 2. Preliminaries 3. Fixed point subgroups 6. Auto-fixed closure Auto-closure in free groups Theorem (Ventura, ’11) Let H � fg F n , given by a finite set of generators. Then, a free-basis for the auto-fixed closure a-Cl F n ( H ) of H is algorithmically computable. Corollary (Ventura, ’11) It is algorithmically decidable whether a given H � fg F n is auto-fixed or not.