The Submonoid Membership Problem for Groups Benjamin Steinberg 1 - - PowerPoint PPT Presentation

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The Submonoid Membership Problem for Groups Benjamin Steinberg 1 - - PowerPoint PPT Presentation

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The Submonoid Membership Problem for Groups Benjamin Steinberg 1 City College of New York bsteinberg@ccny.cuny.edu http://www.sci.ccny.cuny.edu/ benjamin/ June 22, 2013 1 Encompasses joint work with Mark Kambites, Markus Lohrey, Pedro Silva

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SLIDE 1

The Submonoid Membership Problem for Groups Benjamin Steinberg1

City College of New York bsteinberg@ccny.cuny.edu http://www.sci.ccny.cuny.edu/∼benjamin/ June 22, 2013

1Encompasses joint work with Mark Kambites, Markus Lohrey, Pedro Silva

and Georg Zetzsche

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SLIDE 2

Integer programming

  • Integer Programming:
  • Given A ∈ Mmn(Z) and b ∈ Zm, does Ax = b have a solution

x ∈ Nn?

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SLIDE 3

Integer programming

  • Integer Programming:
  • Given A ∈ Mmn(Z) and b ∈ Zm, does Ax = b have a solution

x ∈ Nn?

  • Algebraically speaking, the problem is to determine whether b

is a non-negative linear combination of the columns of A.

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SLIDE 4

Integer programming

  • Integer Programming:
  • Given A ∈ Mmn(Z) and b ∈ Zm, does Ax = b have a solution

x ∈ Nn?

  • Algebraically speaking, the problem is to determine whether b

is a non-negative linear combination of the columns of A.

  • In other words, does b belong to the submonoid of Zm

generated by the columns of A?

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SLIDE 5

Integer programming

  • Integer Programming:
  • Given A ∈ Mmn(Z) and b ∈ Zm, does Ax = b have a solution

x ∈ Nn?

  • Algebraically speaking, the problem is to determine whether b

is a non-negative linear combination of the columns of A.

  • In other words, does b belong to the submonoid of Zm

generated by the columns of A?

  • So integer programming is the submonoid membership

problem for abelian groups.

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SLIDE 6

Integer programming

  • Integer Programming:
  • Given A ∈ Mmn(Z) and b ∈ Zm, does Ax = b have a solution

x ∈ Nn?

  • Algebraically speaking, the problem is to determine whether b

is a non-negative linear combination of the columns of A.

  • In other words, does b belong to the submonoid of Zm

generated by the columns of A?

  • So integer programming is the submonoid membership

problem for abelian groups.

  • Integer programming is well known to be NP-complete.
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SLIDE 7

Integer programming

  • Integer Programming:
  • Given A ∈ Mmn(Z) and b ∈ Zm, does Ax = b have a solution

x ∈ Nn?

  • Algebraically speaking, the problem is to determine whether b

is a non-negative linear combination of the columns of A.

  • In other words, does b belong to the submonoid of Zm

generated by the columns of A?

  • So integer programming is the submonoid membership

problem for abelian groups.

  • Integer programming is well known to be NP-complete.
  • The submonoid membership problem for arbitrary groups is a

non-commutative analogue of integer programming.

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SLIDE 8

Some decision problems

  • Fix a group G and a finite symmetric generating set Σ.
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SLIDE 9

Some decision problems

  • Fix a group G and a finite symmetric generating set Σ.
  • Let π: Σ∗ → G be the canonical projection.
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SLIDE 10

Some decision problems

  • Fix a group G and a finite symmetric generating set Σ.
  • Let π: Σ∗ → G be the canonical projection.
  • Consider the following algorithmic problems for G.
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SLIDE 11

Some decision problems

  • Fix a group G and a finite symmetric generating set Σ.
  • Let π: Σ∗ → G be the canonical projection.
  • Consider the following algorithmic problems for G.
  • The Word Problem:
  • Given w ∈ Σ∗, does π(w) = 1?
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SLIDE 12

Some decision problems

  • Fix a group G and a finite symmetric generating set Σ.
  • Let π: Σ∗ → G be the canonical projection.
  • Consider the following algorithmic problems for G.
  • The Word Problem:
  • Given w ∈ Σ∗, does π(w) = 1?
  • The (Uniform) Generalized Word Problem:
  • Given w, w1, . . . , wn ∈ Σ∗, is π(w) ∈ π(w1), . . . , π(wn)?
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SLIDE 13

Some decision problems

  • Fix a group G and a finite symmetric generating set Σ.
  • Let π: Σ∗ → G be the canonical projection.
  • Consider the following algorithmic problems for G.
  • The Word Problem:
  • Given w ∈ Σ∗, does π(w) = 1?
  • The (Uniform) Generalized Word Problem:
  • Given w, w1, . . . , wn ∈ Σ∗, is π(w) ∈ π(w1), . . . , π(wn)?
  • The (Uniform) Submonoid Membership Problem:
  • Given w, w1, . . . , wn ∈ Σ∗, is π(w) ∈ π({w1, . . . , wn}∗)?
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SLIDE 14

Some decision problems

  • Fix a group G and a finite symmetric generating set Σ.
  • Let π: Σ∗ → G be the canonical projection.
  • Consider the following algorithmic problems for G.
  • The Word Problem:
  • Given w ∈ Σ∗, does π(w) = 1?
  • The (Uniform) Generalized Word Problem:
  • Given w, w1, . . . , wn ∈ Σ∗, is π(w) ∈ π(w1), . . . , π(wn)?
  • The (Uniform) Submonoid Membership Problem:
  • Given w, w1, . . . , wn ∈ Σ∗, is π(w) ∈ π({w1, . . . , wn}∗)?
  • The (Uniform) Rational Subset Membership

Problem:

  • Given w ∈ Σ∗ and a finite automaton A over Σ, is

π(w) ∈ π(L(A ))?

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SLIDE 15

Some decision problems

  • Fix a group G and a finite symmetric generating set Σ.
  • Let π: Σ∗ → G be the canonical projection.
  • Consider the following algorithmic problems for G.
  • The Word Problem:
  • Given w ∈ Σ∗, does π(w) = 1?
  • The (Uniform) Generalized Word Problem:
  • Given w, w1, . . . , wn ∈ Σ∗, is π(w) ∈ π(w1), . . . , π(wn)?
  • The (Uniform) Submonoid Membership Problem:
  • Given w, w1, . . . , wn ∈ Σ∗, is π(w) ∈ π({w1, . . . , wn}∗)?
  • The (Uniform) Rational Subset Membership

Problem:

  • Given w ∈ Σ∗ and a finite automaton A over Σ, is

π(w) ∈ π(L(A ))?

  • Decidability of these problems is independent of Σ.
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SLIDE 16

The generalized word problem

  • The above decision problems were listed in order of difficulty.
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SLIDE 17

The generalized word problem

  • The above decision problems were listed in order of difficulty.
  • It is natural to search for groups distinguishing these problems.
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SLIDE 18

The generalized word problem

  • The above decision problems were listed in order of difficulty.
  • It is natural to search for groups distinguishing these problems.
  • F2 × F2 has undecidable generalized word problem (Mihailova)
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SLIDE 19

The generalized word problem

  • The above decision problems were listed in order of difficulty.
  • It is natural to search for groups distinguishing these problems.
  • F2 × F2 has undecidable generalized word problem (Mihailova)
  • Free solvable groups of derived length ≥ 3 and rank ≥ 2 have

undecidable generalized word problem (Umirbaev).

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SLIDE 20

The generalized word problem

  • The above decision problems were listed in order of difficulty.
  • It is natural to search for groups distinguishing these problems.
  • F2 × F2 has undecidable generalized word problem (Mihailova)
  • Free solvable groups of derived length ≥ 3 and rank ≥ 2 have

undecidable generalized word problem (Umirbaev).

  • Compare: all finitely generated metabelian groups have

decidable generalized word problem (Romanovskii).

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SLIDE 21

The generalized word problem

  • The above decision problems were listed in order of difficulty.
  • It is natural to search for groups distinguishing these problems.
  • F2 × F2 has undecidable generalized word problem (Mihailova)
  • Free solvable groups of derived length ≥ 3 and rank ≥ 2 have

undecidable generalized word problem (Umirbaev).

  • Compare: all finitely generated metabelian groups have

decidable generalized word problem (Romanovskii).

  • The Rips construction produces hyperbolic groups with

undecidable generalized word problem.

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SLIDE 22

Intermezzo: Finite automata

  • A finite automaton A over an alphabet Σ consists of:
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SLIDE 23

Intermezzo: Finite automata

  • A finite automaton A over an alphabet Σ consists of:
  • a finite directed graph with edges labeled by elements of Σ;
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SLIDE 24

Intermezzo: Finite automata

  • A finite automaton A over an alphabet Σ consists of:
  • a finite directed graph with edges labeled by elements of Σ;
  • a distinguished initial vertex;
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SLIDE 25

Intermezzo: Finite automata

  • A finite automaton A over an alphabet Σ consists of:
  • a finite directed graph with edges labeled by elements of Σ;
  • a distinguished initial vertex;
  • a set of final vertices.
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SLIDE 26

Intermezzo: Finite automata

  • A finite automaton A over an alphabet Σ consists of:
  • a finite directed graph with edges labeled by elements of Σ;
  • a distinguished initial vertex;
  • a set of final vertices.
  • The language L(A ) of the automaton consists of all words

labeling a path from the initial vertex to a final vertex.

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SLIDE 27

Intermezzo: Finite automata

  • A finite automaton A over an alphabet Σ consists of:
  • a finite directed graph with edges labeled by elements of Σ;
  • a distinguished initial vertex;
  • a set of final vertices.
  • The language L(A ) of the automaton consists of all words

labeling a path from the initial vertex to a final vertex.

  • A language is called rational if it is accepted by some finite

automaton.

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SLIDE 28

Intermezzo: Finite automata

  • A finite automaton A over an alphabet Σ consists of:
  • a finite directed graph with edges labeled by elements of Σ;
  • a distinguished initial vertex;
  • a set of final vertices.
  • The language L(A ) of the automaton consists of all words

labeling a path from the initial vertex to a final vertex.

  • A language is called rational if it is accepted by some finite

automaton.

  • Examples:
  • The language of geodesic words in a hyperbolic group;
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SLIDE 29

Intermezzo: Finite automata

  • A finite automaton A over an alphabet Σ consists of:
  • a finite directed graph with edges labeled by elements of Σ;
  • a distinguished initial vertex;
  • a set of final vertices.
  • The language L(A ) of the automaton consists of all words

labeling a path from the initial vertex to a final vertex.

  • A language is called rational if it is accepted by some finite

automaton.

  • Examples:
  • The language of geodesic words in a hyperbolic group;
  • The language of geodesic words belonging to a quasiconvex

subgroup of a hyperbolic group.

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SLIDE 30

Rational subsets

  • Let Rat(G) be the collection of rational subsets of G, i.e.,

sets of the form π(L(A )) with A a finite automaton.

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SLIDE 31

Rational subsets

  • Let Rat(G) be the collection of rational subsets of G, i.e.,

sets of the form π(L(A )) with A a finite automaton.

  • Rat(G) is the smallest collection of subsets of G containing

the finite subsets and closed under:

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SLIDE 32

Rational subsets

  • Let Rat(G) be the collection of rational subsets of G, i.e.,

sets of the form π(L(A )) with A a finite automaton.

  • Rat(G) is the smallest collection of subsets of G containing

the finite subsets and closed under:

  • union;
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SLIDE 33

Rational subsets

  • Let Rat(G) be the collection of rational subsets of G, i.e.,

sets of the form π(L(A )) with A a finite automaton.

  • Rat(G) is the smallest collection of subsets of G containing

the finite subsets and closed under:

  • union;
  • product;
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SLIDE 34

Rational subsets

  • Let Rat(G) be the collection of rational subsets of G, i.e.,

sets of the form π(L(A )) with A a finite automaton.

  • Rat(G) is the smallest collection of subsets of G containing

the finite subsets and closed under:

  • union;
  • product;
  • generation of submonoids X → X∗.
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SLIDE 35

Rational subsets

  • Let Rat(G) be the collection of rational subsets of G, i.e.,

sets of the form π(L(A )) with A a finite automaton.

  • Rat(G) is the smallest collection of subsets of G containing

the finite subsets and closed under:

  • union;
  • product;
  • generation of submonoids X → X∗.
  • Examples:
  • finitely generated subgroups;
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SLIDE 36

Rational subsets

  • Let Rat(G) be the collection of rational subsets of G, i.e.,

sets of the form π(L(A )) with A a finite automaton.

  • Rat(G) is the smallest collection of subsets of G containing

the finite subsets and closed under:

  • union;
  • product;
  • generation of submonoids X → X∗.
  • Examples:
  • finitely generated subgroups;
  • finitely generated submonoids;
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SLIDE 37

Rational subsets

  • Let Rat(G) be the collection of rational subsets of G, i.e.,

sets of the form π(L(A )) with A a finite automaton.

  • Rat(G) is the smallest collection of subsets of G containing

the finite subsets and closed under:

  • union;
  • product;
  • generation of submonoids X → X∗.
  • Examples:
  • finitely generated subgroups;
  • finitely generated submonoids;
  • double cosets of finitely generated subgroups.
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SLIDE 38

Examples

  • The automaton
  • g,h
  • recognizes the submonoid {g, h}∗ generated by g, h.
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SLIDE 39

Examples

  • The automaton
  • g,h
  • recognizes the submonoid {g, h}∗ generated by g, h.
  • The automaton
  • g±1

1

,g±1

2

  • g
  • g±1

1

,g±1

2

  • recognizes the double coset g1, g2gg1, g2.
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SLIDE 40

The theorem of Anissimov and Seifert

Theorem (Anissimov, Seifert)

A subgroup H ≤ G belongs to Rat(G) iff H is finitely generated.

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SLIDE 41

The theorem of Anissimov and Seifert

Theorem (Anissimov, Seifert)

A subgroup H ≤ G belongs to Rat(G) iff H is finitely generated.

  • Rational submonoids need not be finitely generated.
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SLIDE 42

The theorem of Anissimov and Seifert

Theorem (Anissimov, Seifert)

A subgroup H ≤ G belongs to Rat(G) iff H is finitely generated.

  • Rational submonoids need not be finitely generated.
  • Rational subsets are not in general closed under complement

and intersection.

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SLIDE 43

The theorem of Anissimov and Seifert

Theorem (Anissimov, Seifert)

A subgroup H ≤ G belongs to Rat(G) iff H is finitely generated.

  • Rational submonoids need not be finitely generated.
  • Rational subsets are not in general closed under complement

and intersection.

  • If Rat(G) is closed under intersection, then G is a Howson

group.

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SLIDE 44

Why rational subsets?

  • Diekert, Guti´

errez and Hagenah showed solving equations with rational constraints over free groups is PSPACE-complete.

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SLIDE 45

Why rational subsets?

  • Diekert, Guti´

errez and Hagenah showed solving equations with rational constraints over free groups is PSPACE-complete.

  • Diekert and Lohrey used this to solve equations and decide

the positive theory for right-angled Artin groups.

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SLIDE 46

Why rational subsets?

  • Diekert, Guti´

errez and Hagenah showed solving equations with rational constraints over free groups is PSPACE-complete.

  • Diekert and Lohrey used this to solve equations and decide

the positive theory for right-angled Artin groups.

  • Dahmani and Guirardel solved equations over hyperbolic

groups with special rational constraints.

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SLIDE 47

Why rational subsets?

  • Diekert, Guti´

errez and Hagenah showed solving equations with rational constraints over free groups is PSPACE-complete.

  • Diekert and Lohrey used this to solve equations and decide

the positive theory for right-angled Artin groups.

  • Dahmani and Guirardel solved equations over hyperbolic

groups with special rational constraints.

  • Dahmani and Groves use rational subsets in their solution to

the isomorphism problem for toral relatively hyperbolic groups.

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SLIDE 48

Why rational subsets?

  • Diekert, Guti´

errez and Hagenah showed solving equations with rational constraints over free groups is PSPACE-complete.

  • Diekert and Lohrey used this to solve equations and decide

the positive theory for right-angled Artin groups.

  • Dahmani and Guirardel solved equations over hyperbolic

groups with special rational constraints.

  • Dahmani and Groves use rational subsets in their solution to

the isomorphism problem for toral relatively hyperbolic groups.

  • The order of g is finite if and only if g−1 ∈ g∗, so decidability
  • f submonoid membership gives decidability of order.
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SLIDE 49

History

Theorem (Benois (1969))

Rational subset membership is decidable for free groups.

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SLIDE 50

History

Theorem (Benois (1969))

Rational subset membership is decidable for free groups.

  • The proof uses an automata theoretic analogue of Stallings

folding.

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SLIDE 51

History

Theorem (Benois (1969))

Rational subset membership is decidable for free groups.

  • The proof uses an automata theoretic analogue of Stallings

folding.

Theorem (Eilenberg, Sch¨ utzenberger (1969))

Rational subset membership in an abelian group is decidable.

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SLIDE 52

History

Theorem (Benois (1969))

Rational subset membership is decidable for free groups.

  • The proof uses an automata theoretic analogue of Stallings

folding.

Theorem (Eilenberg, Sch¨ utzenberger (1969))

Rational subset membership in an abelian group is decidable.

  • It reduces to Integer Programming.
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SLIDE 53

Recent history

  • Decidability of rational subset membership is a virtual

property (Grunschlag 1999).

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SLIDE 54

Recent history

  • Decidability of rational subset membership is a virtual

property (Grunschlag 1999).

  • For every c ≥ 2, there is an r ≫ 1 so that the free nilpotent

group of class c and rank r has undecidable rational subset membership (Roman’kov 1999).

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SLIDE 55

Recent history

  • Decidability of rational subset membership is a virtual

property (Grunschlag 1999).

  • For every c ≥ 2, there is an r ≫ 1 so that the free nilpotent

group of class c and rank r has undecidable rational subset membership (Roman’kov 1999).

  • The decidability of rational subset membership passes through

free products (Nedbaj 2000).

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SLIDE 56

Recent history

  • Decidability of rational subset membership is a virtual

property (Grunschlag 1999).

  • For every c ≥ 2, there is an r ≫ 1 so that the free nilpotent

group of class c and rank r has undecidable rational subset membership (Roman’kov 1999).

  • The decidability of rational subset membership passes through

free products (Nedbaj 2000).

Theorem (Kambites, Silva, BS (2007))

Decidability of rational subset membership is preserved by free products with amalgamation and HNN-extensions with finite edge groups.

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SLIDE 57

A general decidability result

  • Let C be the smallest class of groups containing the trivial

group and closed under:

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SLIDE 58

A general decidability result

  • Let C be the smallest class of groups containing the trivial

group and closed under:

  • Taking finitely generated subgroups;
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SLIDE 59

A general decidability result

  • Let C be the smallest class of groups containing the trivial

group and closed under:

  • Taking finitely generated subgroups;
  • Taking finite index overgroups;
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SLIDE 60

A general decidability result

  • Let C be the smallest class of groups containing the trivial

group and closed under:

  • Taking finitely generated subgroups;
  • Taking finite index overgroups;
  • Free products with amalgamation and HNN extensions with

finite edge groups;

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SLIDE 61

A general decidability result

  • Let C be the smallest class of groups containing the trivial

group and closed under:

  • Taking finitely generated subgroups;
  • Taking finite index overgroups;
  • Free products with amalgamation and HNN extensions with

finite edge groups;

  • Direct product with Z.
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SLIDE 62

A general decidability result

  • Let C be the smallest class of groups containing the trivial

group and closed under:

  • Taking finitely generated subgroups;
  • Taking finite index overgroups;
  • Free products with amalgamation and HNN extensions with

finite edge groups;

  • Direct product with Z.

Theorem (Lohrey, BS (2008))

Every group in the class C has decidable rational subset membership problem.

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SLIDE 63

Right-angled Artin groups: the generalized word problem

  • For Γ a graph, the associated right-angled Artin group is

G (Γ) = V (Γ) | [v, w] : (v, w) ∈ E(Γ).

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SLIDE 64

Right-angled Artin groups: the generalized word problem

  • For Γ a graph, the associated right-angled Artin group is

G (Γ) = V (Γ) | [v, w] : (v, w) ∈ E(Γ).

  • Let

C4 = •

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SLIDE 65

Right-angled Artin groups: the generalized word problem

  • For Γ a graph, the associated right-angled Artin group is

G (Γ) = V (Γ) | [v, w] : (v, w) ∈ E(Γ).

  • Let

C4 = •

  • Then G (C4) = F2 × F2 and so this group has undecidable

generalized word problem.

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SLIDE 66

Right-angled Artin groups: the generalized word problem

  • For Γ a graph, the associated right-angled Artin group is

G (Γ) = V (Γ) | [v, w] : (v, w) ∈ E(Γ).

  • Let

C4 = •

  • Then G (C4) = F2 × F2 and so this group has undecidable

generalized word problem.

  • A graph is chordal if it has no induced cycle of length ≥ 4.
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SLIDE 67

Right-angled Artin groups: the generalized word problem

  • For Γ a graph, the associated right-angled Artin group is

G (Γ) = V (Γ) | [v, w] : (v, w) ∈ E(Γ).

  • Let

C4 = •

  • Then G (C4) = F2 × F2 and so this group has undecidable

generalized word problem.

  • A graph is chordal if it has no induced cycle of length ≥ 4.

Theorem (Kapovich, Myasnikov, Weidmann (2005))

The generalized word problem is decidable for chordal right-angled Artin groups.

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SLIDE 68

Right-angled Artin groups: the rational subset problem

Let P4 = •

  • and C4 = •
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SLIDE 69

Right-angled Artin groups: the rational subset problem

Let P4 = •

  • and C4 = •
  • Theorem (Lohrey, BS (2008))

Let Γ be a graph. Then the following are equivalent:

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SLIDE 70

Right-angled Artin groups: the rational subset problem

Let P4 = •

  • and C4 = •
  • Theorem (Lohrey, BS (2008))

Let Γ be a graph. Then the following are equivalent:

  • 1. rational subset membership is decidable for G (Γ);
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SLIDE 71

Right-angled Artin groups: the rational subset problem

Let P4 = •

  • and C4 = •
  • Theorem (Lohrey, BS (2008))

Let Γ be a graph. Then the following are equivalent:

  • 1. rational subset membership is decidable for G (Γ);
  • 2. submonoid membership is decidable for G (Γ);
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SLIDE 72

Right-angled Artin groups: the rational subset problem

Let P4 = •

  • and C4 = •
  • Theorem (Lohrey, BS (2008))

Let Γ be a graph. Then the following are equivalent:

  • 1. rational subset membership is decidable for G (Γ);
  • 2. submonoid membership is decidable for G (Γ);
  • 3. Γ contains neither an induced C4 nor P4.
slide-73
SLIDE 73

Right-angled Artin groups: the rational subset problem

Let P4 = •

  • and C4 = •
  • Theorem (Lohrey, BS (2008))

Let Γ be a graph. Then the following are equivalent:

  • 1. rational subset membership is decidable for G (Γ);
  • 2. submonoid membership is decidable for G (Γ);
  • 3. Γ contains neither an induced C4 nor P4.

P4 is chordal, yielding our first (but not last!) example of a group with decidable generalized word problem but undecidable submonoid membership problem.

slide-74
SLIDE 74

The direct product of two free monoids

Theorem (Lohrey, BS)

Any group containing a direct product of two free monoids has undecidable rational subset membership problem.

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SLIDE 75

The direct product of two free monoids

Theorem (Lohrey, BS)

Any group containing a direct product of two free monoids has undecidable rational subset membership problem.

  • This is a simple encoding of the Post correspondence problem.
slide-76
SLIDE 76

Submonoids vs. rational subsets

  • The submonoid and rational subset membership problems are

equivalent for right-angled Artin groups.

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SLIDE 77

Submonoids vs. rational subsets

  • The submonoid and rational subset membership problems are

equivalent for right-angled Artin groups.

  • We have no example of a group with decidable submonoid

membership but undecidable rational subset membership.

slide-78
SLIDE 78

Submonoids vs. rational subsets

  • The submonoid and rational subset membership problems are

equivalent for right-angled Artin groups.

  • We have no example of a group with decidable submonoid

membership but undecidable rational subset membership.

  • In fact, we have the following result:
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SLIDE 79

Submonoids vs. rational subsets

  • The submonoid and rational subset membership problems are

equivalent for right-angled Artin groups.

  • We have no example of a group with decidable submonoid

membership but undecidable rational subset membership.

  • In fact, we have the following result:

Theorem (Lohrey, BS (2010))

The submonoid and rational subset membership problems are equivalent for groups with two or more ends.

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SLIDE 80

Submonoids vs. rational subsets

  • The submonoid and rational subset membership problems are

equivalent for right-angled Artin groups.

  • We have no example of a group with decidable submonoid

membership but undecidable rational subset membership.

  • In fact, we have the following result:

Theorem (Lohrey, BS (2010))

The submonoid and rational subset membership problems are equivalent for groups with two or more ends.

  • Recall: a group has 2 or more ends iff it splits over a finite

subgroup.

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SLIDE 81

A simple example

  • Consider G = H ∗ F2 with H non-trivial.
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SLIDE 82

A simple example

  • Consider G = H ∗ F2 with H non-trivial.
  • Assume G has decidable submonoid membership.
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SLIDE 83

A simple example

  • Consider G = H ∗ F2 with H non-trivial.
  • Assume G has decidable submonoid membership.
  • It suffices to prove H has decidable rational subset

membership by the combination theorem.

slide-84
SLIDE 84

A simple example

  • Consider G = H ∗ F2 with H non-trivial.
  • Assume G has decidable submonoid membership.
  • It suffices to prove H has decidable rational subset

membership by the combination theorem.

  • Let A be an automaton over H with state set Q.
slide-85
SLIDE 85

A simple example

  • Consider G = H ∗ F2 with H non-trivial.
  • Assume G has decidable submonoid membership.
  • It suffices to prove H has decidable rational subset

membership by the combination theorem.

  • Let A be an automaton over H with state set Q.
  • Fix a copy of FQ in F2.
slide-86
SLIDE 86

A simple example

  • Consider G = H ∗ F2 with H non-trivial.
  • Assume G has decidable submonoid membership.
  • It suffices to prove H has decidable rational subset

membership by the combination theorem.

  • Let A be an automaton over H with state set Q.
  • Fix a copy of FQ in F2.
  • Encode a transition p

a

− → q by paq−1.

slide-87
SLIDE 87

A simple example

  • Consider G = H ∗ F2 with H non-trivial.
  • Assume G has decidable submonoid membership.
  • It suffices to prove H has decidable rational subset

membership by the combination theorem.

  • Let A be an automaton over H with state set Q.
  • Fix a copy of FQ in F2.
  • Encode a transition p

a

− → q by paq−1.

  • h ∈ L(A ) iff q0hq−1

f

is in the submonoid generated by encodings of transitions.

slide-88
SLIDE 88

A simple example

  • Consider G = H ∗ F2 with H non-trivial.
  • Assume G has decidable submonoid membership.
  • It suffices to prove H has decidable rational subset

membership by the combination theorem.

  • Let A be an automaton over H with state set Q.
  • Fix a copy of FQ in F2.
  • Encode a transition p

a

− → q by paq−1.

  • h ∈ L(A ) iff q0hq−1

f

is in the submonoid generated by encodings of transitions.

  • Here q0 is initial and qf is final.
slide-89
SLIDE 89

Wreath products

  • Let G and H be groups.
slide-90
SLIDE 90

Wreath products

  • Let G and H be groups.
  • G(H) denotes the group of all mappings f : H → G of finite

support.

slide-91
SLIDE 91

Wreath products

  • Let G and H be groups.
  • G(H) denotes the group of all mappings f : H → G of finite

support.

  • The wreath product G ≀ H is the semidirect product G(H) ⋊ H

with respect to the action of H on G(H) by left translation.

slide-92
SLIDE 92

Wreath products

  • Let G and H be groups.
  • G(H) denotes the group of all mappings f : H → G of finite

support.

  • The wreath product G ≀ H is the semidirect product G(H) ⋊ H

with respect to the action of H on G(H) by left translation.

  • I.e., (hf)(h′) = f(h−1h′).
slide-93
SLIDE 93

Lamplighter on a tree

The element cbcb−1cabcb−1ca in Z2 ≀ F2: . . . . . . . . . . . .

a a−1 b−1 b

a−1 b−1 b a b−1 b a a−1 b a a−1 b−1

a−1 b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b a b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b−1

slide-94
SLIDE 94

Lamplighter on a tree

cbcb−1cabcb−1ca: . . . . . . . . . . . . c

a a−1 b−1 b

a−1 b−1 b a b−1 b a a−1 b a a−1 b−1

a−1 b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b a b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b−1

slide-95
SLIDE 95

Lamplighter on a tree

cbcb−1cabcb−1ca: . . . . . . . . . . . . c

a a−1 b−1 b

a−1 b−1 b a b−1 b a a−1 b a a−1 b−1

a−1 b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b a b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b−1

slide-96
SLIDE 96

Lamplighter on a tree

cbcb−1cabcb−1ca: . . . . . . . . . . . . c c

a a−1 b−1 b

a−1 b−1 b a b−1 b a a−1 b a a−1 b−1

a−1 b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b a b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b−1

slide-97
SLIDE 97

Lamplighter on a tree

cbcb−1cabcb−1ca: . . . . . . . . . . . . c c

a a−1 b−1 b

a−1 b−1 b a b−1 b a a−1 b a a−1 b−1

a−1 b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b a b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b−1

slide-98
SLIDE 98

Lamplighter on a tree

cbcb−1cabcb−1ca: . . . . . . . . . . . . c

a a−1 b−1 b

a−1 b−1 b a b−1 b a a−1 b a a−1 b−1

a−1 b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b a b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b−1

slide-99
SLIDE 99

Lamplighter on a tree

cbcb−1cabcb−1ca: . . . . . . . . . . . . c

a a−1 b−1 b

a−1 b−1 b a b−1 b a a−1 b a a−1 b−1

a−1 b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b a b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b−1

slide-100
SLIDE 100

Lamplighter on a tree

cbcb−1cabcb−1ca: . . . . . . . . . . . . c

a a−1 b−1 b

a−1 b−1 b a b−1 b a a−1 b a a−1 b−1

a−1 b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b a b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b−1

slide-101
SLIDE 101

Lamplighter on a tree

cbcb−1cabcb−1ca: . . . . . . . . . . . . c

a a−1 b−1 b

a−1 b−1 b

c

a b−1 b a a−1 b a a−1 b−1

a−1 b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b a b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b−1

slide-102
SLIDE 102

Lamplighter on a tree

cbcb−1cabcb−1ca: . . . . . . . . . . . . c

a a−1 b−1 b

a−1 b−1 b

c

a b−1 b a a−1 b a a−1 b−1

a−1 b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b a b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b−1

slide-103
SLIDE 103

Lamplighter on a tree

cbcb−1cabcb−1ca: . . . . . . . . . . . . c c

a a−1 b−1 b

a−1 b−1 b

c

a b−1 b a a−1 b a a−1 b−1

a−1 b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b a b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b−1

slide-104
SLIDE 104

Lamplighter on a tree

cbcb−1cabcb−1ca: . . . . . . . . . . . . c c

a a−1 b−1 b

a−1 b−1 b

c

a b−1 b a a−1 b a a−1 b−1

a−1 b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b a b−1 b a a−1 b a a−1 b−1 a−1 b−1 b a b−1 b a a−1 b−1

slide-105
SLIDE 105

Rational subsets of wreath products with Z × Z

slide-106
SLIDE 106

Rational subsets of wreath products with Z × Z

Theorem (Lohrey, BS (2009))

The wreath product H ≀ (Z × Z)) has undecidable rational subset membership problem for every non-trivial group H.

slide-107
SLIDE 107

Rational subsets of wreath products with Z × Z

Theorem (Lohrey, BS (2009))

The wreath product H ≀ (Z × Z)) has undecidable rational subset membership problem for every non-trivial group H. Proof idea: The grid-like structure of the Cayley graph of Z × Z allows one to encode a tiling problem.

slide-108
SLIDE 108

Rational subsets of wreath products with Z × Z

Theorem (Lohrey, BS (2009))

The wreath product H ≀ (Z × Z)) has undecidable rational subset membership problem for every non-trivial group H. Proof idea: The grid-like structure of the Cayley graph of Z × Z allows one to encode a tiling problem. A similar idea yields:

slide-109
SLIDE 109

Rational subsets of wreath products with Z × Z

Theorem (Lohrey, BS (2009))

The wreath product H ≀ (Z × Z)) has undecidable rational subset membership problem for every non-trivial group H. Proof idea: The grid-like structure of the Cayley graph of Z × Z allows one to encode a tiling problem. A similar idea yields:

Theorem (Lohrey, BS (2009))

Submonoid membership is undecidable in Z ≀ (Z × Z) and in the free metabelian group of rank 2.

slide-110
SLIDE 110

Submonoids of Z ≀ Z

Theorem (Lohrey, BS, Zetzsche (2012))

The submonoid membership problem for the wreath product Z ≀ Z is undecidable.

slide-111
SLIDE 111

Submonoids of Z ≀ Z

Theorem (Lohrey, BS, Zetzsche (2012))

The submonoid membership problem for the wreath product Z ≀ Z is undecidable. Proof is based on reduction from 2-counter (Minsky) machines:

slide-112
SLIDE 112

Submonoids of Z ≀ Z

Theorem (Lohrey, BS, Zetzsche (2012))

The submonoid membership problem for the wreath product Z ≀ Z is undecidable. Proof is based on reduction from 2-counter (Minsky) machines: . . . . . . −1 −2 −3 1 2 3

slide-113
SLIDE 113

Submonoids of Z ≀ Z

Theorem (Lohrey, BS, Zetzsche (2012))

The submonoid membership problem for the wreath product Z ≀ Z is undecidable. Proof is based on reduction from 2-counter (Minsky) machines: . . . . . . n0 m0

counters at t=0

slide-114
SLIDE 114

Submonoids of Z ≀ Z

Theorem (Lohrey, BS, Zetzsche (2012))

The submonoid membership problem for the wreath product Z ≀ Z is undecidable. Proof is based on reduction from 2-counter (Minsky) machines: . . . . . . m1 n1

counters at t=1

slide-115
SLIDE 115

Submonoids of Z ≀ Z

Theorem (Lohrey, BS, Zetzsche (2012))

The submonoid membership problem for the wreath product Z ≀ Z is undecidable. Proof is based on reduction from 2-counter (Minsky) machines: . . . . . . m2 n2

counters at t=2

slide-116
SLIDE 116

Submonoids of Z ≀ Z

Theorem (Lohrey, BS, Zetzsche (2012))

The submonoid membership problem for the wreath product Z ≀ Z is undecidable. Proof is based on reduction from 2-counter (Minsky) machines:

Corollary

Submonoid membership is undecidable in Thompson’s group F.

slide-117
SLIDE 117

Rational subsets in wreath products: Decidability

Theorem (Lohrey, BS, Zetzsche (2012))

Rational subset membership is decidable in H ≀ V for every finite group H and virtually free group V .

slide-118
SLIDE 118

Rational subsets in wreath products: Decidability

Theorem (Lohrey, BS, Zetzsche (2012))

Rational subset membership is decidable in H ≀ V for every finite group H and virtually free group V .

  • The proof is based on an automaton saturation process.
slide-119
SLIDE 119

Rational subsets in wreath products: Decidability

Theorem (Lohrey, BS, Zetzsche (2012))

Rational subset membership is decidable in H ≀ V for every finite group H and virtually free group V .

  • The proof is based on an automaton saturation process.
  • Termination is guaranteed by the theory of well quasi-orders.
slide-120
SLIDE 120

Rational subsets in wreath products: Decidability

Theorem (Lohrey, BS, Zetzsche (2012))

Rational subset membership is decidable in H ≀ V for every finite group H and virtually free group V .

  • The proof is based on an automaton saturation process.
  • Termination is guaranteed by the theory of well quasi-orders.
  • The languages constructed at each stage form an ascending

chain of ideals with respect to a well quasi-order.

slide-121
SLIDE 121

Rational subsets in wreath products: Decidability

Theorem (Lohrey, BS, Zetzsche (2012))

Rational subset membership is decidable in H ≀ V for every finite group H and virtually free group V .

  • The proof is based on an automaton saturation process.
  • Termination is guaranteed by the theory of well quasi-orders.
  • The languages constructed at each stage form an ascending

chain of ideals with respect to a well quasi-order.

  • No complexity bounds are obtained.
slide-122
SLIDE 122

Open questions

Question

Does there exist a group with decidable submonoid membership and undecidable rational subset membership?

slide-123
SLIDE 123

Open questions

Question

Does there exist a group with decidable submonoid membership and undecidable rational subset membership? This question is equivalent to the following one.

slide-124
SLIDE 124

Open questions

Question

Does there exist a group with decidable submonoid membership and undecidable rational subset membership? This question is equivalent to the following one.

Question

Is decidability of submonoid membership preserved by free products?

slide-125
SLIDE 125

Open questions

Question

Does there exist a group with decidable submonoid membership and undecidable rational subset membership? This question is equivalent to the following one.

Question

Is decidability of submonoid membership preserved by free products?

Question

Is submonoid membership decidable for nilpotent groups?

slide-126
SLIDE 126

Open questions

Question

Does there exist a group with decidable submonoid membership and undecidable rational subset membership? This question is equivalent to the following one.

Question

Is decidability of submonoid membership preserved by free products?

Question

Is submonoid membership decidable for nilpotent groups?

Question

Is it true that rational subset membership is undecidable for G ≀ H whenever G is non-trivial and H is not virtually free?

slide-127
SLIDE 127

The end

Thank you for your Attention!