aa r X i v : . [ m a t h . G R ] F e b Groups with context-free equation solvabilityproblem
Vladimir Yankovskiy ∗ Abstract
We find algebraic conditions on a group equivalent to the position ofits equation solvability problem in the Chomsky Hierarchy. In particular,it is proved that a finitely generated group has a context-free equationsolvability problem if and only if it is finite.MSC 2020: 03D40, 20F10, 20F70
The main result is the following:
Theorem 1.
Suppose that G is a finitely generated group, A — is its finitesymmetric generating set. Then the following statements are equivalent: • n -ary equation solvability problem is context-free for some n > • n -ary equation solvability problem is context-free for all n > • G is finite Also somewhat similar results are proved in the Section 3 for groups withregular equation solvability problem (they are exactly all finite groups) andfor groups with recursively enumerable equation solvability problem (they areexactly all finitely generated subgroups of finitely presented groups).The equation solvability problem is a generalization of the classical wordproblem, first introduced by Dehn in [Deh11]. Classification of groups withrecursively enumerable word problem was obtained by Higman in [Hig61], clas-sification of groups with regular word problem was obtained by Anisimov in[Ani71] and classification of groups with context free word problem was ob-tained by Shupp and Muller in [MS83]. Those results will be formulated in theSection 2.Section 2 deals with the definition of the polynomials on groups and with lan-guage theoretic properties of word problem. ∗ The work of the author was supported by the Russian Foundation for Basic Research,project no. 19-01-00591
Definition 1.
Suppose that G is a finitely generated group and A is its finitesymmetric generating set, { x , x − , ..., x n , x − n } is the set of formal variables andtheir inverses. Then n -ary polynomial over ( G, A ) is a word from F n ( G, A ) :=( A ∪ { x , x − , ..., x n , x − n } ) ∗ .Now we define an additional structure on the polynomials: Definition 2.
Suppose that G is a finitely generated group, and A is its finitesymmetric generating set. Then the interpretation of n -ary polynomials over ( G, A ) is the function i : F n ( G, A ) → ( G n ) G , defined recursively as: i (Λ)( g , ..., g n ) = ei ( aα )( g , ..., g n ) = ai ( α )( g , ..., g n ) i ( x k α )( g , ..., g n ) = g k i ( α )( g , ..., g n ) i ( x − k α )( g , ..., g n ) = g − k i ( α )( g , ..., g n ) Definition 3.
Suppose that G is a finitely generated group, and A is its finitesymmetric generating set. Then word problem in ( G, A ) is the formal language Id ( G, A ) = { α ∈ F ( G, A ) | i ( α ) = e } . Theorem 2. [Ani71]. Id ( G, A ) is regular if and only if G is finite. Theorem 3. [MS83]. Id ( G, A ) is context-free if and only if G is virtually free. Theorem 4. [Hig61]. Id ( G, A ) is recursively enumerable if and only if G is asubgroup of a finitely presented group. Definition 4.
We call ( g , ..., g n ) ∈ G n a solution of α ∈ F n ( G, A ) if i ( α )( g , ..., g n ) = e . Definition 5. n -ary equation solvability problem over ( G, A ) is the language Eq n ( G, A ) of all polynomials from F n ( G, A ) that have a solution.It is not hard to see that Eq ( G, A ) = Id ( G, A ). From this we can derivethe following necessary and sufficient conditions for its regularity and recursiveenumerability. 2 heorem 5.
Suppose that G is a finitely generated group, and A is its finitesymmetric generating set. Then the following statements are equivalent: • Eq n ( G, A ) is regular for some n > ; • Eq n ( G, A ) is regular for all n > ; • G is finite.Proof. If Eq n ( G, A ) is regular for all n >
1, then Eq n ( G, A ) is regular for some n > Eq n ( G, A ) is regular for some n >
1, then Eq n ( G, A ) ∩ A ∗ = Id ( G, A ) is alsoregular. Thus G is regular by the Theorem 2.Suppose that G is finite. Then to conclude the proof we build a deterministicfinite automaton that recognizes Eq n ( G, A ) for any n . In our deterministicfinite automaton the set of states is the set of all functions from G n to G (theirnumber is equal to | G | n | G | ). Transition function ψ : ( G n ) G × F n ( G, A ) → ( G n ) G is defined by formula ψ ( f, a )( g , ..., g n ) = f ( g ) a , a ∈ Af ( g ) g j , a = x j f ( g ) g − j , a = x − j ∀ f ∈ ( G n ) G , g ∈ G, α ∈ F n ( G, A ) , j n The initial state is f ≡ e . The set of terminal states is { α ∈ F n | ∃ g , ..., g n ∈ Gi ( α )( g , ..., g n ) = e } . Theorem 6.
Suppose that G is a finitely generated group, and A is its finitesymmetric generating set. Then the following statements are equivalent: • Eq n ( G, A ) is recursively enumerable for some n > ; • Eq n ( G, A ) is recursively enumerable for all n > ; • G is a subgroup of a finitely presented group.Proof. If Eq n ( G, A ) is recursively enumerable for all n > Eq n ( G, A ) isrecursively enumerable for some n > Eq n ( G, A ) is recursively enumerable for n >
1, then Eq n ( G, A ) ∩ A ∗ = Id ( G, A ) is also recursively enumerable. That means G is a subgroup of afinitely presented group by Theorem 3.Suppose that G is a subgroup of a finitely presented group. That means itsword problem is recursively enumerable by Theorem 4. Then the words from Eq n ( G, A ) are recognized by the following algorithm:Suppose that w is our word. On m -th step for each tuple of words v , ...v n ∈ A ∗ , of length m or less, we run first m steps of Id ( G, A ) recognition algorithmfor w [ x := v , ..., x n := v n ]. If at least one of them finishes before m -th turn w ∈ Eq n ( G, A ). Otherwise we launch the step m + 1.3o find the necessary and sufficient conditions for Eq n ( G, A ) being context-free we first need to pay attention to certain facts that we mention in the nexttwo Sections.
Definition 6.
A set of positive rational numbers Q is context-free if the lan-guage { a m b n | mn ∈ Q } is context-free. Lemma 1. [BPS61].
If a language L is context-free, then there exists an integer p > , called the pumping length, such, that any word s ∈ L of length notexceeding p can be written as s = uvwxy , where | vx | > , | vwx | p and ∀ n ∈ N uv n wx n y ∈ L . Lemma 2.
Any context-free set of rational numbers has finitely many isolatedpoints.Proof.
Suppose that Q ⊂ Q + is a context-free set with infinitely many isolatedpoints. Then L = { a m b n | mn ∈ Q } is context-free. Suppose that p is itspumping length, MN ∈ Q is isolated element of Q such that MN is irreducibleand M + N > p (such element exists by the Pigeonhole Principle, because Q is infinite and there are only finitely many positive rational numbers MN , suchthat M + N p ) ε := inf {| q − MN || q ∈ Q \ { MN }} > n := ⌈ ( M + N ) εN ⌉ + 1 w := a nM b nN Then by the Lemma 1 w = uvxyz , where | vxy | p , | vy | >
1, and uv t xy t z ∈ L ∀ t ∈ N . Suppose that α is the number of instances of a in vy and β is thenumber of instances of b in vy . If v or y contains both a and b , then uv xy z contains ba and therefore does not belong to L .So 3 variants are possible: • If β = 0 then for all t ∈ N a nM +( t − α b nN ∈ L by the Lemma 1. Thatmeans nM +( t − αnN ∈ Q for all t ∈ N . Take t = 2. Then nM + αnN ∈ Q . But | nM + αnN − MN | = αnN M + NnN < ε , because α p M + N . Contradiction. • If α = 0 then for all t ∈ N a nM b nN +( t − β ∈ L by the Lemma 1. Thatmeans ∀ t ∈ N nMnN +( t − β ∈ Q for all t ∈ N . Take t = 2. Then nMnN + β ∈ Q .But | nMnN + β − MN | = ( M + N ) M (( n +1) N + M ) N M nN < ε because α p M + N .Contradiction. 4 If v = a α and y = b β then a nM +( t − α b nN +( t − β ∈ L for all t ∈ N by theLemma 1. That means nM +( t − αnN +( t − β ∈ Q for all t ∈ N . Take t = 2. Then nM + αnN + β ∈ Q . But | nM + αnN + β − MN | = | αN − βM | N ( nN + β ) < ε . It is possible only when αβ = MN . And from that follows p < M + N α + β = | vy | | vxy | p .Contradiction.That way we have shown that L is not context-free. Q.E.D. Definition 7.
Suppose G is a group. Then a, b ∈ G are commensurate if ∃ n, m ∈ Z \ { } such that a n = b m .It is not hard to see, that commensuration is an equivalence relation. Furtherwe denote commensuration class of a as CC ( a ). Theorem 7. [Gro87].
Commensuration classes of hyperbolic groups are of thefollowing types: • set of all finite order elements; • set of all infinite order elements of a maximal virtually-cyclic subgroup. Proof. If Eq n ( G, A ) is context-free for all n >
1, it is context-free for some n > G is finite then by Theorem 4 Eq n ( G, A ) is context-free for all n > Eq n ( G, A ) is context-free for some n >
1, then Eq n ( G, a ) ∩ F = Eq ( G, A )is also context-free.Suppose now that the language Eq ( G, A ) is context-free and G is infinite. Thenthe language Id ( G, A ) = Eq ( G, A ) ∩ A ∗ is also context-free as intersection ofa context-free language with a regular one. That means G is virtually-free(and therefore hyperbolic) by the Teorem 2. Suppose that a is an infinite-order element of G and suppose that H := h CC G ( a ) i . Then H is an in-finite virtually-cyclic group. Suppose that c is a generator of a finite-indexcyclic normal subgroup of H . Now w ∈ F , i ( w ) = c . Consider the language Eq ( G, A ) ∩ { w } ∗ { x } ∗ . It is context-free as intersection of a context-free lan-guage wih a regular one. However Eq ( G, A ) ∩ { w } ∗ { x } ∗ = { w n x m | ∃ b ∈ H b m c n = e } = { w n x m | nm ∈ { nk ∈ Q | ∃ b ∈ H b k = c n } ) } . But { nk ∈ Q | ∃ b ∈ H b k = c n } is discrete as h c i has finite index in H . Contra-diction with the Lemma 2. Q.E.D. 5 cknowledgements I thank my supervisor Anton Klyachko for providing valuable advices regardingthis research.
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On formal properties ofsimple phrase-structure grammars . Zeitschrift f¨ur Phonetik, Sprach-wissenschaft, und Kommunikationsforschung, Vol. 14, 1961, pp. 143–172.[Hig61] Graham Higman.
Subgroups of finitely presented groups . Proceedingsof the Royal Society, Series A, Mathematical and Physical Sciences,Vol. 262, 1961, pp. 455–475.[Ani71] A.V. Anisimov. ¨Uber Gruppen-Sprachen . Kibernetika, Vol. 4, 1971,pp. 18–24.[MS83] D.E. Muller and P.E. Schupp.
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