Stability of Universal Equivalence of Groups under Free Constructions
aa r X i v : . [ m a t h . G R ] A p r STABILITY OF UNIVERSAL EQUIVALENCE OF GROUPSUNDER FREE CONSTRUCTIONS
ANDREW J. DUNCAN, ILYA V. KAZACHKOV, AND VLADIMIR N. REMESLENNIKOV Introduction
In his important paper in [3] J. Stallings introduced a generalisation of amal-gamated products of groups – called a pregroup, which is a particular kind of apartial group. He then defined the universal group U ( P ) of a pregroup P to be auniversal object (in the sense of category theory) extending the partial operationson P to group operations on U ( P ). The universal group turned out to be a versatileand convenient generalisation of classical group constructions: HNN-extensions andamalgamated products. In this respect the following general question arises.Whichproperties of pregroups, or relations between pregroups, carry over to the respectiveuniversal groups? The aim of this paper is to prove that universal equivalence ofpregroups extends to universal equivalence of their universal groups.We begin by some preliminary model-theory results. We refer the reader to [2]for a detailed introduction to model theory. The main goal here is to give a criterionof universal equivalence of two models in the form that best suits our needs.2. Preliminaries
Let L be a language with signature ( C, F, R ), and variables X , where C is aset of constants and F and R are finite sets of functions and relations respectively.In addition each element f of F is associated to a non-negative integer n f , andsimilarly for R .An L -structure M is a 4-tuple: • a non-empty set M ; • a function f M : M n f → M for each f ∈ F ; • a set r M ⊆ M n r for each r ∈ R ; • an element c M for each element c ∈ C .If a is an element of C , F or R we refer to a M as the interpretation of a in M .The subscript M is omitted where no ambiguity arises.We use the language L to write formulas describing the properties of L -structures. Roughly speaking formulas are constructed inductively starting fromconstant symbols from C and variable symbols v , . . . , v n , . . . , using the Booleanconnectives, relations from R , functions from F and the equality symbol ‘=’.More precisely, the set of L -terms is the smallest set T such that: • c ∈ T for each constant symbol c ∈ C ; • each variable symbol v i ∈ T ; • if t , . . . , t n ∈ T and f ∈ F then f ( t , . . . , t n f ) ∈ T . Research supported by EPSRC grant EP/D065275.
We say that Φ is an atomic L -formula if Φ is either • t = t , where t and t are terms or, • r ( t , . . . , t n r ), where r ∈ R and t , . . . , t n r are terms.The set of L -formulas is the smallest set W containing atomic formulas and suchthat • if Φ ∈ W then ¬ Φ ∈ W ; • if Φ and Ψ are in W then Φ ∧ Ψ and Φ ∨ Ψ are in W and • if Φ is in W then ∃ v i Φ and ∀ v i Φ are in W .It is often useful in practice to observe that, asΦ ∨ Ψ ≡ ¬ ( ¬ Φ ∧ ¬ Ψ)and ∀ v i Φ ≡ ¬∃ v i ( ¬ Φ)we can construct all formulas (up to logical equivalence ≡ ) without using ∨ or ∀ .To make induction arguments precise we shall define, for any term or formula s of L , the level l ( s ) and the constants C ( s ) of s . If s is a formula we shall also definethe degree d ( s ) of s . To begin with if t is a term and t = x or t = c , where x is avariable and c a constant, then l ( t ) = 0 and C ( t ) = (cid:26) ∅ , if t = xc, if t = c . If t = f ( t , . . . , t n ), where n = n f and the t i are terms then l ( t ) =max { l ( t ) , . . . , l ( t n ) } + 1, and C ( t ) = ∪ ni =1 C ( t i ).If a is an atomic formula of the form t = t , for terms t and t , then we define l ( a ) = max { l ( t ) , l ( t ) } and C ( t ) = C ( t ) ∪ C ( t ). If r is an n -ary relation and a = r ( t , . . . , t n ) then set l ( a ) = max { l ( t ) , . . . , l ( t n ) } and C ( a ) = ∪ ni =1 C ( t i ). Thedegree of an atomic formula a is defined to be d ( a ) = 0.If Φ = ¬ Ψ or Φ = ∃ x Ψ or ∀ x Ψ then we define l (Φ) = l (Ψ), d (Φ) = d (Ψ) + 1 and C (Φ) = C (Ψ). If Φ = Φ ∧ Φ or Φ ∨ Φ then we define l (Φ) = max { l (Φ ) , l (Φ ) } , d (Φ) = d (Φ ) + d (Φ ) and C (Φ) = C (Φ ) ∪ C (Φ ).We say that a variable v occurs freely in a formula Φ if it is not inside a ∃ v ora ∀ v quantifier, otherwise v is said to be bound. A formula is called a sentence or closed if it has no free variables.Let Φ be a formula with free variables from v = ( v , . . . , v m ) and let ¯ a =( a , . . . , a m ) ∈ M m . We inductively define when Φ holds on ¯ a in a L -structure M (Φ(¯ a ) is true in M or M satisfies Φ(¯ a )), write M | = Φ(¯ a ). • if Φ is t = t , then M | = Φ(¯ a ) if t (¯ a ) = t (¯ a ); • if Φ = r ( t , . . . , t n r ), then M | = Φ(¯ a ) if r ( t (¯ a ) , . . . , t n r (¯ a )) ∈ r M ; • if Φ = ¬ Ψ then
M | = Φ(¯ a ) if M Ψ(¯ a ); • if Φ = Ψ ∧ Ψ then M | = Φ(¯ a ) if M | = Ψ (¯ a ) and M | = Ψ (¯ a ); • if Φ = Ψ ∨ Ψ then M | = Φ(¯ a ) if M | = Ψ (¯ a ) or M | = Ψ (¯ a ); • if Φ = ∃ v m +1 Ψ(¯ v, v m +1 ), then M | = Φ if there exists b ∈ M such that M | = Ψ(¯ a, b ); • if Φ = ∀ v m +1 Ψ(¯ v, v m +1 ), then M | = Φ if for all b ∈ M one has M | =Ψ(¯ a, b )A set of sentences is called a theory . We say that M is a model of a theory T if M | = Φ for all Φ ∈ T . For an L -structure M we denote by Th ( M ) the collection of TABILITY OF UNIVERSAL EQUIVALENCE OF GROUPS UNDER FREE CONSTRUCTIONS3 all sentences that are satisfied by M , Th ( M ) is called the full or elementary theory of M .Every formula Φ of L with free variables ¯ v = ( v , . . . , v k ) is logically equivalentto a formula of the type Q x Q x . . . Q n x n Ψ(¯ x, ¯ v ) , where Q i ∈ {∀ , ∃} , and Ψ(¯ x, ¯ v ) is a boolean combination of atomic formulas invariables from ¯ v ∪ ¯ x . This form is called the prenex normal form of a formula Φ.A sentence Φ is called universal ( existential ) if Φ is equivalent to a formula ofthe form Q x Q x . . . Q n x n Ψ(¯ x ) , where Q i = ∀ ( Q i = ∃ ) for all i , and Ψ(¯ x ) is a boolean combination of atomicformulas in the indicated variables. The collection of all universal (existential)sentences that are satisfied by an L -structure M is called the universal ( existential )theory of M , we denote it by Th ∀ ( M ) ( Th ∃ ( M )). If M and N are L -structuresand Th ∀ ( M ) = Th ∀ ( N ) we say that M and N are universally equivalent and write M ≡ ∀ N . existential equivalence is defined similarly and we write M ≡ ∃ N if M and N are existentially equivalent.Let A be a set of sentences of L and let M and N be models of A with underlyingsets M and N respectively. For subsets S and T of M and N respectively we saythat a map φ : S → T is an L -morphism if the following conditions hold.(i) If c ∈ C ∩ S then c ∈ T and φ ( c ) = c .(ii) If f is an n -ary function (i.e. n f = n ) in F and f ( s , . . . , s n ) ∈ S , for some n -tuple ( s , . . . , s k ) of elements of S , then φ ( f ( s , . . . , s n )) = f ( φ ( s ) , . . . , φ ( s n )) ∈ T. (iii) If r is an n -ary relation in R and ( s , . . . , s n ) ∈ r , for some n -tuple( s , . . . , s n ) of elements of S , then ( φ ( s ) , . . . , φ ( s n )) ∈ r .If φ : S → T is a bijective L -morphism such that φ − is an L -morphism from T to S then we say that φ is an L -isomorphism and that S and T are L -isomorphic or S ∼ = L T . L -isomorphism defines an equivalence relation on the subsets of a model M and we denote by [ S ] the equivalence class of S .Now we restrict attention to finite subsets of models. We denote by F L ( M ) = F ( M ) the set of L -isomorphism equivalence classes of finite subsets of M . We saythat models M and N have equivalent L -isomorphism classes of finite subsets, andwrite F ( M ) ≡ F ( N ), if there exists a bijection θ : F ( M ) → F ( N ) such that,for all finite subsets S ⊆ M , if θ ([ S ]) = [ T ] then there exists an L -isomorphism φ ( S ) → T ′ , for some T ′ ∈ [ T ] (hence for all T ′ ∈ [ T ]). Lemma 2.1. F ( M ) ≡ F ( N ) if and only if, for all finite subsets S ⊆ M , thereexists a subset T ⊆ N such that S ∼ = L T .Proof. If F ( M ) ≡ F ( N ) and S is a finite subset of M then, by definition, S is L -isomorphic to some finite subset of N . Conversely, suppose every finite subsetof M is L -isomorphic to a finite subset of N . For each isomorphism class U offinite subsets of M choose a representative S U , so U = [ S U ]. Similarly choose arepresentative T V for each isomorphism class of finite subsets of N . Consider anisomorphism class U ∈ F ( M ). S U is L -isomorphic to T for some finite subset of N . Let T ′ be the chosen representative of [ T ]. Then S U ∼ = L T ′ . Define θ ( U ) = [ T ′ ].Then θ is a well-defined map from F ( M ) to F ( N ) and straightforward verification A. DUNCAN, I. KAZACHKOV, AND V. REMESLENNIKOV shows that θ is a bijection. By construction, if θ ( U ) = V then S U is L -isomorphicto the representative T ′ of V , so the same goes of any element S ∈ U . Hence F ( M ) ≡ F ( N ). (cid:3) If M and N are models of A then it is easy to see that M ≡ ∃ N if and onlyif M ≡ ∀ N . The following proposition gives a further characterisation of thisproperty, in certain cases. Proposition 2.2.
Assume that L has signature ( C, F, R ) where either (i) C is finite or (ii) R contains a relation δ C and, for each c ∈ C , A contains axioms(a) c ∈ δ C and(b) ∀ x ( x / ∈ δ C = ⇒ x = c ) .Let M and N be models of A . Then M ≡ ∃ N if and only if F ( M ) ≡ F ( N ) .Proof. Assume first that
M ≡ ∃ N . Let S = { m , . . . , m k } be a finite subset of M .Define the formula Φ = ^ ≤ i 1. Let S f, = { ( i , . . . , i n ) ∈ I nk | f ( m i , . . . , m i n ) ∈ S \ C } and S f, = { ( i , . . . , i n ) ∈ I nk | f ( m i , . . . , m i n ) ∈ C } . For each ( i , . . . , i n ) ∈ S f, define s = s ( i , . . . , i n ) to be the integer in I k such that f ( m i , . . . , m i n ) = m s . DefineΦ f, = ^ ( i ,...,i n ) ∈ S f, f ( x i , . . . , x i n ) = f ( m i , . . . , m i n )and Φ f, = ^ ( i ,...,i n ) ∈ S f, f ( x i , . . . , x i n ) = m s ( i ,...,i n ) . Define Φ f = Φ f, ∧ Φ f, . Then M (cid:15) Φ f [ m , . . . , m k ].Let r ∈ R be an n -ary relation, for some n ≥ 1, and let S r = { ( i , . . . , i n ) ∈ I k | ( m i , . . . , m i n ) ∈ r } . TABILITY OF UNIVERSAL EQUIVALENCE OF GROUPS UNDER FREE CONSTRUCTIONS5 Define Φ r = ^ ( i ,...,i n ) ∈ S r r ( x i , . . . , x i n ) ∧ ^ ( i ,...,i n ) / ∈ S r ¬ r ( x i , . . . , x i n ) Then M (cid:15) Φ r [ m , . . . , m k ].Finally define Φ = Φ ∧ Φ ∧ V f ∈ F Φ f ∧ V r ∈ R Φ r . Then M (cid:15) Φ[ m , . . . , m k ]so M (cid:15) ∃ x , . . . , x k Φ. Therefore N (cid:15) ∃ x , . . . , x k Φ and there exist n , . . . , n k ∈ N such that N (cid:15) Φ[ n , . . . , n k ].Set T = { n , . . . , n k } and define φ : S → T by φ ( m i ) = n i , i = 1 , . . . , k . Bydefinition φ is an L -morphism and is a bijection of S and T . Moreover φ − is, byconstruction of Φ, an L -morphism. Hence S ∼ = L T and it follows from Lemma 2.1that F ( M ) ≡ F ( N ).Now suppose that F ( M ) ≡ F ( N ). Write F n for the set of n -ary functions of F . Since F is finite we may assume that F is the union of F n , for n from 1 to K ,for some K ∈ N . Given a finite subset S of M we define the following sequence ofsubsets. Set S = S and having defined S i set S i +1 = S i ∪ [ n =1 ,...,K [ f ∈ F n { f ( m , . . . , m n ) | m j ∈ S i , j = 1 , . . . , n } . Now choose T l ⊆ N such that there is an L -isomorphism φ l from S l to T l , for all l ≥ t of level l with variables among x , . . . , x k and a k -tuple a , . . . , a k of elements of M and set S = { a , . . . , a k } ∪ C ( t ). We claim that t ( a , . . . , a k ) ∈ S l . To see this note that it holds when l = 0, since inthis case t ( a , . . . , a k ) ∈ S . Suppose then that t has level l and that theclaim holds for at all levels below l . Then t = f ( t , . . . , t m ), where l ( t i ) Note that this argument also goes through in the case ( d, l ) = (0 , 0) so by induction(1) holds for formulae Φ of level l and degree 0, for all non-negative integers l .Now let d and l be non-negative integers and assume that (1) holds for formulaeΦ of degree d and level l where either (i) d ≤ d and l = l or (ii) l < l .Suppose then that Φ has level l and degree d + 1. Then either Φ = ¬ Φ orΦ = Φ ∧ Φ , where Φ and Φ have degree at most d and level at most l . IfΦ = ¬ Φ then M (cid:15) Φ ( a , . . . , a m ) if and only if N (cid:15) Φ ( a , . . . , a m ), so the sameholds with Φ in place of Φ . If Φ = Φ ∧ Φ then M (cid:15) Φ( a , . . . , a m ) if and only if M (cid:15) Φ i ( a , . . . , a m ), for i = 1 and 2, if and only if N (cid:15) Φ i ( a , . . . , a m ), for i = 1and 2, if and only if N (cid:15) Φ( a , . . . , a m ). It follows that (1) holds for Φ of level l and any degree d + 1; hence by induction for all ( d, l ). (cid:3) We call an expression of the form t = t , where t and t are terms, an equation .A set S of equations such that every element of S has variables among x , . . . , x m is called a system of equations in m variables. Let S be a system of equations in m variables and let M be a model of L . We say that ( a , . . . , a m ) ∈ M m is a solution of S in M if M (cid:15) s ( a , . . . , a m ), for all s ∈ S . The variety defined by S over M is the set V M ( S ) = { ( a , . . . , a m ) ∈ M m : ( a , . . . , a m ) is a solution of S } . We saythat a model M of L is equationally Noetherian if every system S of equationscontains a finite subset S such that V M ( S ) = V M ( S ). As in [1] we have thefollowing lemma. Lemma 2.3. Let M and N be L -structures. Then, (i) if M is equationally Noetherian and Th ∃ ( N ) ⊆ Th ∃ ( M ) , then N is equa-tionally Noetherian; (ii) if M and N are universally equivalent, M is equationally Noetherian ifand only if N is equationally Noetherian.Proof. Suppose that M is equationally Noetherian and that S is a system of equa-tions in m variables. Choose a subset S ⊆ S such that V M ( S ) = V M ( S ). Let S = { s , . . . , s r } and for each s ∈ S let Φ s be the sentence ∀ x , . . . , x m ( s ∧· · ·∧ s r → s ).Since V M ( S ) = V M ( S ) we have M (cid:15) Φ s and therefore, since under the assump-tions of any of the two statements above Th ∃ ( N ) ⊆ Th ∃ ( M ), we have N (cid:15) Φ s , forall s ∈ S . As S ⊆ S it follows that V N ( S ) ⊆ V N ( S ). If ( b , . . . , b m ) ∈ V N ( S )then, as N (cid:15) Φ s , we have ( b , . . . , b m ) ∈ V N ( S ), so V N ( S ) = V N ( S ). (cid:3) Groups and Pregroups The language of pregroups L pre has signature ( C, F, R ) where C consists of asingle element 1, F consists of a unary function symbol − and R consists of abinary relation D and a ternary relation M . (The usual definition of a pregroupinvolves a product function defined on a subset D ⊂ P × P . Our description oflanguage does not allow F to contain partially defined functions, so we use therelation M instead of this product. We keep the relation D for compatibility withthe usual definition.) A pregroup is a model P of L pre satisfying the followingaxioms.(i) ∀ x, y, z (( x, y, z ) ∈ M → ( x, y ) ∈ D ).(ii) ∀ x, y (( x, y ) ∈ D → ∃ z (( x, y, z ) ∈ M )).(iii) ∀ w, x, y, z (( w, x, y ) ∈ M ∧ ( w, x, z ) ∈ M → y = z ).(iv) ∀ x (( x, , x ) ∈ M ∧ (1 , x, x ) ∈ M ). TABILITY OF UNIVERSAL EQUIVALENCE OF GROUPS UNDER FREE CONSTRUCTIONS7 (v) ∀ x (( x, x − , ∈ M ∧ ( x − , x, ∈ M ).(vi) ∀ x, y, z (( x, y, z ) ∈ M → ( y − , x − , z − ) ∈ M ) . (vii) ∀ a, b, c, r, s, x (( a, b, r ) ∈ M ∧ ( b, c, s ) ∈ M → (( a, s, x ) ∈ M ↔ ( r, c, x ) ∈ M )).(viii) ∀ a, b, c, d, x, y, z (( a, b, x ) ∈ M ∧ ( b, c, y ) ∈ M ∧ ( c, d, z ) ∈ M → ∃ r, s (( a, y, r ) ∈ M ∨ ( y, d, s ) ∈ M )).A pregroup homomorphism is a morphism of L pre -structures and a subpregroup isan L pre -substructure of an L pre -structure. Thus K is a subpregroup of P if andonly if K is a pregroup, K ⊆ P , 1 K = 1 P , D K = D P ∩ ( K × K ) and M K = M P ∩ ( K × K × K ) (from which it follows that the operation of inversion in P extends that in K ).We wish, as in [1] for the group case, to consider pregroups which contain des-ignated copies of some fixed pregroups (or some of their subsets). To this end wemake the following definition. Definition 3.1. Let M be an L -structure and N a subset of M . The diagram of N is the set of all closed atomic formulas, and their negations, which hold in N .Now let S ′ be a fixed multiset of pregroups and, for each L ∈ S ′ , let K L be asubset of L containing 1 L . Let S be the set { K L | L ∈ S ′ } . We define the language of S -pregroups L pre S to be the extension of L pre with signature identical to L pre exceptthat C = ∪ K ∈S { d Kk | k ∈ K } and R contains a unary relation δ S . A K -pregroup is a model P of L pre S satisfying the axioms for a pregroup all the formulas of thediagram of K , for all K ∈ S , and the additional axioms(ix) d Kk ∈ δ S , for all k ∈ K , for all K ∈ S , and(x) ∀ x ( x / ∈ δ S → x = d k ), for all k ∈ K , for all K ∈ S .(There is one axiom of type (ix) and one of type (x) for each k ∈ K and K ∈ S .) A S -pregroup homomorphism is a morphism of L pre S -structures and a S -subpregroupis an L pre S -substructure of an L pre S -structure. A S -pregroup is finitely generated ifit is finitely generated as a model of L pre S . If S consists of a single element K wecall an S -pregroup a K -pregroup and write L pre K instead of L pre S . Lemma 3.2. Let P be a pregroup and a, b, c ∈ P . If ( a, b, c ) ∈ M then ( c, b − , a ) and ( c − , a, b − ) ∈ M .Proof. We have ( a, b, c ) and ( b, b − , ∈ M and, as also ( a, , a ) ∈ M , axiom (vii)implies ( c, b − , a ) ∈ M . Repeating this argument starting with ( c − , c, c, b − , a )and (1 , b − , b ) we see that ( c − , a, b − ) ∈ M . (cid:3) Let S ′ be a fixed multiset of groups and, for each G ∈ S ′ , let K G be a subset of G containing 1 G . Let S be the set { K G | G ∈ S ′ } . The language of S -groups is definedto be the language L S with signature ( C, F, R ), where C = ∪ K ∈S { d Kk | k ∈ K } , F consists of a binary function symbol · and a unary function symbol − and R consists of a unary relation symbol δ S . Then an S -group H is a model of L S satisfying the usual group axioms with respect to · as multiplication and − asinverse in H , as well as all the formulas of the diagram of K , for all K ∈ S , andthe additional axioms(a) d Kk ∈ δ S , for all k ∈ K , K ∈ S , and(b) ∀ x ( x / ∈ δ Σ = ⇒ x = d Kk ), for all k ∈ K , K ∈ S .The class of all S -groups together with the naturally defined S -morphisms forms acategory. A. DUNCAN, I. KAZACHKOV, AND V. REMESLENNIKOV If S consists of a single element K then we refer to K -groups instead of S -groupsand write L K instead of L S . In this case, if K = G we recover the definition of G -group in [1]. Further, if G = K = 1 then we drop the predicate δ S from thelanguage and we have the standard language L of groups. Note that, if G is agroup, a G -group H is equationally Noetherian in the sense defined in the previoussection if and only if it is G -equationally Noetherian in the sense of [1].Notions of universal equivalence, elementary equivalence and equivalence of finitesubsets for S -groups are defined with respect to the language L S ; as are substruc-tures and extensions of S -groups. A S -group H is locally S -discriminated by a S -group N if, given a finite subset F = { h , . . . , h k } of H there is a S -homomorphism(i.e. L S -morphism) from H to N which is injective on F . A S -group H is saidto be finitely generated if there exists a finite subset F of H such that H is gen-erated by F ∪ ∪ K ∈S K . (Thus a finitely generated S -group is a finitely generated L S -model.) If P is any property then a S -group H is said to be locally P if ev-ery non-trivial finitely generated S -subgroup of H has property P . The followingtheorem is proved in [1]. Theorem 3.3 ([1]) . Let G be a group and H and K be G -groups one of whichis G -equationally Noetherian. Then H is locally G -discriminated by K and K islocally G -discriminated by G if and only if K and H are universally equivalent (withrespect to L G ). If a, b are elements of pregroup P and ( a, b ) ∈ D P we write ab for the uniqueelement c such that ( a, b, c ) ∈ M . Following Stallings [3] we define a word of length k over a pregroup P to be a finite sequence ( c , . . . , c k ) of elements of P .If ( c i , c i +1 ) ∈ D then c i c i +1 ∈ P and the word ( c , . . . , c i − , c i c i +1 , c i +1 , . . . , c k ) issaid to be a reduction of ( c , . . . , c k ). The word ( c , . . . , c k ) is said to be reduced if( c i , c i +1 ) / ∈ D , for i = 1 , . . . , k − c = ( c , . . . , c k ) and a = ( a , . . . , a k − ) be words such that ( c , a ) ∈ D ,( a − i − , c i ) and ( a − i − c i , a i ) are in D , for i = 1 , . . . , k − 1, and ( a k − , c k ) ∈ D . Thenthe interleaving c ∗ a of c and a is the word ( d , . . . , d k ) given by d = c a , d i = a − i − c i a i , for i = 1 , . . . , k − 1, and d k = a k − c k . We define a relation ≈ on theset of words by c ≈ d if and only if d = c ∗ a , for some word a . As shown in [3] if c is reduced then so is c ∗ a and the relation ≈ is an equivalence relation on the setof reduced words over P . The universal group U ( P ) of the pregroup P is the set ofequivalence classes of reduced words: the group operation being concatenation ofwords followed by reduction to a reduced word. As P embeds in U ( P ) then, if P isa K -pregroup it follows that U ( P ) is a K -group. A group G may be regarded as apregroup: with D = G × G and M the multiplication table of G . It is shown in [3]that U ( P ) is universal in the sense that, given a group H and a pregroup morphism θ from P to H , there is a unique extension of θ to a group homomorphism from U ( P ) to H . Lemma 3.4. Let P be a pregroup and let ( c , . . . , c m ) and ( d , . . . , d n ) be words.Then ( c , . . . , c m ) ≈ ( d , . . . , d n ) if and only if m = n and ( d − r − · · · d − c · · · c r − , c r ) ∈ D P and ( d − r , d − r − · · · d − c · · · c r ) ∈ D P ,r = 1 , . . . m , and d − m · · · d − c · · · c m = 1 .Proof. Write D = D P . Suppose first that ( c , . . . , c m ) ≈ ( d , . . . , d n ). Then bydefinition m = n and there exists an interleaving ( c , . . . , c m ) ∗ ( a , . . . , a m − ) = TABILITY OF UNIVERSAL EQUIVALENCE OF GROUPS UNDER FREE CONSTRUCTIONS9 ( d , . . . , d m ), for some a i ∈ P . Then (by definition again) with a = a m = 1 wehave ( a i − , c i ) and ( a i − , c i a i ) in D and d i = a i − c i a i . Thus ( c , a ) ∈ D and d = c a . Lemma 3.2 implies that ( d − , c ) ∈ D and d − c = a − .Assume inductively that( d − r − · · · d − c · · · c r − , c r ) ∈ D and ( d − r , d − r − · · · d − c · · · c r ) ∈ D and d − r · · · d − c · · · c r = a − r . As ( a − r , c r +1 ) and ( a − r c r +1 , a r +1 ) ∈ D and( a − r c r +1 ) a r +1 = d r +1 , Lemma 3.2 implies d − r +1 ( a − r c r +1 ) = a − r +1 . Combined withthe inductive hypothesis this shows that the ( r + 1)st version of this hypothesis alsoholds. Hence the statement of the inductive hypothesis holds for r = 1 , . . . , m . As a m = 1 we obtain, from the k th version d − m · · · d − c · · · c m = 1, as required.Conversely, suppose the conditions given in the lemma hold. Then ( d − , c ) ∈ D and so we may define a − = d − c . Two applications of Lemma 3.2 showthat ( c , a ) ∈ D and c a = d . Define a = 1 and suppose that a , . . . , a r have been defined such that ( a − i − , c i ) , ( a − i − c i , a i ) ∈ D a − i = d − i · · · d − c · · · d i and d i = a − i − d i a i , i = 1 , . . . r . Then ( d − r · · · d − c · · · c r , c r +1 ) and( d − r +1 , d − r · · · d − c · · · c r +1 ) ∈ D and we may set a − r +1 = d − r +1 · · · d − c · · · c r +1 = d − r +1 ( a − r c r +1 ). Two applications of Lemma 3.2 give a − r c r +1 a r +1 = d r +1 . Finallywe obtain a − m = d − m · · · d − c · · · c m = 1 so ( c , . . . , c m ) ∗ ( a , . . . , a m − ) is definedand equal to ( d , . . . , d m ) as required. (cid:3) Corollary 3.5. If Q is a subpregroup of a pregroup P then U ( Q ) is a subgroup of U ( P ) . In particular, if P is an S -pregroupthen U ( P ) is an S -group.Proof. To prove the first statement we need to show that if a and b are words over Q then a ≈ b in Q if and only if a ≈ b in P . Suppose that a ≈ b in P . Thenusing Lemma 3.4 and the definition of L pre -substructure we have a ≈ b in Q . Asthe opposite implication is immediate this proves the first part of the corollary. Forthe second statement suppose that K ∈ S and that K is a subset of a pregroup L , as in the definition above. As K ⊆ P we may assume that L ⊆ P and so K ⊆ U ( L ) ⊆ U ( P ). (cid:3) Theorem 3.6. Let P and P be S -pregroups. If P ≡ ∃ P with respect to L pre S then U ( P ) ≡ ∃ U ( P ) with respect to L S .Proof. Let U i = U ( P i ) and D i = D P i , for i = 1 , 2. We shall show that F ( U ) ≡F ( U ) and the theorem will then follow from Proposition 2.2.Let F = { ˜ u , . . . , ˜ u m } be a finite subset of U . For each i choose a representative u i of ˜ u i and write it as a reduced word u i = ( c i , . . . , c im i ) over P . Let S = ∪ mi =1 ∪ m i j =1 { c ij } and for all r ≥ S r +1 = S r ∪ { ab : a, b ∈ S i and ( a, b ) ∈ D } .Let J = max { m i : i = 1 , . . . , m } and define S = S J . As P ≡ ∃ P there is, usingProposition 2.2, an L pre S -isomorphism φ from S to a subset T of P . Note thatsetting T = φ ( S ) we may define T r as we have defined S r , with T in place of S and P in place of P . Then, by definition of isomorphism and by construction of S it follows that φ ( S r ) = T r , for r = 0 , . . . , J , so T = T J . Let c = ( c , . . . , c k )be a word over S (i.e. c i ∈ S , for all i ) and let φ ( c i ) = d i . Then d = ( d , . . . , d k )is a word over T and we define a map θ from words over S to words over T bysetting θ ( c ) = d . In this case, for all such c , we have ( c i , c i +1 ) ∈ D if and only if( d i , d i +1 ) ∈ D , so θ maps reduced words to reduced words. Now let p = ( p , . . . , p m ) and p = ( p , . . . , p m ) be words over S ,with m i ≤ J . Let φ ( p ij ) = q ij , and let θ ( p i ) = q i = ( q i , . . . , q im i ), i =1 , 2. From Lemma 3.4 we have p ≈ p if and only if m = m = k ,( p − ,r − · · · p − , p , · · · p ,r − , p ,r ) and ( p − ,r , p − ,r − · · · p − , p , · · · p ,r − p ,r ) belongto D , for r = 1 , . . . , k , and p − ,k · · · p − , p , · · · p ,k = 1. Since all the elementsof P involved in these conditions belong to S , the conditions hold if and only ifthey hold on replacing p ij with q ij . Hence p ≈ p if and only if q ≈ q . Therefore θ induces a map ˜ θ from equivalence classes of reduced words over S , of length atmost J , to equivalence classes of reduced words over T .Let ˜ S and ˜ T denote the sets of equivalence classes of reduced words of lengthat most J , over S and T respectively. To see that the map that ˜ θ is an L S -morphism from ˜ S to ˜ T consider a word (not necessarily reduced) p = ( p , . . . , p k )over S of length k ≤ J . Let q i = φ ( p i ) and let θ ( p ) = q = ( q , . . . , q k ). Weclaim that for r with 0 ≤ r ≤ k − r reductions which wemay apply to p , resulting in a word p r , if and only if there is a correspondingsequence of r reductions which we may apply to q resulting in a word q r suchthat θ ( p r ) = q r . Moreover p r ∈ S r and q r ∈ T r . This holds trivially for r = 0.Suppose that it holds for 0 , . . . , r , for some 0 ≤ r ≤ k − 2. Let p r = ( p r, , . . . , p r,s )and q r = (( q r, , . . . , q r,s ), with p r ∈ S r and q r ∈ T r and q r = θ ( p r ). We mayapply a reduction to p r if and only if ( p r,i , p r,i +1 ) ∈ D , for some i , in whichcase we may define p r +1 = ( p r, , . . . , p r,i p r,i +1 , . . . p r,s ) and then p r +1 ∈ S r +1 .Since φ is an L pre S -isomorphism this occurs if and only if ( q r,i , q r,i +1 ) ∈ D , inwhich case we may define q r +1 = ( q r, , . . . , q r,i q r,i +1 , . . . q r,s ) and then q r +1 ∈ T r +1 .Since θ ( p r ) = q r it follows that θ ( p r +1 ) = q r +1 and so the claim holds for all r . Now let ˜ p and ˜ p be elements of ˜ S and let p and p be reduced words, oflength at most J , over S representing ˜ p and ˜ p , respectively. Suppose that p is a reduced word (over S ) obtained from the concatenation p p by a sequenceof reductions. Then, in U , we have ˜ p ˜ p = ˜ p , where ˜ p is the equivalence class of p . Let q i = θ ( p i ) and q = θ ( p ). Then, from the above, the concatenation q q reduces to q , which is a reduced word over T . Hence, in U , ˜ q ˜ q = ˜ q , where ˜ q isthe equivalence class of q . Now, in the case where p is a word over S we have˜ θ (˜ p )˜ θ (˜ p ) = ˜ q ˜ q = ˜ q = ˜ θ (˜ p ) = ˜ θ (˜ p ˜ p ), showing that ˜ θ is an L S -morphism. Usingthe result of the first half of this paragraph and the fact that ˜ θ is bijective we canshow that ˜ θ − is also an L S -morphism. In particular ˜ θ restricted to F is an L S isomorphism onto its image. Therefore F ( U ) = F ( U ), as required. (cid:3) Applications In this section we apply Theorem 3.6 to prove that the universal equivalence ofpregroups translates nicely into universal equivalence of free constructions.4.1. Free products. To simplify notation we assume from the outset that wehave two groups A and B whose intersection is the identity element. In this caselet P = A ∪ B and set D = ( A × A ) ∪ ( B × B ). Then P is a pregroup and U ( P ) = A ∗ B . Proposition 4.1. Let A , B , A and B be groups such that A ∩ B = A ∩ B = 1 .If F ( A ) ≡ F ( A ) and F ( B ) ≡ F ( B ) then A ∗ B is existentially equivalent to A ∗ B . TABILITY OF UNIVERSAL EQUIVALENCE OF GROUPS UNDER FREE CONSTRUCTIONS11 Proof. Let P = A ∪ B and P = A ∪ B be two pregroups as above. Let S be afinite subset of P in the language L pre . Then S = ( S ∩ A ) ∪ ( S ∩ B ) = S A ∪ S B .Let S ′ A and S ′ B be two finite subsets of A and B in the language of groups L ,isomorphic to S A and S B , respectively. Then S ′ = S ′ A ∪ S ′ B is a subset of P isomorphic to S in the language L pre . By Proposition 2.2, P ≡ ∃ P in the language L pre , and by Theorem 3.6 A ∗ B ≡ ∃ A ∗ B in the language L . (cid:3) Free Products with Amalgamation. Again it simplifies notation to assumethat A and B are C -groups which intersect in the designated copy of the subgroup C , where C = 1. In this case let P = A ∪ B and set D = ( A × A ) ∪ ( B × B ). Then P is a C -pregroup and U ( P ) = A ∗ C B . Proposition 4.2. Let A , B , A and B be C -groups such that A ∩ B = A ∩ B = C . If F L C ( A ) ≡ F L C ( A ) and F L C ( B ) ≡ F L C ( B ) then the group A ∗ C B isexistentially equivalent to A ∗ C B in the language L C and, a fortiori, in thelanguage L .Proof. Let P = A ∪ B and P = A ∪ B , be two C -pregroups as above. Let S bea finite subset of P in the language L pre C . Let S A = S ∩ A and S B = S ∩ B so S = S A ∪ S B . Let S ′ A and S ′ B be two finite subsets of A and B in the languageof C -groups L C , isomorphic to S A and S B , respectively. Then S ′ = S ′ A ∪ S ′ B isa subset of P isomorphic to S in the language L pre C . By Proposition 2.2, P ≡ ∃ P in the language L pre C , and by Theorem 3.6 A ∗ C B ≡ ∃ A ∗ C B in the language L C . (cid:3) HNN-Extensions. Given a group G and an isomorphism θ : C → C , where C and C are subgroups of G , let t be a symbol not in G and(2) P = G ∪ t − G ∪ Gt ∪ t − Gt. Let P be the set of equivalence classes of the equivalence relation generated by t − ht = θ ( h ), for all h ∈ C . Set D = [ ε ,ε ,ε =0 , t − ε Gt ε × t − ε Gt ε ⊆ P × P. (The equivalence relation means that ( p, θ ( c )) and ( θ ( c ) , p ) belong to D for all c ∈ C and p ∈ P .) Then P and D constitute a pregroup and it can be verified that C and C embed in P . Hence P is an S -pregroup, where S = { C , C } . Moreover U ( P ) is the HNN-extension h G, t | t − ct = θ ( c ) , c ∈ C i , which is an S -group (withconstants C and θ ( C ) = C ). Proposition 4.3. Let A and A be S -groups, where S = { C , C } and θ : C → C is an isomorphism. If F L S ( A ) ≡ F L S ( A ) then the group G = h A , t | t − ct = θ ( c ) , c ∈ C i is existentially equivalent, in the language L S and in the language L ,to the group G = h A , t | t − ct = c, c ∈ C i .Proof. Let P and P be the two S -pregroups corresponding to A and A , respec-tively, as defined above, and let P , and P , be the underlying sets, as in (2).Let S be a finite subset of P in the language L pre S . Let ˆ S ⊆ P , be the union ofall the equivalence classes of elements of S . Then ˆ S is a disjoint union of 4 sets, S = ˆ S ∩ A , S = ˆ S ∩ t − A , S = ˆ S ∩ A t and S = ˆ S ∩ t − A t . To obtaincorresponding sets in A define T = S , T = tS , T = S t − and T = tS t − .By hypothesis there exist subsets T ′ i ⊆ A , such that T i ∼ = L S T ′ i , for i = 1 , . . . , Set S ′ = T ′ , S ′ = t − T ′ , S ′ = T ′ t and S ′ = t − T ′ t . Define ˆ S ′ = S ′ ∪ S ′ ∪ S ′ ∪ S ′ .The L S -isomorphisms between the T i ’s and the T ′ i ’s induce a bijection from ˆ S toˆ S ′ and by construction this isomorphism factors through the equivalence relationson P , and P , to give an L pre S -isomorphism between S and the quotient S ′ of ˆ S ′ in P . Applying Proposition 2.2 and Theorem 3.6, G is universally equivalent to G in the language L S (and consequently in the language L ). (cid:3) References [1] G. Baumslag, A. Myasnikov, and V.N. Remeslennikov, Algebraic geometry over groups. I.Algebraic sets and ideal theory, J. Algebra (1999), 16-79.[2] D. Marker Model Theory: An Introduction. Springer, 2002.[3] J. Stallings, Group Theory and Three-Dimensional Manifolds. Yale Mathematical Mono-graphs , Yale Univ. Press, 1971. School of Mathematics and Statistics, University of Newcastle-upon-Tyne,Newcastle-Upon-Tyne NE1 7RU, United Kingdom E-mail address : [email protected] Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St.West, Montreal, Quebec H3A 2K6, Canada E-mail address : [email protected] Institute of Mathematics (Russian Academy of Science), 13 Pevtsova St., Omsk,644099, Russia E-mail address ::