aa r X i v : . [ m a t h . L O ] A p r DESCRIBING GROUPS
MENG-CHE “TURBO” HO
Abstract.
We study two complexity notions of groups – the syntactic com-plexity of a computable Scott sentence and the m -degree of the index set ofa group. Finding the exact complexity of one of them usually involves find-ing the complexity of the other, but this is not always the case. Knight etal. [CHKM06], [CHK + Q . In some of these cases, wealso show that the sentence we give are optimal. In the last section, we alsoshow that d-Σ ( ∆ in the complexity hierarchy of pseudo-Scott sentences,contrasting the result saying d- Σ = ∆ in the complexity hierarchy of Scottsentences, which is related to the boldface Borel hierarchy. Introduction
One important aspect of computable structure theory is the study of the computability-theoretic complexity of structures. Historically, there are many natural questionsof this flavor even outside the realm of logic. For example, the word problem forgroups asks: for a given finitely-generated group, is there an algorithm that candetermine if two words are the same in the group? It was shown in [Rab60] thatthere is such an algorithm if and only if the group is computable in the sense ofcomputability theory.In this work, we will study the computability-theoretic complexity of groups.Among many different notions of complexities of a structure, we look at the quan-tifier complexity of a computable Scott sentence and the complexity of the indexset.1.1.
Background in recursive structure theory.
Instead of just using the first-order language, we will work in L ω ,ω . This is the language where we allow count-able disjunctions and countable conjunctions in addition to the usual first-orderlanguage. An classic theorem of Scott shows that this gives all the expressivepower one needs for countable structures. Theorem 1.1 (Scott, [Sco65]) . Let L be a countable language, and A be a countablestructure in L . Then there is a sentence in L ω ,ω whose countable models are exactlythe isomorphic copies of A . Such a sentence is called a Scott sentence for A . To work in a computability setting, this is not good enough, because we alsowant the sentence to be computable in the following way:
Date : July 2, 2018.
Definition 1.2.
We say a set is computably enumerable ( c.e. , or recursively enu-merable , r.e. ) if there is an algorithm that enumerates the elements of the set.We say a sentence (or formula) in L ω ,ω is computable if all the infinite conjunc-tions and disjunctions in it are over c.e. sets. Similarly, we define a computableScott sentence to be a Scott sentence which is computable.All the L ω ,ω sentences and formulas we mention in this chapter will be com-putable, so we will say Scott sentence instead of computable Scott sentence.We say a structure is computable if its atomic diagram is computable. We alsoidentify a structure with its atomic diagram. However, the effective Scott theoremis not true, that is, not all computable structures have a computable Scott sentence.We say an L ω ,ω formula is Σ or Π if it is finitary (i.e. no infinite disjunction orconjunction) and quantifier free. For α >
0, a Σ α formula is a countable disjunctionof formulas of the form ∃ xφ where φ is Π β for some β < α . Similarly, a Π α formulais a countable conjunction of formulas of the form ∀ xφ where φ is Σ β for some β < α . We say a formula is d-Σ α if it is a conjunction of a Σ α formula and a Π α formula. The complexity of Scott sentences of groups will be one of the main topicsthroughout this chapter.Another complexity notion we will study is the following: Definition 1.3.
For a structure A , the index set I ( A ) is the set of all indices e such that φ e gives the atomic diagram of a structure B with B ∼ = A .There is a connection between the two complexity notions that we study: Proposition 1.4.
For a complexity class Γ , if we have a computable Γ Scott sen-tence for a structure A , then the index set I ( A ) in Γ . This proposition and many examples lead to the following thesis:For a given computable structure A , to calculate the precise com-plexity of I ( A ), we need a good description of A , and once we havean “optimal” description, the complexity of I ( A ) will match thatof the description.In this chapter, we focus on the case where the above-mentioned structures aregroups. The thesis is shown to be false in [KM14], where they found a subgroup of Q with index set being d-Σ which cannot have a computable d-Σ Scott sentence.However, in the case of finitely-generated groups, the thesis is still open, and thegroups we considered give further evidence for the thesis in this case. For morebackground in computable structure theory, we refer the reader to [AK00].1.2.
Groups.
We fix the signature of groups to be {· , − , } . Throughout thechapter, we will often identify elements (words) in the free group F k = F ( x , . . . , x k )with functions from G k → G , by substituting x i by the corresponding elements from G , and do the group multiplication in G . Here we restate the relation between wordproblem and computability of the group: Theorem 1.5 ([Rab60]) . A finitely-generated group is computable if and only if ithas solvable word problem.
In this chapter, all the groups we consider will be computable.
ESCRIBING GROUPS 3
History.
Scott sentences and index sets for many classes of groups have beenstudied, for example, reduced abelian p -groups [CHKM06], free groups [CHK + D ∞ , and torsion-freeabelian groups of rank 1 [KS]. We will not list all the results, but will mentionmany of them as needed.1.4. Overview of results.
For the reader’s convenience, we summarize the mainresults of each section: • (Section 2) Every polycyclic group (including the nilpotent groups) has acomputable d-Σ Scott sentence, and the index set of a finitely-generatednon-co-Hopfian nilpotent group is m -complete d-Σ . • (Section 3) Certain finitely-generated solvable groups, including ( Z /d Z ) ≀ Z , Z ≀ Z , and the solvable Baumslag–Solitar groups BS (1 , n ), have computabled-Σ Scott sentences and their index sets are m -complete d-Σ . • (Section 4) The infinitely-generated free nilpotent group has a computableΠ Scott sentence and its index set is m -complete Π . • (Section 5) We give an example of a subgroup of Q whose index set is m -complete Σ , achieving an upper bound of such groups given in [KS]. • (Section 6) We give another example of a subgroup of Q which has bothcomputable Σ and computable Π pseudo-Scott sentences, but has nocomputable d-Σ pseudo-Scott sentence, contrasting a result in the Borelhierarchy of Mod( L ).2. Finitely-generated nilpotent and polycyclic groups
In this section, we will focus on finitely-generated groups, especially nilpotentand polycyclic groups. A priori, even if a structure is computable, it might nothave a computable Scott sentence. However, the following theorem says that acomputable finitely-generated group always has a computable Scott sentence.
Theorem 2.1 (Knight, Saraph, [KS]) . Every computable finitely-generated grouphas a computable Σ Scott sentence.
If we think of nilpotent and polycyclic groups as classes of “tame” groups, thenthe abelian groups are the “tamest” groups. Using the fundamental theorem offinitely-generated abelian groups, which says that every finitely-generated abeliangroup is a direct sum of cyclic groups, one can obtain the following theorem sayingthat the previous computable Scott sentences are not optimal in the case of abeliangroups:
Theorem 2.2 (Knight, Saraph, [KS]) . Let G be an infinite finitely-generatedabelian group. Then G has a computable d- Σ Scott sentence. Furthermore, I ( G ) is m -complete d- Σ . This theorem, together with some other results in [KS], leads to the question:Does every finitely-generated group have a computable d-Σ Scott sentence? Gen-eralizing the previous theorem, we show that this is true for polycyclic groups,and also prove completeness for certain classes of groups. We start by giving thedefinition for several group-theoretic notions that are used in the discussion.
Definition 2.3.
For two subgroups
N, M of G , we write [ N, M ] to be the subgroupof G generated by all commutators [ n, m ] with n ∈ N and m ∈ M . MENG-CHE “TURBO” HO
For a group G , we inductively define G = G and G k +1 = [ G k , G ]. We call G k the k -th term in the lower central series . A group is called nilpotent if G k +1 is thetrivial group for some k , and the smallest such k is called the nilpotency class ofthe group. Note that G is abelian if its nilpotency class equals 1.We also inductively define Z ( G ) = 1 and Z k +1 ( G ) = { x ∈ G | ∀ y ∈ G, [ x, y ] ∈ Z i ( G ) } . We call Z k ( G ) the k -th term in the upper central series . It is well-knownthat a group G is nilpotent if and only Z k ( G ) = G for some k . In this case, thesmallest such k is equal to to the nilpotency class of the group.We define the free nilpotent group of rank m and class p by N p,m = F ( m ) /F ( m ) p +1 ,where F ( m ) is the free group of m generators. Definition 2.4.
For a group G , we inductively define G (0) = G and G ( k +1) =[ G ( k ) , G ( k ) ]. We call G ( k ) the k -th term in the derived series . In the case when k = 1, this is the derived subgroup G ′ = G (1) of G . A group is called solvable if G ( k ) = 1 for some k , and the smallest such k is called the derived length of thegroup. Note that G is abelian if its derived length equals 1. Definition 2.5.
A polycyclic group is a solvable group in which every subgroup isfinitely-generated.By definition, every polycyclic group is solvable. It is well known that everyfinitely-generated nilpotent group is polycyclic. It is also known that all poly-cylic groups have solvable word problem, and thus are computable. By contrast,an example of a finitely-presented solvable but non-computable group is given in[Har81].We start by giving a sentence saying a tuple generates a subgroup isomorphic to agiven finitely-generated computable group G . We first fix a presentation h a | R i of G , where R is normally closed. The solvability of the word problem of G then says R is computable. Throughout this chapter, we will write h x i ∼ = G to be shorthandfor ^ w ( a ) ∈ R w ( x ) = 1 ∧ ^ w ( a ) / ∈ R w ( x ) = 1 . Since R is computable, the sentence is also computable, and we see it is Π .And since x satisfies all the relations of a and nothing more, this sentence implies h x i ∼ = h a i = G . However, this actually says more – this sentence requires that thesetwo groups are generated in the same way. Thus, for instance, if a is a centralelement in G , then so is x . This will be a useful observation later. This also impliesthat the choice of presentation is relevant. In most of our discussion, the choicewill be implicit, which is usually the standard presentation (i.e. the one given inthe definition.)The following is a very useful lemma for finding a computable Scott sentence forfinitely-generated groups. We will use this lemma for both polycyclic and solvablegroups. Lemma 2.6 (Generating set lemma) . In a computable group G , if there is a non-empty computable Σ formula φ ( x ) such that every x ∈ G satisfying φ is a gener-ating tuple of the group, then G has a computable d- Σ Scott sentence.Proof.
Consider the Scott sentence which is the conjunction of the following:(1) ∀ x (cid:20) φ ( x ) → ∀ y W w w ( x ) = y (cid:21) ESCRIBING GROUPS 5 (2) ∃ x [ φ ( x ) ∧ h x i ∼ = G ]In (1), w ranges over all words in x .Note that (1) is Π and (2) is Σ , thus the conjunction is d-Σ . To see this is aScott sentence, pick a group H satisfying the sentence. Then pick a tuple x ∈ H that satisfies the second conjunct. The first conjunct then says H is generated by x , thus is isomorphic to G . (cid:3) We now are ready to state and prove our theorem about polycyclic groups, whichgeneralizes the result in [KS] about infinite finitely-generated abelian groups.
Theorem 2.7.
Every polycyclic group G has a computable d- Σ Scott sentence.Proof.
We will show the claim that there is a d-Σ formula φ ( x ) such that every x ∈ G satisfying φ is a generating set of the group. We induct on the derived lengthof G .When the derived length is 1, i.e. G is abelian, the statement of the theoremwas proved in [KS], but for the inductive hypothesis, we need to find φ for G . Forsimplicity, we think of G additively in this case. By the fundamental theorem ofabelian groups, suppose G ∼ = Z n ⊕ T , where T is the torsion part of G , and | T | = k .Let χ ( y ) be the (finitary) sentence saying the k -tuple y satisfies the atomic diagramof T . Then we consider φ ( x, y ) to be χ ( y ) ∧ ( ^ m> mx = 1) ∧ ( ^ det( M ) = ± ∀ z ^ h i j i∈ k n M z = x + h y i j i ) . Here, we use two tuples x and y for clarity, but one can concatenate them into justone tuple x .The first conjunct says that y is exactly the k torsion elements in the group. Thesecond conjunct says x is torsion-free. In the third conjunct, we are thinking z , x ,and h y i j i as row vectors, and M ranges over all n × n matrices with entries in Z anddeterminant not equal to ±
1. Thus V h i j i∈ k n M z = x + h y i j i is really saying M z = x modulo T . So, working modulo T and again thinking of the x i ’s as row vectors in Z ∼ = G/T , the third conjunct forces x , as an n × n matrix, to have determinant ± x is a basis of G modulo T , and so every x, y satisfying φ will generate thegroup G . Finally, we see that the sentence is Π , thus proving the induction base.Now we prove the induction step. Assume the claim is true for all polycyclicgroups with derived length less than that of G . In particular, the derived subgroup G ′ has a computable d-Σ formula φ G ′ as described in the claim. In G , G ′ is definedby the computable Σ formula G ′ ( x ) ≡ ∃ s _ w ∈ ( F | s | ) ′ x = w ( s ) . Thus, we may relativize φ by replacing ∃ xθ ( x ) by ∃ x ( G ′ ( x ) ∧ θ ( x )), ∀ xθ ( x ) by ∀ x ( G ′ ( x ) → θ ( x )), and adding one more conjunct V i G ′ ( x i ). This does not increasethe complexity of the sentences. Furthermore, every element of G satisfying therelativized version ˜ φ G ′ of φ G ′ generates G ′ in G . As in the base case, suppose G/G ′ ∼ = Z n ⊕ T , where T is the torsion part, and let χ ( y ) to be the atomic diagramof T . We consider φ ( x, y, z ) to be the conjunction of the following:(1) ( V m> mx ˆ = 1) ∧ ( V det( M ) = ± ∀ z V h i j i∈ k n M z ˆ = x + h y i j i ) MENG-CHE “TURBO” HO (2) ˆ χ ( y )(3) ˜ φ G ′ ( z )Notice that in (1) we still think of G/G ′ additively for clarity, while we shouldreally think of it multiplicatively since G is no longer abelian. This is very similarto the sentence in the base case, but everything is relativized. We write a ˆ= b todenote that ∃ g ( G ′ ( g ) ∧ a = bg ), i.e. a and b are equal in the quotient group, andthis is Σ . And we write a ˆ = b to denote the negation of a ˆ= b , which is Π . So,the complexity of (1) is still Π . For ˆ χ ( y ), again we replace all the = and = in χ bythe relativized versions ˆ= and ˆ =, hence making it d-Σ . The relativization doesn’tincrease the complexity of φ G ′ , thus the whole conjunct is d-Σ .Now (2) says y is T in G/G ′ , (1) says x generates Z n in G/G ′ , and (3) says z generates G ′ in G . Thus, x, y , and z together generate G , hence proving the claim.The theorem now follows from the generating set lemma (Lemma 2.6). (cid:3) We now turn our attention to index sets. We give some results on the complete-ness of index sets of nilpotent groups, but we need a group-theoretic lemma and adefinition first.
Proposition 2.8 (Finitely-generated nilpotent group lemma) . Every finitely-generatedinfinite nilpotent group has infinite center. In particular, its center is isomorphicto Z × A for some abelian group A .Proof. We induct on the nilpotency class. The statement is obvious when thenilpotency class is 1.Suppose N is a finitely-generated nilpotent group with finite center. It sufficesto show that N is finite. Let the order of the center Z ( N ) be k . Then Z ( N ) k = 1.Let the upper central series of N be 1 = Z ( N ) ⊳Z ( N ) ⊳Z ( N ) ⊳ · · · ⊳Z p ( N ) = N .For g ∈ Z ( N ) and h ∈ N , one has [ g, h ] ∈ Z ( N ). Thus, using the identity [ xy, z ] =[ y, z ] x [ x, z ], we have [ g k , h ] = [ g, h ] k = 1, and so g k ∈ Z ( N ), i.e., Z ( N ) /Z ( N ) hasexponent dividing k .Now consider M = N/Z ( N ). We have Z ( M ) = Z ( N ) /Z ( N ), thus has exponentdividing k . Since N is finitely-generated and nilpotent, so is M , and so is Z ( M ).Hence Z ( M ) is finite.But the nilpotency class of M is less than that of N , so by induction hypothesis, M is finite. Then | N | = | M | · | Z ( N ) | is also finite. (cid:3) Definition 2.9.
A group is co-Hopfian if it does not contain an isomorphic propersubgroup.Consider G to be a finitely-generated non-co-Hopfian group. Then let φ : G → G to be an injective endomorphism from G onto one of its proper isomorphic sub-groups. Then we can form the direct system G φ / / G φ / / G φ / / · · · , andwrite the direct limit as ˆ G . Since every finite subset of ˆ G is contained in some finitestage, ˆ G is not finitely-generated, thus is not isomorphic to G . This observationwill be used later.We’re now ready to prove the completeness result: Theorem 2.10.
The index set of a finitely-generated nilpotent group is Π -hard.Furthermore, the index set of a non-co-Hopfian finitely-generated nilpotent group N is d- Σ -complete. ESCRIBING GROUPS 7
Proof.
We start by proving the second statement. Fix φ to be an injective endo-morphism of N onto one of its proper isomorphic subgroups. Then we apply theconstruction above to obtain ˆ N .For a d-Σ set S , we write S = S r S , where S ⊇ S are both Σ sets, andwe let S ,s and S ,s be uniformly computable sequences of sets such that n ∈ S i ifand only if for all but finitely many s , n ∈ S i,s . Then we construct G n ∼ = ˆ N, n / ∈ S N, n ∈ S r S N × Z , n ∈ S ∩ S . To build the diagram of G n , at stage s , we build a finite part of G n and a partialisomorphism to one of these three groups based on whether n / ∈ S ,s , n ∈ S ,s r S ,s ,or n ∈ S ,s ∩ S ,s . It is clear how to build the partial isomorphisms, since all thegroups are computable, so we only need to explain how we can change betweenthese groups when S ,s and S ,s change.To change from N to ˆ N , we apply φ . Note that at every finite stage, the resultinggroup will still be isomorphic to N , but in the limit, it will be ˆ N if and only if weapply φ infinitely often, i.e., n / ∈ S , as desired.To change from N to N × Z , we simply create a new element a that has infiniteorder and commutes with everything else. To change from N × Z to N , we choosean element b of infinite order in Z ( N ) by the finitely-generated nilpotent grouplemma (Lemma 2.8). We collapse the new element a by equating it with a bigenough power of b . Again, this will result in the limiting group being N if wecollapse b infinitely often, i.e. n / ∈ S , and N × Z if we collapse b only finitely often,i.e. n ∈ S .The second statement follows from doing only the Π part of the above argument,i.e. constructing G n ∼ = ( N, n / ∈ S N × Z , n ∈ S . (cid:3) Note that here we do not have a completeness result for the class of co-Hopfianfinitely-generated nilpotent groups. For a discussion about this ad-hoc class ofgroups, we refer the readers to [Bel03]. However, “most” finitely-generated nilpo-tent groups are non-co-Hopfian, including the finitely-generated free nilpotent groups,and we have the following:
Corollary 2.11.
The index set of a finitely-generated free nilpotent group is d- Σ -complete.Note. Using the nilpotent residual property (Lemma 4.4), we can show the d-Σ completeness result for free nilpotent groups within the class of free nilpotentgroups, provided that the number of generators is more than the step of the group.For the definition and more discussion on the complexity within a class of groups,we refer the reader to [CHK + MENG-CHE “TURBO” HO
Proposition 2.12.
The index set of a computable finitely-generated non-co-Hopfiangroup is Σ -hard. On the other hand, a computable finitely-generated co-Hopfiangroup G has a computable d- Σ Scott sentence.Proof.
The first statement is proved by running the Σ part of the argument inTheorem 2.10.For the second statement, consider the computable Π formula φ ( x ) ≡ h x i ∼ = G .This is a non-empty formula, and since G is co-Hopfian, every realization of φ in G generates G . Thus by the generating set lemma (Lemma 2.6), G has a computabled-Σ Scott sentence. (cid:3) Some examples of finitely-generated solvable groups
In this section, we continue to look at the bigger, but still somewhat tame, classof finitely-generated solvable groups. Note that even though the class of solvablegroups is closed under subgroups, the class of finitely-generated solvable groups isnot. This leads to an inherent difficulty when dealing with solvable groups, namelya group could possibly contain a higher-complexity subgroup. For example, thelamplighter group, which we shall define later and prove to have a computabled-Σ Scott sentence, contains a subgroup isomorphic to Z ω , whose index set is m -complete Π .We start this section with the definition of the (regular, restricted) wreath prod-uct, which is a technique often used in group theory to construct counterexamples: Definition 3.1.
For two groups G and H , we define the wreath product G ≀ H of G by H to be the semidirect product B ⋊ H , where the base group B is the directsum of | H | copies of G indexed by H , and the action of H on B is by shifting thecoordinates by left multiplication.One important example of a finitely-generated solvable group is the lamplightergroup. It is usually defined as the wreath product Z / Z ≀ Z , but we will be lookingat two generalizations of it, ( Z /d Z ) ≀ Z and Z ≀ Z . Theorem 3.2.
The lamplighter groups L d = ( Z /d Z ) ≀ Z each have computable d- Σ Scott sentences. Furthermore, their index sets are m -complete d- Σ .Proof. To find the Scott sentence, we will use the generating set lemma (Lemma2.6). Consider the formula φ ( a, t ) ≡ ( h a, t i ∼ = L d ) ∧ ( ∀ s ^ i> a t = a ( s i ) ) ∧ ( ∀ b ^ k Y i ( b k i ) t i = a ) . In the second conjunct, s ranges over the group elements. In the second infiniteconjunction, k ranges over all sequences in Z indexed by Z and has only finitelymany, but at least two, nonzero entries. We first observe that the standard generatorsatisfies this formula, so φ does not define the empty set.Now let ( a, t ) be a tuple satisfying φ . The first conjunct implies that a is in thebase group, and t is not in the base group. The second conjunct says that if wethink of t as an element of the semidirect product L d ∼ = ( Z /d Z ) Z ⋊ Z , then the Z -coordinate of t is ±
1. The third coordinate then says that a does not have morethan one nonzero entries, and hence (by the first conjunct) the only nonzero entrymust be co-prime to d . Thus, a generates a copy of Z /d Z . Using conjugation by t to generate the other copies of Z /d Z , we see a together with t generate the whole ESCRIBING GROUPS 9 group. So by the generating set lemma (Lemma 2.6), L d has a computable d-Σ Scott sentence.To show completeness of the index set, fix φ to be an injective endomorphism of L d onto one of its proper isomorphic subgroups, say mapping the standard genera-tors ( a, t ) to ( a, t ). Let ˆ L d be the direct limit of L d φ / / L d φ / / L d φ / / · · · .For a d-Σ set S , we write S = S r S , where S ⊇ S are both Σ sets, andwe let S ,s and S ,s be uniformly computable sequences of sets such that n ∈ S i ifand only if for all but finitely many s , n ∈ S i,s . Then we construct G n ∼ = ˆ L d , n / ∈ S L d , n ∈ S r S ( Z /d Z ) ≀ Z , n ∈ S ∩ S . As in the nilpotent case (Theorem 2.10), we build a partial isomorphism of oneof these groups in stages. To change between L d and ˆ L d is the same as before, weapply φ whenever n / ∈ S ,m , and keep building L d otherwise.To change from L d to ( Z /d Z ) ≀ Z , we create a new element s to be the othergenerator of Z , and equate it with a big enough power of t to change back. Thiswill result in the limiting group being L d if we collapse s infinitely often, i.e. n / ∈ S ,and ( Z /d Z ) ≀ Z otherwise. (cid:3) Theorem 3.3.
Let L = Z ≀ Z . Then L has a computable d- Σ Scott sentence, and I ( L ) is m -complete d- Σ .Sketch of proof. The proof is essentially the same as above. The Π formula willbe φ ( a, t ) ≡ ( h a, t i ∼ = L ) ∧ ( ∀ s ^ i> a t = a ( s i ) ) ∧ ( ∀ b ^ l,k Y i ( b k i ) t li = a ) . The only difference is that we will also allow k to have only one nonzero entry whichis not ±
1, in addition to k ’s which have at least two nonzero entries. This is to ruleout the case where a is a power of the standard generator.For completeness, we will construct G n ∼ = ˆ L, n / ∈ S L, n ∈ S r S Z ≀ Z , n ∈ S ∩ S . (cid:3) Remark.
In fact, this theorem can be generalized to Z n ≀ Z m . However, we willomit the proof in the interest of space. We have to add into φ ( a, t ) extra conjunctsto make sure that a generates a copy of Z n and t generates Z m modulo the basegroup, and the extra conjuncts are similar to what we did in the polycyclic case(Theorem 2.7). In proving completeness, we use the direct limit and Z n ≀ Z ( m +1) asthe alternate structures.We now look at another class of groups, the Baumslag–Solitar groups, which arevery closely related to the lamplighter groups. Indeed, in [Sta06], it was shownthat BS (1 , n ) converges to Z ≀ Z as n → ∞ . We shall see great similarity in thearguments used for these groups also. Definition 3.4.
The
Baumslag–Solitar groups BS ( m, n ) are two-generator one-relator groups given by the presentation: BS ( m, n ) = h a, b | ba m b − = a n i Note that BS ( m, n ) ∼ = BS ( n, m ). Theorem 3.5. BS ( m, n ) is solvable if and only if | m | = 1 or | n | = 1 , in whichcase it is also not polycyclic and its derived length is 2. Theorem 3.6.
For each n , the solvable Baumslag–Solitar group BS (1 , n ) has acomputable d- Σ Scott sentence. Furthermore, its index set is m -complete d- Σ forevery n.Proof. BS (1 , n ) has the semidirect product structure B ⋊Z where B = Z [ n , n , . . . ] = { xy : y | n k for some k } , and the action of 1 ∈ Z on B is by multiplication by n . Againwe are abusing notation by writing BS (1 , n ) multiplicatively but B additively.For finding the Scott sentence, we consider the formula φ ( a, t ) ≡ ( h a, t i ∼ = BS (1 , n )) ∧ ( ∀ b ^ gcd( i,n )=1 b i = a ) . This is not empty because the standard generators (1 , , ∈ B ⋊ Z satisfy it.For a tuple ( a, t ) satisfying φ , the first conjunct guarantees that a is in the basegroup and t , as an element of the original Baumslag–Solitar group BS (1 , n ) = B ⋊Z ,has the Z coordinate being 1 in the semidirect product, because of the inclusionof the formula tat − = a d . The second conjunct guarantees that the B coordinateof a is xy for some x and y both dividing some power of n . Thus appropriatelyconjugating a by t , we see y ′ ∈ h a, t i for some y ′ , thus 1 ∈ h a, t i , and hence a and t generate the whole group. By the generating set lemma (Lemma 2.6), we obtaina computable d-Σ Scott sentence for BS (1 , n ).To show completeness, we first observe BS (1 , n ) is not co-Hopfian. Indeed, let p be a prime not dividing n , then consider the endomorphism sending a to a p andfixing b . This is injective but not surjective because, for instance, it misses theelement 1 in B . Thus, we construct G n ∼ = \ BS (1 , n ) , n / ∈ S BS (1 , n ) , n ∈ S r S ( B Z ) ⋊ ( Z ) , n ∈ S ∩ S , where in ( B Z ) ⋊ ( Z ), B Z is the direct sum of countably many copies of B , indexedby Z , and the action of the first coordinate of Z is by multiplying by n (to eachcoordinate), and the action of the second coordinate is by shifting the copies of B .The same argument as above will show this construction gives d-Σ completenessof I ( BS (1 , n )). (cid:3) Infinitely-generated free nilpotent groups
We will now turn our attention to infinitely-generated groups. In this section, westart with a natural continuation of Section 2, showing that the infinitely-generatedfree nilpotent groups N p, ∞ have a computable Π Scott sentence, and their indexsets are m -complete Π . We start by stating the following result for p = 1: ESCRIBING GROUPS 11
Theorem 4.1 ([CHKM06]) . The infinitely-generated free abelian group Z ω has acomputable Π Scott sentence. Furthermore, I ( Z ω ) is m -complete Π . To find a computable Scott sentence for the infinitely-generated free nilpotentgroup, we give a lemma analogous to the generating set lemma (Lemma 2.6).
Lemma 4.2 (Infinite generating set lemma) . Suppose G ∼ = h a , a , . . . | R i , where R is a normal subgroup of F ω . Let R i be R ∩ F a ,...,a i ⊂ F ω . If there are h γ k i k ∈ ω such that (1) γ k ( x ) implies h x i ∼ = h a , a , . . . , a k | R k i modulo the theory of groups. (2) G | = ∃ x γ ( x ) . (3) G | = ^ k ( ∀ x , . . . , x k [ γ k ( x , . . . , x k ) → ∀ y _ l ≥ k +1 ∃ x k +1 , . . . , x l γ l ( x , . . . , x l ) ∧ z ∈ h x , . . . , x l i ]) (“Every γ k ( x ) can be ‘extended’ ”. In a countable group,this implies x can be extended to a basis.)Then φ , the conjunction of the group axioms and the sentences (2) and (3), is aScott sentence of G .Proof. By assumption, G models φ . Let H be a countable group modeling φ . Wefirst choose x by (2). Note that (3) allows us to extend any x satisfying γ k togenerate any element in the group H . Thus, we enumerate H , and iterativelyextend x to generate the whole group H . If we consider the relations that hold onthe infinite limiting sequence x , the relation on x , . . . , x k is exactly R k by (1), thusthe group H = h x i ∼ = h a | R i = G . (cid:3) Corollary 4.3. N p, ∞ has a computable Π Scott sentence.Proof.
Let γ k ( y ) ≡ ( h y i ∼ = N p,k ) ∧ ( ∀ z V det( M ) = ± M z ˆ = y ), where in the secondconjunct the inequality is relativized (as in the polycyclic case, Theorem 2.7) to theabelianization H/H ′ and we are abusing notation and thinking of the abelianizationadditively. So for M z we mean matrix multiplication, thinking of each z i as a rowvector.To show the γ k ’s satisfy the extendibility condition, fix x satisfying γ k . Workingin the abelianization, after truncating the columns in which x has no nonzero entries, we write x in its Smith normal form, i.e. find invertible k × k and n × n matrices S and T such that SxT has all but the ( i, i )-th entries being zero. The second conjunctof γ k guarantees that all the ( i, i )-th entries are actually 1. Thus, x can be extendedto a basis of Z ∞ = ab( N p, ∞ ). By a theorem of Magnus ([MKS04, Lemma 5.9]), thisimplies that x can be extended to a basis of N p, ∞ , thus satisfying the extendibilitycondition.The γ k ’s are Π , and a direct counting shows that the Scott sentence we obtainfrom the previous lemma is Π . (cid:3) For completeness of the index set of N p, ∞ , we generalize the technique from theabelian case, but we will need the following group-theoretic lemma. Lemma 4.4 (Nilpotent residual property) . For n, m ≥ p , N p,n is fully residually- N p,m . I.e., for every finite subset S ⊂ N p,n , there exists a homomorphism φ : N p,n → N p,m such that φ is injective on S . Proof.
Baumslag, Myasnikov, and Remeslennikov in [BMR99] showed that anygroup universally equivalent to (i.e., having the same first-order universal theoryas) a free nilpotent group N p,m is fully residually- N p,m . Timoshenko in [Tim00]showed that N p,n and N p,m are universally equivalent for n, m ≥ p . Combiningthese two results, we prove the desired lemma.This is a combination of the result of Baumslag, Myasnikov, and Remeslennikov[BMR99] that any group universally equivalent to (i.e., having the same first-orderuniversal theory as) a free nilpotent group N p,m is fully residually- N p,m , and theresult of Timoshenko [Tim00] that N p,n and N p,m are universally equivalent for n, m ≥ p . (cid:3) Corollary 4.5. I ( N p, ∞ ) is m -complete Π for every p .Proof. Recall that COF, the index set of all cofinite c.e. sets, is m-complete Σ .We will reduce the complement of COF to I ( N p, ∞ ).We construct G n uniformly in n . We first fix an infinite set of generators a , a , . . . and p distinguished generators b , b , . . . , b p distinct from the a i ’s, andwe start the construction by constructing b . At a finite stage, if we see some nat-ural number k being enumerated into W n , we collapse a k by taking the subgroup N p,m ⊂ N p, ∞ generated by all the generators that have been mentioned so far.Having the b means m > p , so by the nilpotent residual property (Lemma 4.4), wecan embed what we have constructed so far into N p,m − with the same generatorsexcept a k .Thus, in the limit, if n ∈ COF, then we will collapse all but finitely many a i ’s, hence the limiting group will be a finitely-generated (free) nilpotent groupnot isomorphic to N p, ∞ ; and if n / ∈ COF, we will still have infinitely many of the a i ’s, and the limiting group will be isomorphic to N p, ∞ . This shows I ( N p, ∞ ) is m -complete Π . (cid:3) Remark.
In the corollary, one actually can prove that I ( N p, ∞ ) is m -complete Π within the class of free nilpotent groups. The interested reader can compare thisto the same result for infinitely-generated free abelian groups in [CHKM06].5. A subgroup of Q In this section, we will look at a special subgroup of Q . Knight and Saraph[KS, §
3] considered subgroups of Q , distinguishing between cases by looking at thefollowing invariants. Definition 5.1.
We write P to be the set of primes. Let G be a computablesubgroup of Q . Without loss of generality, we will assume 1 ∈ G , otherwise we cantake a subgroup of Q isomorphic to G containing 1. We define:(1) P ( G ) = { p ∈ P : G | = p ∤ } (2) P fin ( G ) = { p ∈ P : G | = p | p k ∤ k } (3) P ∞ ( G ) = { p ∈ P : G | = p k | k } Remark. (1) Define P k ( G ) = { p ∈ P : G | = p k | p k +1 ∤ } . Then twosubgroups G , H of Q are isomorphic if and only if P k ( G ) = ∗ P k ( H ) forevery k , with equalities holding on cofinitely many of k , and P ∞ ( G ) = P ∞ ( H ). S = ∗ T means S and T only differ by finitely many elements.(2) Since G is computable, P is Π , P fin ∪ P ∞ is Σ , P fin is Σ , and P ∞ isΠ . ESCRIBING GROUPS 13
Dividing the subgroups of Q into cases by these invariants, Knight and Saraphdetermined the upper and lower bound of complexities of Scott sentences and theindex sets in some cases. The case we consider here is when P is infinite, P fin isfinite (and thus, without loss of generality, empty), and P ∞ is infinite. This is case5 in [KS], and they have the following results: Theorem 5.2 ([KS]) . Let G be a computable subgroup of Q with | P | = ∞ , P fin = ∅ , and | P ∞ | = ∞ . Then (1) G has a computable Σ Scott sentence. (2) I ( G ) is d- Σ -hard. (3) If P ∞ is low, then I ( G ) is d- Σ . (4) If P ∞ is not high , then I ( G ) is not m -complete Σ . It was not known that if there is a subgroup of Q as in the theorem that has m -complete Σ index set, thus achieving the upper bound in (1). In Proposition5.4, we shall give such an example.Also, the following theorem shows that such a group does not have a computabled-Σ Scott sentence unless P is computable. Theorem 5.3 ([KM14]) . Let G be a computable subgroup of Q with | P | = ∞ , P fin = ∅ , and | P ∞ | = ∞ , and suppose P ∞ is not computable. Then G does nothave a computable d- Σ Scott sentence.
Thus, when P ∞ is low but not computable, this gives a negative answer to theconjecture that the complexity of the index set should equal to the complexityan optimal Scott sentence. Continuing in this direction, we give an example of asubgroup in this case where the two complexities do equal each other, and are bothΣ . Proposition 5.4.
Let K be the halting set. Let G ⊆ Q be a subgroup such that ∈ G , P ∞ ( G ) = { p n ∈ P | n ∈ K } , and P fin ( G ) = ∅ . Then I ( G ) is m -complete Σ .Proof. Fix n . We construct G n so that G n ∼ = G if and only if n ∈ COF.For every s , we can recursively find the index k s = e of a program such that φ e ( e ) = ( ↓ , if φ n ( s ) ↓↑ , if φ n ( s ) ↑ . Now we construct G n by making p k s | k s . We also make p i divideevery element if we see i ∈ K .Now we verify G n ∼ = G if and only if n ∈ COF. We first observe that P ∞ ( G n ) = { p i | i ∈ K } . It’s also clear from construction that P fin ( G n ) = { p k s | k s / ∈ K } . But k s ∈ K if and only if φ n ( s ) ↓ , thus P fin ( G ) = { p k s | φ n ( s ) ↑} .Now, P ∞ ( G n ) = { p i | i ∈ K } = P ∞ ( G ). Thus G n ∼ = G iff P fin ( G n ) = ∗ P fin ( G ) = ∅ iff P fin ( G n ) = { p k s | φ n ( s ) ↑} is finite iff n ∈ COF. (cid:3)
Remark.
Note that this argument works for any X ≡ m K . It is natural to then askthat whether we can find a Turing-degree based characterization of when the indexset will be m -complete Σ . In the next section, we will show this cannot be found. Complexity hierarchy of pseudo-Scott sentences
In this section, we continue looking at subgroups of Q as above. We first give thedefinition of a pseudo-Scott sentence, which, just like a Scott sentence, identifies astructure, but only among the computable structures. Note that every computableScott sentence is a pseudo-Scott sentence. Definition 6.1. A pseudo-Scott sentence for a structure A is a sentence in L ω ,ω whose computable models are exactly the computable isomorphic copies of A .Similar to the case of computable Scott sentences, a pseudo-Scott sentence of astructure yields a bound on the complexity of the index set of the structure.We shall give an example of a group which has a computable Σ pseudo-Scottsentence and a computable Π pseudo-Scott sentence, but no computable d-Σ pseudo-Scott sentence. This is related to a question about the effective Borel hier-archy in Mod( L ).Consider the complexity hierarchy of (computable pseudo-)Scott sentences. Since α r ( β r γ ) = ( α ∧ ¬ β ) ∨ ( α ∧ γ ), we see that the hierarchy collapses in the sensethat the complexity classes k -Σ n are all the same for k ≥ n +1 and d-Σ n are thesame or not for regular, computable, and pseudo-Scott sentences. The complexityhierarchy of Scott sentences (computable Scott sentences, respectively) is relatedto the boldface (effective, respectively) Borel hierarchy on the space Mod( L ), see[Vau75] and [VB07]. In [Mil78], it was shown that ∆ n +1 and d- Σ n are the samein the boldface case, i.e. if a structure has a Σ n +1 Scott sentence and a Π n +1 Scottsentence, then it also has a d-Σ n Scott sentence. This gives a positive answer tothe question for Scott sentences. However, we will prove that this is not true in thecomplexity hierarchy of computable pseudo-Scott sentences by giving a subgroup G ⊂ Q which has a computable Σ pseudo-Scott sentence and a computable Π pseudo-Scott sentence, but no computable d-Σ pseudo-Scott sentence. This givesa negative answer to the question for pseudo-Scott sentences. The question ofwhether ∆ n +1 and d-Σ n are the same in the effective case (the complexity hierarchyof computable Scott sentence) remains open.We start by strengthening a result in [KM14]. The first part of the proof wherewe construct the theory T is unchanged. Lemma 6.2.
Fix a non-computable c.e. set X , and let G ⊂ Q be a subgroup suchthat ∈ G , P ∞ ( G ) = { p i | i ∈ X } , and P fin ( G ) = ∅ . Then G does not have acomputable d- Σ pseudo-Scott sentence.Proof. Suppose G has a computable d-Σ pseudo-Scott sentence φ ∧ ψ , where φ is computable Π and ψ is computable Σ . Let α be a computable Π sentencecharacterizing the torsion-free abelian groups A of rank 1 such that P ∞ ( A ) ⊆ X .By [KM14, Lemma 2.3], α ⊢ φ , thus we can replace φ by α in the pseudo-Scottsentence.Also, again by [KM14], we can assume ψ has the form ∃ x χ ( x ) where x is asingleton and χ ( x ) is a c.e. conjunction of finitary Π formulas.Now we consider the first-order theory T in L with an extra constant symbol c to be union of the following sentences:(1) axioms of torsion-free abelian groups,(2) ∀ x ∃ y py = x for each p ∈ X , ESCRIBING GROUPS 15 (3) ρ i ( c ) for every finitary Π conjunct ρ i ( x ) of the Π sentence χ ( x ).Now we show that [KM14, Lemma 2.4] is still true: Claim.
For every i / ∈ X , there is some k such that T ⊢ p ki ∤ c .Proof of claim. Suppose this is not true. There is n / ∈ X such that T ∪ { p kn | c } ∞ k =1 is consistent. Take a model H | = T ∪ { p kn | c } ∞ k =1 , and let C ⊂ H be the subgroupconsisting of rational multiples of c .Let K ⊂ Q be the computable group with 1 ∈ K , P ∞ ( K ) = { p i | i ∈ X } ∪ { p n } ,and P fin ( K ) = ∅ . Then K is isomorphic to a subgroup of C . By interpreting theconstant symbol c in K as the preimage of c ∈ C , K is a substructure of H . Thusall the finitary Π statements ρ i ( c ) are also true in K .So, K is a torsion-free rank 1 abelian group satisfying T , thus K | = φ ∧ ψ . Since K is computable, it is isomorphic to G , but P ∞ ( G ) = P ∞ ( K ), a contradiction. (cid:3) Now we have a computable theory T such that for every i / ∈ X , there is some k such that T ⊢ p ki ∤ c . Therefore, the complement of X is c.e., and this contradictsthe assumption that X is non-computable c.e. (cid:3) We also need the following lemma:
Lemma 6.3.
There exists a c.e. set X ⊆ ω with X ≡ T ′ satisfying the following:There is a uniformly c.e. sequence S n so that if W n ⊃ X , W n = ∗ X , then S n is aninfinite c.e. subset of W n r X .Proof. Write I i = [ i ( i +1)2 , ( i +1)( i +2)2 ). Note that | I i | = i + 1. Consider the followingrequirements: • R i : X ∩ I i = ∅ if and only if i ∈ ′ • Q k : build an infinite c.e. S k ⊂ W k r X if W k ⊃ X and W k = ∗ X If at some stage R i sees i ∈ ′ , then it puts some element of I i that is not yetblocked by higher priority Q k ’s into X . Q k will attempt to put n into S k whenever n ∈ W k r X at stage s . Suppose n ∈ I i . If i < k , then Q k does nothing. If i > k , then Q k puts n into S k , and blocks R i from enumerating n into X , but Q k will also block itself from enumerating otherelements of I i into S k .Note that for each R i , at most i elements of I i will be blocked, because for each I i , every higher priority Q k will block at most one element. Thus R i can alwayssatisfy the requirement.Also, if W k ⊃ X and W k = ∗ X , then Q k will eventually enumerate infinitelymany numbers into S k , since after enumerating finitely many of them, there areonly finitely many things blocked by Q k itself and finitely many higher priority R i ’s. Lastly, S k will be disjoint from X because whenever n is enumerated into S k ,the R i ’s will be blocked from enumerating it into X . (cid:3) Theorem 6.4.
There exists a group with both computable Σ and computable Π pseudo-Scott sentences (i.e. ∆ ), but no computable d- Σ pseudo-Scott sentence.Proof. Choose X as in the previous lemma. Consider the subgroup G ⊂ Q with P fin ( G ) = ∅ and P ∞ ( G ) = X . By [KM14], G has a computable Σ (pseudo-)Scott sentence. By Lemma 6.2, G does not have a computable d-Σ pseudo-Scottsentence. Let φ be the conjunction of V S k ∃ g V p ∈ S k p ∤ g , “ G is a subgroup of Q ”, and“ P ∞ ( G ) ⊇ X ”.Say H | = φ be a computable group. Then P ∞ ( H ) ∪ P fin ( H ) is c.e., so let n be such that W n = P ∞ ( H ) ∪ P fin ( H ). Note that W n ⊃ X . If H ≇ G , then W n = ∗ X . Now consider S n ⊂ Y r X , and the corresponding conjunct in φ whichsays ∃ g V p ∈ S Y p ∤ g . But every element in H is divisible by all but finitely manyelements from W n , and S n is an infinite subset of W n , so H cannot model thisexistential sentence, a contradiction. Thus φ is a computable Π pseudo-Scottsentence of G X . (cid:3) Remark.
Note that in the first countable conjunction, the set of indices of S Y for Y ⊃ X and Y = ∗ X is not c.e. However, the set of indices of S Y for all Y ⊂ ω is c.e., even computable, by construction, and G X still models φ for this biggerconjunction.Note that the two groups in Proposition 5.4 and Theorem 6.4 both have P fin = ∅ and P ∞ ≡ T ′ . However, one of them has index set being m -complete Σ , whilethe other has index set being ∆ . This tells us that we cannot hope to give aTuring-degree based characterization of which combinations of P , P fin , and P ∞ give m -complete Σ index sets and which do not. Acknowledgments.
The author had a lot of useful input and stimulating conver-sations during the course of the work, including with Julia Knight, Alexei Myas-nikov, Arnold Miller, and Steffen Lempp. Special thanks to Uri Andrews and toTullia Dymarz for being such supportive and fantastic advisors.
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