Metabelian groups: full-rank presentations, randomness and Diophantine problems
Albert Garreta, Leire Legarreta, Alexei Miasnikov, Denis Ovchinnikov
aa r X i v : . [ m a t h . G R ] J un Metabelian groups: full-rank presentations,randomness and Diophantine problems
Albert Garreta, Leire Legarreta, Alexei Miasnikov, Denis Ovchinnikov
Abstract
We study metabelian groups G given by full rank finite presentations x A | R y M in the variety M of metabelian groups. We prove that G isa product of a free metabelian subgroup of rank max t , | A | ´ | R |u anda virtually abelian normal subgroup, and that if | R | ď | A | ´ G is undecidable, while it is decidable if | R | ě | A | .We further prove that if | R | ď | A | ´ G all, but one, factors are virtually abelian. Since finite presentations havefull rank asymptotically almost surely, finitely presented metabelian groupssatisfy all the aforementioned properties asymptotically almost surely. Contents Introduction
In this paper we study finitely generated metabelian groups G given by fullrank finite presentations G “ x a , . . . , a n | r , . . . , r m y M in the variety M ofmetabelian groups, random metabelian groups in the few relators model, and theDiophantine problem in such groups. We prove that the Diophantine problemin the group G above is undecidable if m ď n ´ m ě n (thecase m “ n ´ m ď n ´ G all, but one, factors are virtually abelian; and, finally, G has a rathernice structure, namely, G is a product of two subgroups G “ HL , where H isa free metabelian group of rank max t n ´ m, u and L is a virtually abeliannormal subgroup of G . The class of metabelian groups admitting full rankpresentations is rather large. Indeed, it turns out that for fixed n and m afinite presentation x a , . . . , a n | r , . . . , r m y M has full rank asymptotically almostsurely. In particular, random metabelian groups (in the few relators model) havefull rank presentations asymptotically almost surely. Hence, they asymptoticallyalmost surely satisfy all the properties mentioned above. Let A “ t a , . . . , a n u be a finite alphabet, A ´ “ t a ´ , . . . , a ´ n u , A ˘ “ A Y A ´ , p A ˘ q ˚ the set of all (finite) words in A ˘ , and R “ t r , . . . , r m u a finite subsetof p A ˘ q ˚ . We fix this notation for the rest of the paper.A pair p A, R q is called a finite presentation , we denote it by x A | R y or x a , . . . , a n | r , . . . , r m y . If V is a variety or a quasivariety of groups then afinite presentation x A | R y determines a group G “ F V p A q{xx R yy , where F V p A q is a free group in V with basis A and xx R yy is the normal subgroup of F V p A q generated by R . In this case we write G “ x A | R y V . The relation matrix M p A, R q of the presentation x A | R y is an m ˆ n integral matrix whose p i, j q -thentry is the sum of the exponents of the a j ’s that occur in r i . It was introducedby Magnus in [16] (see also [15], Chapter II.3, for its ties to relation modulesin groups). The number d “ | A | ´ | R | , if non-negative, is called the deficiency of the presentation x A | R y (see [15], Chapter II.2 for a short survey on groupswith positive deficiency). The matrix M p A, R q has full rank if its rank is equalto min t| A | , | R |u , i. e., it is the maximum possible.We showed in [10] that if a finitely generated nilpotent group G admits afull-rank presentation, then G is either virtually free nilpotent (provided thedeficiency d ě d “ d ď M of all metabelian groups by full rank pre-sentations also have a rather restricted structure, as witnessed by the followingresult. Theorem 1.1.
Let G be a metabelian group given by a full-rank presentation G “ x A | R y M . Then there exist two finitely generated subgroups H and K of G such that:1. H is a free metabelian group of rank max p| A | ´ | R | , q , . K is a virtually abelian group with | R | generators, and its normal closure L “ K G in G is again virtually abelian;3. G “ x H, K y “ LH .Moreover, there is an algorithm that given a presentation G “ x A | R y M findsa free basis for the subgroup H and a generating set in | R | generators for thesubgroup K . The result above complements the Generalized Freiheissatz for M . In [22] Ro-manovskii proved that if a metabelian group G is given in the variety M by afinite presentation x A | R y M of deficiency d ě
1, then there is a subset of gener-ators A Ď A with | A | “ d which freely generates a free metabelian subgroup H “ x A y . Theorem 1.1 shows that if the presentation x A | R y M has full rankthen there is a free metabelian of rank d subgroup H of G and, in addition,there are virtually abelian subgroups K and L as described in items 2) and 3)above, such that G “ HL . Two remarks are in order here. First, the subgroup H in Theorem 1.1 is not necessary equal to x A y for a suitable A Ď A as in theRomanovskii’s result. However, for a given full rank presentation G “ x A | R y M of G one can find algorithmically another full rank presentation of G , which isin Smith normal form (see Section 2.1), such that the subgroup H is, indeed,generated by a suitable A Ď A and K is generated by A r A . Second, evenif the presentation G “ x A | R y M is in Smith normal form, but it is not of fullrank, then the subgroups K and L as in Theorem 1.1 may not necessarily exist(see details in Section 2.1).In another direction, we showed in [10] that in any direct decomposition of anilpotent group G given in the nilpotent variety N c , c ě
2, by a finite full rankpresentation of deficiency ě
1, all, but one, direct factors are finite.A similar result holds in the variety M as well. Theorem 1.2.
Let G be a finitely generated metabelian group given by a full-rank presentation G “ x A | R y M such that | R | ď | A | ´ . Then in any directdecomposition of G all, but one, direct factors are virtually abelian. In Section 3 we study the Diophantine problem in finitely generated metabeliangroups given by full rank presentations. This is a continuation of research in [7]and [10].Recall, that the
Diophantine problem in an algebraic structure A (denoted D p A q ) is the task to determine whether or not a given finite system of equationswith constants in A has a solution in A . D p A q is decidable if there is an algorithmthat given a finite system S of equations with constants in A decides whether ornot S has a solution in A . Furthermore, D p A q is reducible to D p M q , for anotherstructure M , if there is an algorithm that for any finite system of equations S in A computes a finite system of equations S M in M such that S has a solutionin A if and only if S M has a solution in M .3ote that due to the classical result of Davis, Putnam, Robinson and Matiya-sevich, the Diophantine problem D p Z q in the ring of integers Z is undecidable[4, 18]. Hence if D p Z q is reducible to D p M q , then D p M q is also undecidable.To prove that D p A q reduces to D p M q for some structures A and M it suf-fices to show that A is interpretable by equations (or e-interpretable ) in M .E-interpretability is a variation of the classical notion of the first-order inter-pretability, where instead of arbitrary first-order formulas finite systems of equa-tions are used as the interpreting formulas (see Definition 3.3 for details). Themain relevant property of such interpretations is that if A is e-interpretable in M then D p A q is reducible to D p M q by a polynomial time many-one reduction(Karp reductions). Theorem 1.3.
Let G be a metabelian group given by a full-rank presentation G “ x A | R y M . Then the following hold:1. If | R | ď | A | ´ then the ring of integers Z is e-interpretable in G , and theDiophantine problem in G is undecidable.2. If | R | ě | A | then the Diophantine problem of G is decidable (in fact, thefirst-order theory of G is decidable). This result is analogous to the one obtained for nilpotent groups in [10].
Remark 1.4.
For the case of deficiency 1 (i.e., m “ n ´ G remains an interesting open problem. The recentwork [14] proves decidability of the Diophantine problem in BS p , n q “ x a , a | a n ´ “ r a , a sy . Also some deficiency 1 presentations define cyclic groups, whichhave decidable Diophantine problem [6]. Conjecture 1.5.
Let G be a metabelian group given by a full-rank presentation G “ x A | R y M . If | A | ´ | R | “ then the Diophantine problem in G is decidable. The first notion of genericity or a random group in the class of finitely presentedgroups is due to Gromov [11], where he introduced what is now known as thefew relators model. A slightly different approach was suggested by Olshanskii[21] and Arzhantseva and Olshanskii [1].Nowadays, the few relators model can be described as follows. Let m, n befixed positive integers. Consider, in the notation above, the set S p n, m q of allfinite presentations x A | R y with | A | “ n, | R | “ m . For a given positive integer ℓ consider a finite subset S p n, m, ℓ q of S p n, m q which consists of all presentations x A | R y P S p n, m q , where each relator in R has length precisely ℓ . Now fora given property of groups P consider a subset S P p n, m, ℓ q of S p n, m, ℓ q of allpresentations x A | R y P S p n, m, ℓ q which define groups that satisfy P . Theproperty P is termed p n, m q -generic iflim ℓ Ñ8 | S P p n, m, ℓ q|| S p n, m, ℓ q| “ . | X | denotes the cardinality of a set X . In this event we also say sometimesthat P holds for x A | R y asymptotically almost surely as ℓ Ñ 8 .In the book [11] Gromov stated that the group property of being hyperbolicis p n, m q -generic for all n and m . Later Olshanskii [21] and Champetier [2] gaverigorous proofs of this result. We refer to [20] for a survey on random finitelypresented groups and to Kapovich and Schupp [13] on group theoretic models ofrandomness and genericity.Observe that the few relators model described above concerns classical finitelypresented groups. However, this approach can be utilized as well for finitelypresented groups in any fixed variety of groups V , in particular, for finitelygenerated metabelian groups given in the variety M of all metabelian groups byfinite presentations x A | R y M . Note that every finitely generated group in M has a finite presentation in M .For the variety of nilpotent groups N c (of a fixed nilpotency class c ), thisapproach has been studied recently in [3, 10]. Other models of randomness forthe groups in N c can be found in [5, 7]. To the best of our knowledge, there isno study of random metabelian groups (in any model) prior to this paper.The following result is fundamental to our approach. Theorem 1.6. [10] Let R be a set of m words of length ℓ in an alphabet A ˘ “t a ˘ , . . . , a ˘ n u , i.e. each word is obtained by successively concatenating randomlychosen letters from A ˘ with uniform probability. Then M p A, R q has full rank(i.e. rank p M p A, R qq “ min t n, m u ) asymptotically almost surely as ℓ Ñ 8 . Corollary 1.7.
Let x A | R y “ x a , . . . , a n | r , . . . , r m y be a presentation whereall relators r i have length ℓ ą . Then the presentation x A | R y has full rankasymptotically almost surely as ℓ tends to infinity. One of the main appeals of full-rank presentations is that they occur asymp-totically almost surely in the few-relators model for random groups in any variety V , in particular, in the variety M . Hence, random metabelian groups (in thefew relators model) have full rank presentations asymptotically almost surely.Therefore, all the properties described above for metabelian groups given byfull rank presentations are generic in the class of finitely presented metabeliangroups. We refer to Section 4 for precise statements of the results. Throughout the paper we use the following notation.Let G be a group. By γ i p G q we denote the i -th term of the lower centralseries of G , that is γ p G q “ G and γ i ` p G q “ r G, γ i p G qs for all i ě
1. By G p i q we denote the i -th term of the derived series of G , that is G p q “ G and G p i ` q “ r G p i q , G p i q s .We denote by N c and M the families of nilpotent groups of nilpotency classat most c , and of metabelian groups, respectively. In general, we refer to books515, 23] for the standard facts and notation in group theory and to [19] for basicnotions regarding varieties. In this section we obtain some structural results on full rank metabelian groups,in particular, Theorem 1.1.To prove this theorem we need some results on presentations in Smith normalform.Recall (see, for example, [25]), that an integer matrix A “ p a ij q is in Smithnormal form if there is some integer r ě d i “ a ii , ď i ď r are positive, A has no other nonzero entries, and d i divides d i ` for 1 ď i ă r . Definition 2.1.
A finite presentation x A | R y is said to be in Smith normalform if the relation matrix M p A, R q is in Smith normal form. Proposition 2.2.
For any finite presentation x A | R y there exists a finite pre-sentation in Smith normal form x A | R y , with | A | “ | A | , | R | “ | R | , and rank p M p A, R qq “ rank p M p A , R qq , such that for any variety V the groups G “ x A | R y V and G “ x A | R y V are isomorphic. Moreover, such a pre-sentation x A | R y and an isomorphism G Ñ G can be found algorithmically.Proof. The presentation x A | R y can be obtained by repeatedly applying Nielsentransformations on the tuples A and R . Indeed, note that such a Nielsen trans-formation has the effect in the relation matrix M p A, R q of adding or subtractingtwo columns or two rows, respectively. It is known (see [25]) that one can find afinite sequence of elementary row and column operations that transform M p A, R q into its Smith normal form. Performing the corresponding Nielsen transforma-tion on x A | R y one gets the required presentation x A | R y . The details of thisprocedure can be found in [10].The complexity of finding a presentation in Smith normal form and otheralgorithmic considerations regarding full-rank presentations and presentationsin Smith normal form will be studied in upcoming work. Lemma 2.3.
Let G be a metabelian group given by a full-rank presentation inSmith normal form: G “ x A | R y M “ x a , . . . , a n | a α “ c , . . . , a α m m “ c m y M , where c i P r F p A q , F p A qs , α i P Z r t u for all i “ , . . . , m , and m ď n . Then thefollowing holds:1. K “ x a , . . . a m y is a virtually abelian group, and its normal closure L “ K G in G is again virtually abelian;2. H “ x a m ` , . . . , a n y is a free metabelian group of rank n ´ m ;3. G “ x H, K y “ HL . roof. We show first that K “ x a , . . . a m y is virtually abelian. Note that K X G is abelian. Set N “ α ¨ ¨ ¨ α m . Then for all g P K one has g N P K X G . Hence K { K X G is finite, so K is virtually abelian. Similarly, for L “ K G the subgroup L X G is normal in L and abelian. The quotient L { L X G is abelian, of period N , and finitely generated (since L “ K p L X G qq , hence finite. It follows that L is abelian-by-(finite abelian). Note that L might not be finitely generated.Now we show that x a m ` , . . . , a n y is a free metabelian group of rank n ´ m .Assume that n ´ m ě
2. By Romanovski’s aforementioned result [22] (see thediscussion after Theorem 1.1), we know that there exists a subset A Ď A with | A | “ | A | ´ | R | such that x A y is free metabelian freely generated by A . Weclaim that A “ t a m ` , . . . , a n u . Indeed, otherwise there exists a i P A such that a ti P G for some t P Z r t u . Note that | A | ě
2, so there is a j P A with i ‰ j .It follows then that r a ti , r a i , a j ss “
1, a contradiction with the fact that A freelygenerates x A y as a free metabelian group.If n ´ m “ a n Ñ a i Ñ i “ , . . . , n ´ G Ñ Z . Hence, a n has infinite order in G , and x a n y is a freemetabelian group of rank 1. We note in passing, that in the case when m “ n ´ A that generates an infinite cyclic group (i.e., a free metabeliangroup of rank 1) is not necessarily unique, e.g., in BS p , n q – x a , a | a n ´ “r a , a sy , both a and a generate an infinite cyclic subgroup. Proof of Theorem 1.1.
Let G be a metabelian group given by a full-rank presen-tation G “ x A | R y M . By Proposition 2.2 one can find algorithmically anotherpresentation G x A | R y M “ x a , . . . , a n | r , . . . , r m y M of G which is in Smithnormal form.If m ď n then Lemma 2.3 applied to the presentation x A , | R y gives sub-groups H and K with the required properties. From these we obtain the requiredsubgroups in G by inverting the isomorphism x A | R y “ x A | R y .On the other hand, if m ą n , then G is a quotient of the full-rank metabeliangroup with zero deficiency G “ x a , . . . , a n | r , . . . , r n y (since M p A , R q is inSmith normal form, this is a full-rank presentation), and by the previous caseit follows that G is virtually abelian. Since G is a quotient of G , G is alsovirtually abelian. Remark 2.4.
Note, that if the presentation in Lemma 2.3 is in Smith normalform, but not of full rank, then by the Generalized Freiheissatz [22] the free sub-group H “ x A y of G exists, but there might not exists corresponding subgroups K and L for H . Indeed, let G “ x a , a , a , a | r a , a s “ , r a , a s “ y . Then a , a generate a free metabelian group of rank 2, but G { H G is freemetabelian of rank 2, so there is no a virtually abelian subgroup L , such that G “ HL . Remark 2.5.
Finally, note that Theorem 1.1 or Lemma 2.3 may not reveal fullythe structure of G and how it is related to the subgroups H and K . For example,7onsider G “ x a , a , a , a | a m “ r a , a s , a k “ r a , a sy . Lemma 2.3 tells us that x a , a y is freely generated by a , a and x a , a y is avirtually abelian group. On the other hand, it is easy to check that x a , a y – BS p , m ` q , x a , a y – BS p , k ` q , and G is the free metabelian product oftwo Baumslag-Solitar groups, which is not clear directly from the decomposition G “ HL . In this section we prove our main result on direct decomposition of metabeliangroups given by full rank presentations, namely Theorem 1.2.
Proof of Theorem 1.2.
We showed in [10] that if H is a finitely generated nilpo-tent group of class c ě H “ x A | R y N c with | R | ď | A | ´
1, then in any direct decomposition of H all, but one, directfactors are finite. We will use this fact in our proof. Let now G be a finitelygenerated metabelian group given by a full-rank presentation G “ x A | R y M such that | R | ď | A | ´
1. Assume G “ G ˆ ¨ ¨ ¨ ˆ G k for some k ě G i , i “ , . . . , k . Let π be the natural projection of G onto H “ G { γ p G q .The quotient H admits the full-rank presentation x A | R y N , so by the resultmentioned above all, but one, say π p G k q , of the groups π p G q , . . . , π p G k q arefinite. Hence ker π X G i has finite index in G i for all i “ , . . . , k ´
1. However ker π is abelian since ker π “ γ p G q ď r G, G s . In this section we introduce the technique of interpretability by systems of equa-tions. It is nothing else than the classical model-theoretic technique of inter-pretability (see [12, 17]), restricted to systems of equations (equivalently, positiveexistential formulas without disjunctions). In [8, 9] we used this technique tostudy the Diophantine problem in different classes of solvable groups and rings.In what follows we often use non-cursive boldface letters to denote tuples ofelements: e.g. a “ p a , . . . , a n q . Furthermore, we always assume that equationsmay contain constants from the algebraic structure in which they are considered. Definition 3.1.
A set D Ď M m is called definable by systems of equations in M , or e-definable in M , if there exists a finite system of equations, sayΣ D p x , . . . , x m , y , . . . , y k q , in the language of M such that for any tuple a P M m ,one has that a P D if and only if the system Σ D p a , y q on variables y has a solutionin M . In this case Σ D is said to e-define D in M . Remark 3.2.
Observe that, in the notation above, if D Ď M m is e-definablethen it is definable in M by the formula D y Σ D p x , y q . Such formulas are called positive primitive , or pp-formulas. Hence, e-definable subsets are sometimes8alled pp-definable. On the other hand, in number theory such sets are usuallyreferred to as Diophantine ones. And yet, in algebraic geometry they can bedescribed as projections of algebraic sets. Definition 3.3.
An algebraic structure A “ p A ; f, . . . , r, . . . , c, . . . q is called e-interpretable in another algebraic structure M if there exists n P N , a subset D Ď M n and an onto map (called the interpreting map) φ : D ։ A , such that:1. D is e-definable in M .2. For every function f “ f p x , . . . , x n q in the language of A , the preim-age by φ of the graph of f , i.e. the set tp x , . . . , x k , x k ` q | φ p x k ` q “ f p x , . . . , x k qu , is e-definable in M .3. For every relation r in the language of A , and also for the equality relation “ in A , the preimage by φ of the graph of r is e-definable in M .The following is a fundamental property of e-interpretability. Intuitively itstates that if A is e-interpretable in M , then any system of equations in A canbe ‘encoded’ as a system of equations in M . Lemma 3.4.
Let A be e-interpretable in M with an interpreting map φ : D ։ A (in the notation of the Definition 3.3). Then for every finite system of equations S p x q in A , there exists a finite system of equations S ˚ p y , z q in M , such that if p b , c q is a solution to S ˚ p y , z q in M , then b P D and φ p b q is a solution to S p x q in A . Moreover, any solution a to S p x q in A arises in this way, i.e. a “ φ p b q for some solution p b , c q to S ˚ p y , z q in M , for some i “ , . . . , k . Furthermore,there is a polynomial time algorithm that constructs the system S ˚ p y , z q whengiven a system S p x q .Proof. It suffices to follow step by step the proof of Theorem 5.3.2 from [12],which states that an analogue of the above holds when A is interpretable by firstorder formulas in M . One needs to replace all first order formulas by systemsof equations.Now we state two key consequences of Lemma 3.4. Corollary 3.5. If A is e-interpretable in M , then D p A q is reducible to D p M q .Consequently, if D p A q is undecidable, then D p M q is undecidable as well. Corollary 3.6. e-interpetability is a transitive relation, i.e., if A is e-intepretablein A , and A is e-interpretable in A , then A is e-interpretable in A . The following is a key property of e-interpretability that is used below.
Proposition 3.7 ([9]) . Let H be a normal subgroup of a group G . If H is e-definable in G (as a set) then the natural map π : G Ñ G { H is an e-interpretationof G { H in G . Consequently, D p G { H q is reducible to D p G q . .2 The Diophantine problem in metabelian groups given by fullrank presentations We next discuss the Diophantine problem in metabelian groups admitting a full-rank presentation. We will need the following result regarding the same problemin nilpotent groups:
Theorem 3.8 ([10]) . Let G be a finitely generated nonabelian nilpotent group ad-mitting a full rank presentation of deficiency at least (i.e. there are at least twomore generators than relations). Then the ring of integers Z is e-interpretablein G , and, in particular, the Diophantine problem of G is undecidable. Next we recall the definition of (finite) verbal width. This notion is conve-niently related to definability by equations, as we see in Proposition 3.9.Let w “ w p x , . . . , x m q be a word on an alphabet of variables and its in-verses t x , . . . , x m u ˘ . The w -verbal subgroup of a group G is defined as w p G q “x w p g , . . . , g m q | g i P G y , and G is said to have finite w -width if there existsan integer n , such that every g P w p G q can be expressed as a product of atmost n elements of the form w p g , . . . , g m q ˘ . Hence, w p G q is e-definable in G through the equation x “ ś ni “ w p y i , . . . , y im q w p z i , . . . , z im q ´ on variables x and t y ij , z ij | ď i ď n, ď j ď m u . Proposition 3.9.
Let G be a group and let H be a normal verbal subgroup of G with finite verbal width. Then the quotient G { H is e-interpretable in G .Proof. By the argument above the subgroup H is e-definable in G . Now theresult follows from Proposition 3.7. Proof of Theorem 1.3.
Let G “ x A | R y M be a full rank presentation of ametabelian group G . To prove 1) note first that since γ p G q ě r G , G s the2-nilpotent quotient G { γ p G q admits a presentation x A | R y N in the variety N of nilpotent groups of class ď
2. In particular, the group G { γ p G q has afull-rank presentation of deficiency at least 2. By Theorem 3.8, the ring Z ise-interpretable in G { γ p G q . It is known that in a finitely generated metabeliangroup any verbal subgroup has finite width [24], in particular, the subgroup γ p G q has finite width in G . Hence by Proposition 3.9 the group G { γ p G q ise-interpretable in G . It follows from transitivity of e-interpretations that Z ise-interpretable in G , so the Diophantine problem in G is undecidable.To prove the second statement of the theorem observe that if | R | ě | A | thenby Theorem 1.1 the group G is virtually abelian. Hence the Diophantine problemin G is decidable (see [6]). In this section we study random metabelian groups in the few-relators model.More precisely, we consider group presentations G “ x a , . . . , a n | r , . . . , r m y M “ x A | R y M (1)10n the variety of metabelian groups M , where the set of generators A “ t a , . . . , a n u is fixed, the number of relations m is also fixed, and R is a set of m words oflength ℓ in the alphabet A ˘ chosen randomly and uniformly, as explained inSection 1.3. We then study the asymptotic properties of G as ℓ tends to infinity.As we mentioned in the introduction the key observation here is that due toTheorem 1.6 a finite presentation in the variety of all groups (hence, any variety)has full rank asymptotically almost surely. Theorem 4.1.
Let n, m P N , and let G be a finitely generated metabelian groupgiven by a presentation x A | R y M “ x a , . . . , a n | r , . . . , r m y M , where all relators r i have length ℓ . Then the following holds asymptotically almost surely as ℓ Ñ 8 :There exist two finitely generated subgroups H and K of G such that:1. H is a free metabelian group of rank max p| A | ´ | R | , q ,2. K is a virtually abelian group with | R | generators, and its normal closure L “ K G in G is again virtually abelian;3. G “ x H, K y “ LH .Moreover, in this case there is an algorithm that given a presentation G “ x A | R y M finds a free basis for the subgroup H and a generating set in | R | generatorsfor the subgroup K .Proof. It follows from Theorems 1.6 and 1.1.
Theorem 4.2.
Let n, m P N , and let G be a finitely generated metabelian groupgiven by a presentation x A | R y M “ x a , . . . , a n | r , . . . , r m y M , where all relators r i have length ℓ . Then the following hold asymptotically almost surely as ℓ Ñ 8 :1. If | R | ď | A | ´ then the ring of integers Z is interpretable in G by systemsof equations, and the Diophantine problem in G is undecidable.2. If | R | ě | A | then the Diophantine problem of G is decidable (in fact, thefirst-order theory of G is decidable).Proof. It follows from Theorems 1.6 and 1.3.
Theorem 4.3.
Let n, m P N , and let G be a finitely generated metabelian groupgiven by a presentation x A | R y M “ x a , . . . , a n | r , . . . , r m y M , where all relators r i have length ℓ . Assume n ě m ´ . Then the following holds asymptoticallyalmost surely as ℓ Ñ 8 : in any direct decomposition of G all, but one, directfactors are virtually abelian.Proof. It follows from Theorems 1.6 and 1.2.
This work was supported by the Mathematical Center in Akademgorodok.Additionally, the first named author was supported by the ERC grant PCG-336983. The first and second named authors were supported by the BasqueGovernment grant IT974-16, and by the Ministry of Economy, Industry andCompetitiveness of the Spanish Government Grant MTM2017-86802-P.11
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