Non-orderability of random triangular groups by using random 3CNF formulas
aa r X i v : . [ m a t h . G R ] F e b NON-ORDERABILITY OF RANDOM TRIANGULAR GROUPSBY USING RANDOM 3CNF FORMULAS
DAMIAN ORLEF
Abstract.
We show that a random group Γ in the triangular binomial model Γ( n, p ) is a.a.s.not left-orderable for p ∈ ( cn − , n − / − ε ) , where c, ε are any constants satisfying ε > , c > (1 /
8) log / ≈ . . We also prove that if p ≥ (1 + ε )(log n ) n − for any fixed ε > ,then a random Γ ∈ Γ( n, p ) has a.a.s. no non-trivial left-orderable quotients. We proceed byconstructing 3CNF formulas, which encode necessary conditions for left-orderability and thenproving their unsatisfiability a.a.s. Introduction
The triangular binomial model of random groups is defined as follows.Fix p : N → [0 , . Given n ∈ N , let S = { s , s , . . . , s n } be a set of n generators. A randomgroup in the triangular binomial model Γ( n, p ) is given by the presentation h S | R i , where R isa random subset of the set of all cyclically reduced words of length 3 over S ∪ S − , with each wordincluded in R independently with probability p ( n ) .Given a property P of groups or presentations, we say that a random group in the model Γ( n, p ) satisfies P asymptotically almost surely ( a.a.s. ) if lim n →∞ P (cid:0) Γ ∈ Γ( n, p ) satisfies P (cid:1) = 1 . This model (a variant of) was introduced by Żuk in [Żuk03]. Basic properties of randomtriangular groups vary with p as described in the following two theorems. Every bound on p isassumed to hold for almost all n . Theorem 1.1 ([Żuk03, Theorem 3], [AŁŚ14, Theorem 1]) . For any fixed ε > , if p < n − / − ε ,then a random group in Γ( n, p ) is a.a.s. infinite, torsion-free, and hyperbolic, while also there existsa constant C > such that if p > Cn − / , then a random group in Γ( n, p ) is a.a.s. trivial. For sufficiently small values of p , random triangular groups are actually free. Theorem 1.2 ([AŁŚ15, Theorems 1, 2, 3]) . There exist constants c , c > such that a randomgroup in Γ( n, p ) is a.a.s. free if p < c n − , and a.a.s. non-free or trivial if p > c n − . In this article we explore left-orderability of random triangular groups. A particular consequenceof Theorem 1.2 is that if p < c n − , then a random group in Γ( n, p ) is a.a.s. left-orderable (see[DNR14, Section 1.2.3]). We show that this statement is optimal up to multiplication of p bya constant. Theorem A.
Let c > (1 /
8) log / ≈ . be a constant. If p > cn − , then a randomgroup in Γ( n, p ) is a.a.s. trivial or not left-orderable. In particular, for any fixed ε > , if p ∈ ( cn − , n − / − ε ) , then a random group in Γ( n, p ) is a.a.s. not left-orderable. Increasing p , we are able to show even more. Theorem B.
Suppose that p ≥ (1 + ε )(log n ) n − for any fixed ε > . Then a random groupin Γ( n, p ) has a.a.s. no non-trivial left-orderable quotients. Theorem B is optimal up to a constant factor. By [AŁŚ15, Lemma 12], if p ≤ (1 / n ) n − ,then a.a.s. there exists a generator s ∈ S such that neither s nor s − belongs to any relation in R , allowing for a homomorphism φ : h S | R i ։ Z with φ ( s ) = 1 , and hence a random group in Γ( n, p ) has Z as its quotient a.a.s.Note that if p ≥ n d − , for any fixed d > / , then p satisfies the hypothesis of Theorem B andhence its conclusion applies to random triangular groups in the Żuk’s model at densities d > / (see [AŁŚ15, Section 3] for details). On the other hand, random groups in the Gromov densitymodel do not have non-trivial left-orderable quotients at any density d ∈ (0 , , as shown in [Orl17].A countable group is left-orderable if and only if it admits a faithful action on the real line byorientation-preserving homeomorphisms (see [DNR14, Section 1.1.3]). Hence Theorem A (resp. B)is, equivalently, a statement about non-existence of faithful (resp. non-trivial) actions of randomtriangular groups on the real line. Note that some constraints were previously known for actionsof random triangular groups on the circle. Namely, by [AŁŚ15, Theorem 3], there exists a constant C ′ > such that if p ≥ C ′ (log n ) n − , then a random group Γ ∈ Γ( n, p ) has a.a.s. Kazhdan’sproperty (T), and hence by a result of Navas, [Nav02], every action of Γ on the circle by orientation-preserving diffeomorphisms of class C α ( α > / ) has a finite image.Another consequence of our proof of Theorem A is a new proof of the second part of Theorem 1.2stating that a random group in Γ( n, p ) is a.a.s. non-free or trivial if p > cn − for some constant c > . However, our argument shows that this is the case for c > (1 /
8) log / ≈ . , whichis not an optimal result. In [AŁŚ15] this statement is formally shown to hold for c ≥ , butthe method presented there works as soon as c > / . , as we briefly explain now. First,if p ≥ C ′ (log n ) n − , then non-freeness or triviality follows from property (T). If p is smaller,then in particular p < n − / − ε for a fixed ε > . This guarantees that the natural presentationcomplex of a random Γ ∈ Γ( n, p ) is a.a.s. aspherical, and hence the Euler characteristic of Γ is a.a.s. equal to χ (Γ) = 1 − n + | R | . If in addition p > cn − for a fixed c > / , then a.a.s. | R | = 8 n p (1 + o (1)) > n , hence χ (Γ) > and Γ is not a free group a.a.s. Outline of the proofs.
In the proofs of our results we use 3CNF propositional formulas, i.e.formulas of form ( a ∨ b ∨ c ) ∧ . . . ∧ ( a k ∨ b k ∨ c k ) , where each a i , b i , c i is either of form x or ¬ x for a Boolean variable x .To prove Theorem A for smaller values of p , we translate a random triangular presentation Γ = h S | R i into a random 3CNF formula Φ R constructed as the conjunction of the expressions ofform (cid:0) x ε i i ∨ x ε j j ∨ x ε k k (cid:1) ∧ (cid:0) x − ε i i ∨ x − ε j j ∨ x − ε k k (cid:1) , corresponding to relators r = s ε i i s ε j j s ε k k ∈ R , where x , x , . . . , x n are Boolean variables andour convention is that x = x and x − = ¬ x . Every left-order on Γ a.a.s. induces a truthassignment satisfying Φ R . We show, however, that a.a.s. Φ R is unsatisfiable, hence Γ is notleft-orderable. For larger values of p and in the proof of Theorem B , we consider a collectionof similarly constructed random 3CNF formulas Φ R,A , indexed by certain subsets A ⊆ S , suchthat a.a.s. every non-trivial left-orderable quotient of a random Γ ∈ Γ( n, p ) leads to satisfiabilityof at least one of those formulas. In this case we prove that a.a.s. none of the formulas Φ R,A aresatisfiable and the desired conclusion follows.We note that the constant (1 /
8) log / ≈ . in Theorem A can probably be improved byusing the existing research on the random 3-NAESAT problem. To see the connection, let Φ ′ R bea random formula constructed as the conjunction of the clauses x ε i i ∨ x ε j j ∨ x ε k k , taken for each r = s ε i i s ε j j s ε k k ∈ R . We say that Φ ′ R is Not-All-Equal-satisfiable ( NAE-satisfiable ) if there existsa truth assignment η such that in every clause x ε i i ∨ x ε j j ∨ x ε k k at least one, but not all, of the literals x ε i i , x ε j j , x ε k k is true under η . It is straightforward to see that Φ R is satisfiable if and only if Φ ′ R isNAE-satisfiable. When p = cn − , a formula Φ ′ R is similar to a uniformly random 3CNF formulaon n variables with ⌊ cn ⌋ clauses, which is known to be a.a.s. not NAE-satisfiable if c is largerthan a certain threshold α (e.g. [COP12]). The conclusions of Theorem A should thus hold forany c > α/ . In this article we do not aim to give the optimal constant for Theorem A, whichallows us to provide a more self-contained proof. Organisation.
In Section 2 we introduce our notation, recall basic properties of left-orderablegroups and construct propositional formulas central to our arguments. In Section 3 we prove
ON-ORDERABILITY OF RANDOM TRIANGULAR GROUPS 3
Theorem A under the additional assumption that p < n − / − ε . The remaining case is a directconsequence of Theorem B, which we prove in Section 4. Acknowledgements.
The author would like to thank Piotr Nowak and Piotr Przytycki forvaluable discussions and suggestions. The author was partially supported by the European Re-search Council (ERC) under the European Union’s Horizon 2020 research and innovation pro-gramme (grant agreement no. 677120-INDEX) and by (Polish) Narodowe Centrum Nauki, UMO-2018/30/M/ST1/00668. 2.
Preliminaries
Most of the notions we use depend implicitly on n . By o ( f ( n )) we denote any function g ( n ) such that g ( n ) /f ( n ) → as n → ∞ . Throughout, we assume Γ = h S | R i is group given bya presentation with the set of formal generators S = { s , s , . . . , s n } and R ⊆ W , where W isthe set of all cyclically reduced words of length 3 over S ∪ S − . We denote by ι : F ( S ) ։ Γ the associated epimorphism of the free group generated by S . For any A ⊆ S define R A = { s ε i i s ε j j s ε k k ∈ R : s i , s j , s k ∈ A, ε i , ε j , ε k ∈ {− , }} to be the set of words over A ∪ A − , belonging to R . Definition 2.1.
We say that a group G is left-ordered by a linear order ≤ if ≤ is invariant underleft-multiplication by G , i.e. satisfies the condition a ≤ b = ⇒ ga ≤ gb for all a, b, g ∈ G . We saythat G is left-orderable if it is left-ordered by some linear order ≤ .Given a group G left-ordered by ≤ , we use < and > as usual shorthands. The followingproperties of left-orderable groups are straightforward consequences of Definition 2.1. Remark 2.2.
If a group G is left-ordered by ≤ , then the following hold.(1) For every non-trivial g ∈ G , g > G if and only if g − < G .(2) If g , g , . . . , g m ∈ G are such that g i > G for every i , then also g g . . . g m > G .Now we define the propositional formulas central to our arguments. Let x , x , . . . , x n beBoolean variables. We adopt a convention that x i = x i and x − i = ¬ x i for all i . For every r = s ε i i s ε j j s ε k k ∈ W with ε i , ε j , ε k ∈ {− , } , set φ r = (cid:0) x ε i i ∨ x ε j j ∨ x ε k k (cid:1) ∧ (cid:0) x − ε i i ∨ x − ε j j ∨ x − ε k k (cid:1) . For any A ⊆ S , let Φ R,A = ^ r ∈ R A φ r and let Φ R = Φ R,S . The definition of Φ R,A is justified by the following.
Proposition 2.3.
Suppose q : Γ ։ Q is an epimorphism onto a left-orderable group Q , such that ker( qι ) ∩ A = ∅ . Then Φ R,A is satisfiable.
Of our special interest is the case when Q = Γ , q = id Γ and A = S . Corollary 2.4.
Suppose Γ is left-orderable and ker( ι ) ∩ S = ∅ . Then Φ R is satisfiable.Proof of Proposition 2.3. Let Q be left-ordered by ≤ . We construct a truth assignment η , satis-fying Φ R,A , as follows. Let x i be any variable such that s i ∈ A . Then qι ( s i ) = 1 Q . Set η ( x i ) = ( T if qι ( s i ) > Q ,F if qι ( s i ) < Q . Function η extends naturally to all propositional formulas over variables x i , for which s i ∈ A .By Remark 2.2(1), for every such variable x i and every ε ∈ {− , } , the value of η ( x εi ) representsthe validity of the statement qι ( s εi ) > Q .Now consider any r = s ε i i s ε j j s ε k k ∈ R A . Since Q = qι ( r ) = qι ( s ε i i ) qι ( s ε j j ) qι ( s ε k k ) , by Re-mark 2.2(2) at least one of the elements qι ( s ε i i ) , qι ( s ε j j ) , qι ( s ε k k ) is not greater than Q and hence DAMIAN ORLEF η satisfies the formula x − ε i i ∨ x − ε j j ∨ x − ε k k . Similarly, as Q = qι ( r − ) = qι (cid:0) s − ε k k (cid:1) qι ( s − ε j j ) qι ( s − ε i i ) ,we establish that η satisfies x ε i i ∨ x ε j j ∨ x ε k k and so it satisfies φ r .Finally, η satisfies Φ R,A , being a conjunction of formulas φ r for r ∈ R A . (cid:3) Non-left-orderability for lower probabilities
In this section we prove Theorem A under the additional assumption that p < n − / − ε for some ε > . The case of larger p follows from Theorem B, which is proved independently in Section 4.Throughout, P is the probability function in the model Γ( n, p ) .Our choice of an upper bound for p is motivated by the following fact. Lemma 3.1 ([AŁŚ15, Corollary 10]) . Suppose p < n − / − ε for a constant ε > . Then a.a.s. ker( ι ) ∩ S = ∅ in Γ( n, p ) . Before proving Theorem A we need to establish two simple lemmas, which let us pass toa different model of randomness for the set of relators R in computations of probabilities. Firstwe need some asymptotic control over the size of the set R . Lemma 3.2.
Suppose p > n − ε for some ε > . Then | R | ∈ (cid:0) (1 − δ )8 pn , (1 + δ )8 pn (cid:1) a.a.s.for some δ = δ ( n ) = o (1) .Proof. Under the model Γ( n, p ) , | R | has the binomial distribution B ( | W | , p ) , so that E | R | = p | W | and Var | R | = p (1 − p ) | W | . Let δ = ( p | W | ) − / .By the Chebyshev’s inequality, P (cid:0)(cid:12)(cid:12) | R | − E | R | (cid:12)(cid:12) ≥ δ E | R | (cid:1) ≤ Var | R | δ ( E | R | ) = p (1 − p ) | W | δ p | W | ≤ δ p | W | = δ. Now note that | W | = 8 n (1 + o (1)) , as n (2 n − n − ≤ | W | ≤ (2 n ) and hence p | W | = 8 pn (1 + o (1)) > n ε (1 + o (1)) , so that δ = o (1) .We have | R | ∈ ((1 − δ ) E | R | , (1 + δ ) E | R | ) a.a.s. Since E | R | = 8 pn (1 + o (1)) , δ can be adjustedso that the desired conclusion holds. (cid:3) Let c = (1 /
8) log / . From now on to the end of the section we assume that p ∈ ( cn − , n − / − ε ) ,where ε > and c > c . Let δ be such that the conditions of Lemma 3.2 hold and denote I δ = ((1 − δ )8 pn , (1 + δ )8 pn ) ∩ N .Fix m and let r , r , . . . , r m be independent random uniform words picked from W . Let P m bethe associated probability function and let R m = { r , r , . . . , r m } . The following lemma enablesus to work with R m in place of R . Lemma 3.3.
There exists θ = θ ( n ) = o (1) such that, for every property P of subsets of W ,and every m ∈ I δ , P (cid:0) R satisfies P (cid:12)(cid:12) | R | = m (cid:1) ≤ (1 + θ ) P m (cid:0) R m satisfies P (cid:1) . Proof. In Γ( n, p ) , conditional on | R | = m , the set R is a random uniform subset of W , of size m .Let D m be the event that r , r , . . . , r m are pairwise distinct. Conditional on D m , R m is alsoa random uniform subset of W , of size m . Hence P (cid:0) R satisfies P (cid:12)(cid:12) | R | = m (cid:1) = P m (cid:0) R m satisfies P (cid:12)(cid:12) D m (cid:1) ≤ P m (cid:0) R m satisfies P (cid:1) P m (cid:0) D m (cid:1) . It suffices to show that P m (cid:0) D m (cid:1) ≥ − o (1) for every m ∈ I δ , with the bound o (1) dependingonly on n . ON-ORDERABILITY OF RANDOM TRIANGULAR GROUPS 5
We first note that m ≤ δ ) p n ≤ δ ) n / − ε = o ( n ) . As | W | = 8 n (1 + o (1)) ,we have m / | W | ≤ o (1) . Finally, by the Bernoulli’s inequality P m (cid:0) D m (cid:1) = (cid:18) − | W | (cid:19) (cid:18) − | W | (cid:19) . . . (cid:18) − m − | W | (cid:19) ≥ (cid:18) − m − | W | (cid:19) m − ≥ − ( m − | W | ≥ − m | W | ≥ − o (1) , proving our claim. (cid:3) Proof of Theorem A for p < n − / − ε . Let Γ ∈ Γ( n, p ) be a random triangular group. Considerthe following event in Γ( n, p ) . A = (cid:8) Γ is left-orderable, ker( ι ) ∩ S = ∅ and | R | ∈ I δ . (cid:9) By Lemmas 3.1 and 3.2, it suffices to show that P ( A ) → as n → ∞ .If A holds, then by Corollary 2.4 the formula Φ R is satisfiable. We bound P ( A ) as follows,using Lemma 3.3. P (cid:0) A (cid:1) ≤ X m ∈ I δ P (cid:0) Φ R is satisfiable and | R | = m (cid:1) = X m ∈ I δ P (cid:0) Φ R is satisfiable (cid:12)(cid:12) | R | = m (cid:1) P (cid:0) | R | = m (cid:1) ≤ (1 + θ ) X m ∈ I δ P m (cid:0) Φ R m is satisfiable (cid:1) P (cid:0) | R | = m (cid:1) ≤ (1 + θ ) max m ∈ I δ P m (cid:0) Φ R m is satisfiable (cid:1) (1)Fix m ∈ I δ . Let E be the set of all n truth assignments η : { x , x , . . . , x n } → { T, F } . Usingthe independence of r , r , . . . , r m , we obtain P m (cid:0) Φ R m is satisfiable (cid:1) ≤ X η ∈E P m (cid:0) η satisfies Φ R m (cid:1) = X η ∈E P m η satisfies m ^ i =1 φ r i ! = X η ∈E P m (cid:0) η satisfies φ r i for i = 1 , , . . . , m (cid:1) = X η ∈E m Y i =1 P m (cid:0) η satisfies φ r i (cid:1) = X η ∈E (cid:0) P m (cid:0) η satisfies φ r (cid:1)(cid:1) m . (2)Now fix η . For any 3 pairwise distinct s i , s j , s k ∈ S , there exist exactly 2 triples ( ε i , ε j , ε k ) ∈ {− , } ,and hence exactly 2 words of form w = s ε i i s ε j j s ε k k , such that η does not satisfy φ w = (cid:0) x ε i i ∨ x ε j j ∨ x ε k k (cid:1) ∧ (cid:0) x − ε i i ∨ x − ε j j ∨ x − ε k k (cid:1) . Hence in total there are at least n ( n − n −
2) = 2 n (1 + o (1)) such words w ∈ W that η does not satisfy φ w , so P m ( η satisfies φ r ) ≤ − n | W | (1 + o (1)) = 1 − n n (1 + o (1)) = 34 + o (1) , where the bound o (1) depends only on n . From (1) and (2) this leads to DAMIAN ORLEF P ( A ) ≤ (1 + θ ) max m ∈ I δ P m (cid:0) Φ R m is satisfiable (cid:1) ≤ (1 + θ ) max m ∈ I δ n (cid:18)
34 + o (1) (cid:19) m ≤ (1 + θ ) 2 n (cid:18)
34 + o (1) (cid:19) (1 − δ )8 pn = (1 + θ ) (cid:18)
34 + o (1) (cid:19) (1 − δ )8 pn ! n . As p > cn − with c > c , we have P ( A ) ≤ (1 + θ ) (cid:18)
34 + o (1) (cid:19) (1 − δ )8 c ! n . Finally, note that the base of the power tends to (cid:18) (cid:19) c < (cid:18) (cid:19) c = 2 (cid:18) (cid:19) log / = 1 , as n → ∞ , hence P ( A ) → . (cid:3) Non-left-orderability of quotients
In this section we prove Theorem B. The condition that p is slightly larger than (log n ) n − is dictated by the following lemma, which ensures that a positive proportion of elements of S represent non-trivial elements after passing to a non-trivial quotient Q . Lemma 4.1.
Suppose that p ≥ (1 + ε )(log n ) n − for some ε > . Then there exists a constant α ∈ (0 , such that a random group Γ ∈ Γ( n, p ) a.a.s. satisfies the following property: ( ⋆ ) For every non-trivial epimorphism q : Γ ։ Q , | ker( qι ) ∩ S | < αn. Proof.
Choose α ∈ (0 , so that α (1 + ε ) > . For every subset A ⊆ S , define in Γ( n, p ) the event V A = { There exists a non-trivial epimorphism q : Γ ։ Q such that ker( qι ) ∩ S = A. } and the set of words P A = { abc : a, b ∈ A, c ∈ S \ A } ⊆ W . We claim that V A ⊆ { P A ∩ R = ∅} . Indeed, suppose that V A holds and w = abc ∈ P A . Then qι ( w ) = qι ( a ) qι ( b ) qι ( c ) = qι ( c ) = 1 Q and hence w / ∈ R .Note that if | A | ≥ αn , then | P A | = | A | · | A | · ( n − | A | ) ≥ α n ( n − | A | ) , so that P (cid:0) V A (cid:1) ≤ P (cid:0) P A ∩ R = ∅ (cid:1) = (1 − p ) | P A | ≤ (1 − p ) α n ( n −| A | ) ≤ e − pα n ( n −| A | ) , where we use the bound e x ≥ x , true for x ∈ R . Also note that P (cid:0) V S (cid:1) = 0 .Now we can bound the probability that the property ( ⋆ ) does not hold as follows. ON-ORDERABILITY OF RANDOM TRIANGULAR GROUPS 7 P (cid:0) Γ ∈ Γ( n, p ) does not satisfy property ( ⋆ ) (cid:1) ≤ X A ⊆ S : αn ≤| A |≤ n − P (cid:0) V A (cid:1) ≤ X A ⊆ S : αn ≤| A |≤ n − e − pα n ( n −| A | ) ≤ X A ⊆ S : | A |≤ n − e − pα n ( n −| A | ) = n − X k =0 (cid:18) nk (cid:19) e − pα n ( n − k ) = n − X k =0 (cid:18) nk (cid:19)(cid:16) e − pα n (cid:17) n − k = (cid:16) e − pα n (cid:17) n − ≤ (cid:16) e e − pα n (cid:17) n − e e log n − pα n − . To finish the proof of the lemma, it suffices to show that the last expression tends to 0 as n → ∞ .This is true as the topmost exponent satisfies lim sup n →∞ (cid:0) log n − pα n (cid:1) ≤ lim sup n →∞ (cid:0) log n − (1 + ε )(log n ) n − α n (cid:1) = lim sup n →∞ (cid:0) − α (1 + ε ) (cid:1) log n = −∞ . (cid:3) Proof of Theorem B.
Let Γ ∈ Γ( n, p ) be a random triangular group and let α ∈ (0 , be a numbersatisfying Lemma 4.1. Define in Γ( n, p ) the event S = { There exists a non-trivial epimorphism q : Γ ։ Q with a left-orderable Q and | ker( qι ) ∩ S | < αn } . It suffices to prove that P ( S ) → as n → ∞ . Suppose S holds and let A = S \ ker( qι ) . Wehave | A | ≥ (1 − α ) n and by Proposition 2.3 the formula Φ R,A is satisfiable.Let η : { x i : s i ∈ A } → { T, F } be a truth assignment satisfying Φ R,A . For every s i ∈ A , choose ε i ∈ {− , } so that η ( x ε i i ) = T . Introduce the set P A,η = { s ε i i s ε j j s ε k k : s i , s j , s k ∈ A } , of cyclically reduced words, of cardinality | P A,η | ≥ (1 − α ) n . By design, if w ∈ P A,η , then η doesnot satisfy φ w , so that w / ∈ R A and hence P A,η ∩ R = P A,η ∩ R A = ∅ .For fixed A and η we have P ( P A,η ∩ R = ∅ ) ≤ (1 − p ) (1 − α ) n . Denoting by E A the set of all truth assignments η : { x i : s i ∈ A } → { T, F } , we can bound P ( S ) ≤ X A ⊆ S :(1 − α ) n ≤| A | X η ∈E A P ( P A,η ∩ R = ∅ ) ≤ n n (1 − p ) (1 − α ) n ≤ n e − p (1 − α ) n = e n log 4 − p (1 − α ) n . As p ≥ (log n ) n − , we have e n log 4 − p (1 − α ) n ≤ e n log 4 − (1 − α ) n log n = e n (log 4 − (1 − α ) log n ) → as n → ∞ and hence P ( S ) → as n → ∞ . (cid:3) DAMIAN ORLEF
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