On the length of non-solutions to equations with constants in some linear groups

We show that for any finite-rank free group $\Gamma$, any word-equation in one variable of length $n$ with constants in $\Gamma$ fails to be satisfied by some element of $\Gamma$ of word-length $O(\log (n))$. By a result of the first author, this logarithmic bound cannot be improved upon for any finitely generated group $\Gamma$. Beyond free groups, our method (and the logarithmic bound) applies to a class of groups including $\mathrm{PSL}_d(\mathbb{Z})$ for all $d \geq 2$, and the fundamental groups of all closed hyperbolic surfaces and $3$-manifolds. Finally, using a construction of Nekrashevych, we exhibit a finitely generated group $\Gamma$ and a sequence of word-equations with constants in $\Gamma$ for which every non-solution in $\Gamma$ is of word-length strictly greater than logarithmic.


Introduction
For Γ any group and any g ∈ Γ, there is a unique homomorphism π g : Γ * Z → Γ restricting to the identity on Γ and sending a specified generator x of Z to g.We call w a mixed identity (or an identity with constants) for Γ if w is nontrivial but π g (w) = e Γ for all g ∈ Γ. Equivalently, we may define a word-map w : Γ → Γ by w(g) = π g (w); w is then a mixed identity for Γ iff w −1 (e Γ ) = Γ.
We describe Γ as MIF (mixed identity-free) if there are no mixed identities for Γ in Γ * Z. Nonabelian free groups are MIF (a fact due to Baumslag [3]), as more generally are all torsion-free nonelementary Gromov hyperbolic groups (see [1] and the references therein).Being MIF imposes significant structural restrictions on Γ: an MIF group has no nontrivial finite conjugacy classes, and admits no nontrivial decomposition as a direct product.Moreover, MIF is inherited by nontrivial normal subgroups.
Given a finite generating set S for Γ, if Γ is MIF then for any w ∈ Γ * Z there exists g ∈ Γ of minimal word-length |g| S satisfying w(g) = e.Intuitively, the greater this minimal word-length should be, the harder it is to verify that w is not a mixed identity for Γ.The goal of this note is to study how |g| S can depend on the length of w.Following [4], we formalize this as follows.The complexity of w ∈ Γ * Z (with respect to S) is given by: χ S Γ (w) = min{|g| S : g ∈ Γ, w(g) = e Γ } (with the convention that χ S Γ (w) = +∞ if w is a mixed identity for Γ).Here |g| S denotes the word-length of g with respect to the generating set S. The MIF growth function M S Γ : N → N ∪ {+∞} of Γ (with respect to S) is then given by: Intuitively, a group of slow MIF growth is "strongly MIF": it is easy to verify that a given w is not a mixed identity for Γ, because there exists a witness of short word-length.As the dependence of M S Γ on S is slight, we suppress it from our notation for the remainder of the Introduction (with the understanding that all implied constants may depend on S).
Since MIF places such powerful structural restrictions on a group, it is no surprise that groups of slow MIF growth are difficult to identify, and indeed, there are no groups of bounded MIF growth.
In light of Theorem 1.1, we shall refer to a finitely generated group Γ as sharply MIF if its MIF growth function M Γ satisfies M Γ (n) ≪ log(n).Prior to this note, we did not have any examples of sharply MIF groups.In fact the only previously existing explicit upper bound on MIF growth was: if Γ is a finite-rank nonabelian free group then Employing different methods from those of [4], we improve this result for free groups, and extend it to many other groups besides.
Since free groups are Kleinian, we can consequently improve upon the result from [4].
Corollary 1.3 (Theorem 3.4).Let Γ be a free group of finite rank at least 2. Then Γ is sharply MIF.
More generally, Theorem 1.2 implies the following.
Corollary 1.4.Let M be either (i) a closed orientable surface of genus at least 2, and (ii) a closed orientable hyperbolic 3-manifold.Then Γ is sharply MIF.
As these two corollaries illustrate, there are large families of torsion-free Gromov hyperbolic groups with logarithmic MIF growth.One may ask whether these are instances of a more general phenomenon.Question 1.5.Let Γ be a torsion-free nonelementary Gromov hyperbolic group.Must Γ be sharply MIF?
Of necessity, any proof of an affirmative answer to Question 1.5 would require radically different methods from ours, since we make essential use of the abundant supply of finite quotients of Kleinian groups.As such, there is no obvious way to apply our proof-strategy to groups which are not known to be residually finite.
More generally still, if K is an algebraically closed field, and G is an algebraic group defined over K, then for any w ∈ G * Z, the associated word-map w : G → G is a morphism of algebraic varieties, so that w −1 (e G ) ⊆ G is Zariski-closed.Therefore, if G is MIF then so is any Zariski-dense subgroup Γ ≤ G.Here we focus on the case G = PGL d (C): by Theorem 5 of [16] Then Γ is sharply MIF.
Here Ad denotes the adjoint representation of G on the associated Lie algebra.It is well known that every nonelementary Kleinian group satisfies the conclusion of Theorem 1.6 with d = 2, so Theorem 1.2 follows immediately.Other examples which clearly satisfy the conditions of Theorem 1.6 are the following (see Section 3 below).
Example 1.7.For every d ≥ 2, PSL d (Z) is sharply MIF; so too is every finitely generated subgroup Γ of PSL d (Z) which is Zariski-dense in PGL d (C).
There has been much interest in recent years in so-called thin groups, that is, subgroups of an arithmetic group which have infinite index but are nevertheless Zariski-dense in the ambient algebraic group.Specifically, there are many interesting open questions asking which group-theoretic properties a thin group must satisfy [7].Theorem 1.6 shows that, beside PSL d (Z) itself, any finitely generated thin subgroup of PSL d (Z) must be sharply MIF.
Remark 1.8.One could also define MIF growth in terms of words with constants in k variables, for any k ∈ N, by considering the complexity in Γ of elements of Γ * F k , where F k is a free group of rank k (with the complexity of w ∈ Γ * F k being defined in terms of an appropriate metric on Γ k ).However, whether or not the the group is MIF, and the asymptotic behaviour of the MIF growth function will be the same, irrespective of which (fixed) value of k we use to define it (see Lemma 2.2 of [5]); for instance, whether or not Γ is sharply MIF does not depend on the value of k.For simplicity we therefore restrict to the one-variable case.
All of our upper bounds on MIF growth are based on the existence of a rich supply of finite quotients of Γ which have no short mixed identities, and with good expansion properties.The lengths of mixed identities for finite groups were studied by the authors in [5], where the following result was proved (see Theorem 3 of [5], of which the following is a special case).Theorem 1.9.There exists c > 0 such that for all primes p and all d ≥ 2, PSL d (p) has no mixed identity of length ≤ cp.
The basic proof-strategy of all our upper bounds on M Γ in this note is the same.To ease the understanding of the general proof to follow, let us sketch the argument in the special case of Γ a nonabelian free group, with finite free basis S. First, we faithfully represent Γ as a subgroup of SL 2 (Z).Let For p a prime number, let π p : SL 2 (Z) → SL 2 (p) be reduction modulo p.By the Prime Number Theorem, there exists p of moderate size such that all π p (c i ) ∈ SL 2 (p) are still non-central.By Theorem 1.9, there exists g ∈ SL 2 (p) for which w(g We would like to lift g to g ∈ Γ, so that w(g) = e.By deep results on the expansion properties of SL 2 (p), not only is the restriction of π p to Γ surjective (so that such a lift g exists), but the Cayley graph of SL 2 (p) with respect to generating set π p (S) has small diameter, so that we can choose that lift g to have small word-length with respect to S.
In the setting of Theorem 1.6, we will be working over a number field K, so instead of reductions modulo a rational prime, we shall be considering reductions π P modulo a prime element P in the ring of integers of K.If d ≥ 3, the surjectivity of π P is not such a straightforward matter as it is for d = 2; the technical hypothesis (1) is assumed in Theorem 1.6, so that surjectivity is guaranteed, by an appropriate "Strong Approximation" Theorem.
In light of our results as described above, one might wonder whether in fact every finitely generated MIF group is sharply MIF.It transpires that this is not the case.It was noted already in [4] (Remark 9.3) that to find a counterexample, it would be sufficient to find a finitely generated MIF group of subexponential word growth.Following circulation of an earlier version of our results, N. Matte Bon kindly pointed out to us that a construction of Nekrashevych, based on ideas from topological dynamics, provides such a group.As such we have the following.
Theorem 1.10.There exists a finitely generated MIF group which is not sharply MIF.

Linear groups over number fields
Let K be a number field, with |K : Q| = D. Let C 0 > 0 (to be chosen).Let P n be the set of rational primes p satisfying C 0 n < p ≤ C 0 n 2 , and let Q n ⊆ P n be the subset of those primes which split completely over K.By Chebotarev's Density Theorem the proportion of rational primes at most Cn 2 which split completely over K is at least 1/2D (for all n larger than a constant depending only on K), hence the product of all such primes is at least p(C 0 n 2 /2D).Thus: and the conclusion follows.
Let O K be the ring of integers of K. Let σ 1 , . . ., σ D : K ֒→ C be the distinct field embeddings of K into C. Recall that the norm of an element x ∈ K is given by: Thus for p a rational prime and P ∈ O K a K-prime dividing p, p divides N K/Q (P), hence µ(P) ≥ p 1/D .Let Γ ≤ SL d (K) be generated by the finite set S, which we may assume to be symmetric and to contain I d .Let Z = x , so that S ∪ {x} generates Γ * Z.Our main technical result is as follows.
suppose that K and Γ satisfy condition (1).There exists C = C(S) > 0 such that the following holds.Let: for some c i ∈ Γ \ Z(Γ) and integers a i with a i = 0 for 1 ≤ i ≤ k − 1. Suppose that |w| S∪{x} = n.Then there exists g ∈ Γ such that |g| S ≤ C log(n) and w(g) / ∈ Z(Γ).
The rest of this Section is devoted to the proof of Theorem 2.2, and the deduction of Theorem 1.6 from it.In what follows, we continue to assume that the hypotheses of Theorem 2.2 hold.Note that since |w| S∪{x} = n, we have |c i | S ≤ n for 1 ≤ i ≤ k.We may take n to be larger than a quantity depending only on K and d, which will be chosen in the course of the proof.
For each s ∈ S, we may choose s * ∈ M d (O K ) and δ(s) ∈ O K such that s = (1/δ(s))s * in M d (K).We can then extend to the whole of Γ, by choosing, for each g ∈ Γ, a representative g = s 1 • • • s |g|S of g as a word in S of minimal length, and setting: so that for all g ∈ Γ, g = (1/δ(g))g * .Let M the maximal value of µ over the nonzero elements of {s * i,j : s ∈ S, 1 ≤ i, j ≤ d} ∪ {δ(s) : s ∈ S}.
Proof.We show that µ takes value at most (dM ) r on all entries of products of the form (by induction on r).The second claim holds by submultiplicativity of µ.
If p splits completely over K, then for any K-prime P ∈ O K dividing p, reduction modulo P induces a surjective ring homomorphism π P : O K → F p .If moreover P does not divide any element of {δ(s) : s ∈ S}, then every nonzero entry of every element of Γ is invertible modulo P, and π P induces a group homomorphism Γ → SL d (p), which we also denote by π P .
For G a group, T ⊆ G a generating set and r ≥ 0 let: denote the closed ball of radius r about the identity in the word-metric induced by T on G. Proposition 2.5.Let Q n be as in Lemma 2.1.There exists p ∈ Q n and a K-prime P ∈ O K dividing p, such that: (i) P does not divide any element of {δ(s) : s ∈ S}; (ii) For the proof we require the following well-known auxiliary fact.
Proof of Proposition 2.5.Since S is finite, there are only finitely many K-primes P ∈ O K for which (i) fails.We claim that there are only finitely many P satisfying (i) but not (ii).For d ≥ 3, Theorem 10.5 of [17] applies (by ( 1)).
The case d = 2 is described in [11] Section 3; we outline the argument here for the reader's convenience.By the Tits alternative, there exist a, b ∈ Γ freely generating a nonabelian free subgroup of Γ.
. If π P is not surjective, then by Lemma 2.6, π P (g) = I 2 .This contradicts Lemma 2.4 for n sufficiently large (in terms of S).Let B be the set of all K-primes for which either (i) or (ii) fails, so that |B| is bounded in terms of S alone.Suppose for a contradiction that for all p ∈ Q n \ B and all K-primes P ∈ O K dividing p, there exists 1 ≤ i ≤ k such that π P (c i ) ∈ Z(SL d (p)).Thus for all s ∈ S, By contrast, since all the c i are non-central in Γ and S generates Γ, we have for all 1 ≤ i ≤ k that there exists s ∈ S such that 0 = sc i − c i s ∈ M d (K).Multiplying through by δ(c i )δ(s), we have 0 , but all entries of s * c * i − c * i s * are divisible by P. Thus, taking the product of all nonzero entries of all matrices s * c * i − c * i s * we obtain an element z ∈ O K which is divisible by all of the P. Thus, as observed above p|N K/Q (P)|N K/Q (z) for each p ∈ Q n \ B, so that: Taking c sufficiently large in Lemma 2.1 (which in turn requires C 0 to be chosen sufficiently large), this contradicts (4) for n sufficiently large (in terms of S).
If G is a finite group, recall that the diameter of G with respect to T is: Proposition 2.7.There exists C = C(S) > 0 such that for all but finitely many rational primes p which completely split in K, for all primes P of O K which divide p, π P (S) generates SL d (p) and diam(SL d (p), π P (S)) ≤ C log(p).
The proof rests on the following deep result, due to Helfgott [9], Breuillard-Green-Tao [6] and Pyber-Szabó [13].Here AAA = {abc : a, b, c ∈ A}.We also require a Lemma, the proof of which is similar to that of Proposition 2.7.
Proof of Proposition 2.7.First, as in Proposition 2.5, we may assume that P does not divide any element of {δ(s) : s ∈ S} and that π P (S) generates SL 2 (p).Next, Γ has exponential word growth, that is, there exists δ = δ(S) > 0 such that for all r > 0, |B S (r)| ≥ (1 + δ) r (this follows, for instance, from the Tits alternative, since Γ has a nonabelian free subgroup).Let c be as in Lemma 2.4 and let: Proof of Theorem 2.2.Let p ∈ Q n and P be as in Proposition 2.5.Let w(x) ∈ Γ * Z be as in (2).For 1 . By Proposition 2.5, all c i are nontrivial, so that: is nontrivial.Choosing C 0 sufficiently large in the definition of Q n , as we may, Theorem 1.9 applies to w(x), so there exists g ∈ PSL d (p) such that w(g) = e.Let g ∈ SL d (p) be a lift of g.Then noting that π P (B S (r)) = B πP (S) (r), and applying Proposition 2.7, there exists g ∈ B S (C log(p)) such that π P (g) = g.Since π P is surjective and π P (w(g)) is not central in SL d (p), w(g) is not central in Γ.Finally, since p ∈ Q n , log(p) = O(log(n)), as desired.
Proof of Theorem 1.6.Let Γ be as in the statement of the Theorem, and let S be a finite generating set for Γ.

Applications
As a first application of Theorem 1.In the remainder of this Section we deduce Theorem 1.2 and Corollaries 1.3 and 1.4 from Theorem 1.6.

Kleinian groups
We refer to [15] for general background on Kleinian group.Our exposition in this Subsection is closely inspired by the proof of Theorem 1.2 from [11].Theorem 3.2.Let Γ be a finitely generated torsion-free nonelementary Kleinian group.Then Γ is sharply MIF.
As noted in [15], examples of torsion-free nonelementary Kleinian group include all fundamental groups of closed orientable hyperbolic 2-and 3-manifolds.As such, Corollary 1.4 follows immediately from Theorem 3.2.
In proving Theorem 3.2 is helpful to recall the following well-known characterization of Zariski-density in the case of SL 2 .Theorem 3.3.For Γ ≤ SL 2 (C), the following are equivalent.
(i) Γ is Zariski-dense in SL 2 (C); Recall that an action of a group G on a set Ω is highly transitive if, for every positive integer n, and every two n-tuples x 1 , . . ., x n and y 1 , . . ., y n of distinct points of Ω, there exists g ∈ G such that g(x i ) = y i for 1 ≤ i ≤ n.It is easy to see that if the action of G on Ω is highly transitive, then so is the restriction of that action to any finite-index subgroup of G. Proposition 4.2.Let G and α be as in Definition 4.1.Let x ∈ X and suppose the orbit Orb(x) of x in X is infinite.Then the action of T (α) on Orb(x) is highly transitive.
An action of a group on X is minimal if every orbit of the action is dense in X.
Theorem 4.3 (Theorem 8.1 of [12]).There exists a finitely generated infinite group F of subexponential word growth with a faithful continuous minimal action α : F → Homeo(X), such that T (α) ∼ = F .Moreover the derived subgroup [F, F ] of F is simple, and has finite index in F .
Proof of Theorem 1.10.Let F be as in Theorem 4.3.Our example shall be Γ = [F, F ]. Being of finite index in F , Γ is also a finitely generated group of subexponential growth.As noted in [4] Remark 9.3, no group of subexponential growth can be sharply MIF.It therefore suffices to verify that Γ is MIF.
Let x ∈ X and set Ω = Orb(x), a countably infinite set.Then by Proposition 4.2 and F ∼ = T (α), F admits a highly transitive action on Ω, hence so does Γ.Moreover, since Ω is dense in X, under this action no nontrivial element of Γ has finite support on Ω.Finally, we apply Theorem 5.9 of [10]: if Γ is not MIF, then Γ contains a normal subgroup isomorphic to the group A of all finitely supported even permutations of N. Since Γ is simple, we have Γ ∼ = A.This is a contradiction, as Γ is finitely generated and A is not.Remark 4.4.As noted in the Introduction to [12], the construction given therein yields an upper bound on the word-growth of Γ of the form f (n) = C 1 exp n/ exp(C 2 √ log n) .By the argument of [4] Remark 9.3, we obtain a lower bound for M Γ which is approximately an inverse function to n 2 f (n).The lower bound thus obtained is stronger than log(n) log log(n) C , but weaker than log(n) 1+ 1 C , for every C > 0. Presumably this lower bound is far from sharp; MIF growth for topological full groups should be investigated elsewhere.

Lemma 2 . 1 .
For all c > 0, if C 0 is chosen sufficiently large (depending on c and D) then for all n ≫ D 1, p∈Qn p ≥ exp(cn 2 ) Proof.By the Prime Number Theorem, the product p(x) of all rational primes at most x satisfies: lim x→∞ log(p(x))/x = 1.

Lemma 2 . 4 .
There exists c = c(S) > 0 such that the following holds.Let p ∈ Q n and let P ∈ O K be a K-prime dividing p. Suppose P does not divide any element of {δ(s) : s ∈ S}.Then the restriction of π P to B S (c log(p)) is injective.Proof.Note that if h, k ∈ Γ satisfy π P (h) = π P (k) then g = hk −1 ∈ ker(π P ) and |g| S ≤ |h| S + |k| S .Hence suppose that I d = g ∈ ker(π P ).Then 0 = X = g * − δ(g)I d ∈ M d (O K ) has all entries divisible by P. Letting x ∈ O K be a nonzero entry of X, we therefore have µ(x) ≥ p 1/D .On the other hand, by Lemma 2.3, µ(x) ≤ (dM ) |g|S + M |g|S ≤ ((d + 1)M ) |g|S , and the conclusion follows.

1 .
By contrast, by Lemma 2.3, µ takes value at most 2(dM ) n+1 on the entries of s * c * i − c * i s * .Since k ≤ n, there are at most d 2 n nonzero elements of O K which are entries of one of the s * c * i − c * i s * , and as such:
, PGL d (C) is MIF.Our most general result makes gives sharp bounds on the MIF growth of certain finitely generated Zariski-dense subgroups of PGL d (C) (see Theorem 2.2 below).Let d ≥ 2 and let K be a number field.Let Γ ≤ PSL d (K) be finitely generated and Zariski-dense in PGL d (C).If d ≥ 3 suppose that: 6, we return to Example 1.7, and prove that PSL d (Z) is sharply MIF.We shall be applying Theorem 1.6 with K = Q, so that hypothesis (1) holds automatically.Finite generation of PSL d (Z) is classical.Finally, since PSL d (Z) is a lattice in PSL d (R), it is Zariski-dense by the Borel Density Theorem.Remark 3.1.Of course, the preceding argument extends to any finitely generated subgroup of PSL d (Z) which is Zariski-dense in PSL d (R).In particular, any finite-index subgroup of PSL d (Z) is sharply MIF.