Words, Hausdorff dimension and randomly free groups
aa r X i v : . [ m a t h . G R ] J un WORDS, HAUSDORFF DIMENSION AND RANDOMLYFREE GROUPS
MICHAEL LARSEN AND ANER SHALEV
Abstract.
We study fibers of word maps in finite and residually finitegroups, and derive various applications. Our main result shows that,for any word 1 = w ∈ F d there exists ǫ > w : Γ d → Γ haveHausdorff dimension at most d − ǫ .We conclude that profinite groups G := ˆΓ, Γ as above, satisfy noprobabilistic identity, and therefore they are randomly free , namely, forany d ≥
1, the probability that randomly chosen elements g , . . . , g d ∈ G freely generate a free subgroup (isomorphic to F d ) is 1. This solves anopen problem from [DPSS].Additional applications and related results are also established. Forexample, combining our results with recent results of Bors, we concludethat a profinite group in which the set of elements of finite odd orderhas positive measure has an open prosolvable subgroup. This may beregarded as a probabilistic version of the Feit-Thompson theorem.2010 Mathematics Subject Classification.
Primary 20E26, 20P06; Secondary 20D06,20G40.ML was partially supported by NSF grant DMS-1401419. AS was partially supportedby ERC advanced grant 247034, BSF grant 2008194, ISF grant 1117/13 and the VinikChair of mathematics which he holds. Introduction
In the past few decades there has been considerable interest in the theoryof word maps, see for instance [Bo, LiS, La, Sh, LS1, LST, GT, N2], as wellas Segal’s monograph [Se] and the references therein. Many of these worksfocus on the image of word maps on finite simple groups and on a relatedWaring type problem. There is also increasing interest in fibers of wordmaps and related problems, see [LS2, LS3, B1, B2].In this paper we prove various results showing that fibers of word mapsare small, not just in a measure-theoretic sense, but also in the strongersense of Hausdorff dimension. Our results apply for a wide family of finiteand infinite groups, well beyond the family of finite simple groups.The main result of this paper is the following.
Theorem 1.1.
Let Γ be a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors. Let d ≥ and let w ∈ F d be a non-trivial word. Then there exists ǫ > depending only on w such thatthe Hausdorff dimension of any fiber of the associated word map w : Γ d → Γ is at most d − ǫ . Here F d denotes a free group of rank d freely generated by x , . . . x d . Aword w = w ( x , . . . , x d ) ∈ F d (which we write in reduced form) gives rise toa word map w : Γ d → Γ on any group Γ, which is induced by substitution.By an upper composition factor of Γ we mean a composition factor ofsome finite quotient Γ / ∆ of Γ, where ∆ is a normal subgroup of Γ of finiteindex (and we assume ∆ is open if Γ is a profinite group).For a residually finite group Γ and d ≥
1, we define the
Hausdorff dimen-sion of a subset S ⊆ Γ d byHdim( S ) = lim inf ∆ log | S ∆ d / ∆ d | log | Γ / ∆ | , where ∆ ranges over the finite index normal subgroups of Γ (and again ifΓ is profinite we also assume that ∆ is open). Thus Hdim(Γ d ) = d andTheorem 1.1 states that, under the assumptions of the theorem we haveHdim( w − ( g )) ≤ d − ǫ for every g ∈ Γ.In fact the proof of Theorem 1.1 gives a bit more; see Theorem 5.2 for aneffective finitary version.These results may be regarded as a far-reaching extension of the mainresult of [LS2], stating that, for w as above there exists ǫ = ǫ ( w ) > T is a large enough finite simple group then the fibers of w : T d → T have size at most | T | d − ǫ . Here and throughout this paper, by a finite simplegroup we mean a non-abelian finite simple group.We now list several consequences of Theorem 1.1. The first one deals withlinear groups and strengthens results from [LS3]. ORDS, HAUSDORFF DIMENSION AND RANDOMLY FREE GROUPS 3
It is easy to see, using strong approximation (see [N1, Pi, We]), that afinitely generated linear group which is not virtually solvable has infinitelymany finite simple groups of Lie type as upper composition factors. ApplyingTheorem 1.1, we deduce the following.
Theorem 1.2.
Let Γ be a finitely generated linear group over any field.Suppose Γ is not virtually solvable. Then the fibers of any word map on Γ d induced by a non-trivial word w ∈ F d have Hausdorff dimension less then d . The next consequence of our main result deals with probabilistic identi-ties. We need more notation. Given a word w ∈ F d and a finite group G , let p w,G denote the associated probability distribution on G . Thus, for g ∈ G we have p w,G ( g ) = | w − ( g ) | / | G | d .Recall that a word 1 = w ∈ F d is said to be a probabilistic identity of aresidually finite group Γ if there exists δ > H of Γ, we have p w,H (1) ≥ δ . This amounts to saying that, in the profinitecompletion G of Γ, the probability (with respect to the normalized Haarmeasure on G ) that w ( g , . . . , g d ) = 1 where g , . . . , g d ∈ G are randomelements is positive.For example, w = x is a probabilistic identity of the infinite dihedralgroup Γ = D ∞ .It follows from [Ma] that a residually finite group which satisfies the prob-abilistic identity x is finite-by-abelian-by-finite. A similar conclusion holdsfor the probabilistic identity [ x , x ], as follows from the earlier paper [Ne],which is applied in [Ma]. However, very little is known about groups satis-fying more general probabilistic identities.Now, let Γ be a residually finite group with infinitely many non-isomorphicnon-abelian upper composition factors, and let 1 = w ∈ F d . By Theorem1.1 we see that Γ has arbitrarily large finite quotients H such that p w,H (1) ≤| H | − ǫ , which tends to 0. This implies the following new result. Theorem 1.3.
Let Γ be a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors. Then Γ does not satisfyany probabilistic identity. This result generalizes Theorem 1.2 in [LS3], showing that a finitely gener-ated linear group which satisfies a probabilistic identity is virtually solvable.It also enables us to solve an open problem regarding randomly free groups.We say that a profinite group G is randomly free if, for every d ≥
1, theprobability that a d -tuple of elements of G freely generates a free subgroupis 1. We use freely generate in the sense of abstract group theory, i.e., nonon-trivial word evaluated at the chosen elements should give the identity. Aresidually finite group is said to be randomly free if its profinite completionis randomly free.See [Ep] and [Sz] for earlier results on groups in which almost all subgroupsare free.We need the following straightforward observation. MICHAEL LARSEN AND ANER SHALEV
Lemma 1.4.
A residually finite group Γ is randomly free if and only if itdoes not satisfy any probabilistic identity. To show this, let G be the profinite completion of Γ. Note that g , . . . , g d ∈ G freely generate a free subgroup of G if and only if w ( g , . . . , g d ) = 1 forevery 1 = w ∈ F d . Now, suppose Γ does not satisfy any probabilistic identity.Then the probability that w ( g , . . . , g d ) = 1 ( g i ∈ G ) is 0 for any such word w . As Haar measure is σ -additive, the probability that there exists w = 1such that w ( g , . . . , g d ) = 1 is also 0. Thus, g , . . . , g d ∈ G freely generatea free subgroup with probability 1, proving that Γ is randomly free. Thereverse implication is trivial.Combining the above lemma with Theorem 1.3 we obtain the following. Theorem 1.5.
Let Γ be a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors. Then Γ is randomly free. This immediately implies (using strong approximation) the following prob-abilistic Tits alternative, which is the main result of [LS3].
Corollary 1.6.
A finitely generated linear group is either virtually solvableor randomly free.
Theorem 1.5 (applied to profinite groups G ) solves a problem raised in2003 in [DPSS] (see Problem 7 there). A partial solution, assuming G hasinfinitely many alternating groups as upper composition factors, was givenin 2005 by Ab´ert, see [Ab, Theorem 1.7]. Hence it remains to prove thetheorem assuming G has infinitely many simple groups of Lie type as uppercomposition factors.A recent work of Bors [B1] can be used to handle the case where G hasclassical groups of unbounded rank as composition factors. Indeed Theorem1.1.2 there implies Theorem 1.3 in this case. Hence it remains to handlethe case where G has infinitely many groups of Lie type of bounded rank asupper composition factors.After discussing various consequences of Theorem 1.1, let us now statesome results of independent interest on finite groups, which play a crucialrole in its proof. The first result deals with almost simple groups of Lie typeof bounded rank. Theorem 1.7.
For any non-trivial word w = w ( x , . . . , x d ) and any positiveinteger r , there exist N, ǫ > depending only on w and r such that, if T isa finite simple group of Lie type of order ≥ N and rank ≤ r , then for all g , . . . , g d , g ∈ Aut ( T ) , (cid:12)(cid:12) { ( t , . . . , t d ) ∈ T d | w ( t g , . . . , t d g d ) = g } (cid:12)(cid:12) ≤ | T | d − ǫ . Combining this with results from [LS2] and [B1], we show that, for any1 = w ∈ F d there exist N, ǫ > G ofsize at least N and any g ∈ G we have p w,G ( g ) ≤ | G | − ǫ —see Corollary 4.6below. This extends the main result of [LS2] from simple groups to almost ORDS, HAUSDORFF DIMENSION AND RANDOMLY FREE GROUPS 5 simple groups. This extension is by no means routine, and it occupies alarge part of our paper.Combining the result above with other tools we deduce the following moregeneral theorem on so-called semisimple finite groups.
Theorem 1.8.
Let G be a finite group such that T k ≤ G ≤ Aut ( T k ) forsome k ≥ and a finite simple group T . Suppose w = 1 is a word. Thenthere exist constants N = N ( w ) , ǫ = ǫ ( w ) > depending only on w suchthat, if | T | ≥ N , then for any g ∈ G we have p w,G ( g ) ≤ | T k | − ǫ . Note that, if G is any finite group with a chief factor G /G ∼ = T k , then G has a semisimple quotient G/C G ( G /G ) (the section centralizer) lyingbetween T k and its automorphism group, to which Theorem 1.8 may beapplied.The next result enables us to pass from any finite group with a largenon-abelian composition factor to a large semisimple quotient K as above,in which the chief factor T k is almost as large as K . Proposition 1.9.
For any δ > there exists f = f ( δ ) > such that if G is a finite group with a non-abelian composition factor of order ≥ f , then G has a quotient K with a composition factor T of order ≥ f such that T k ≤ K ≤ Aut ( T k ) and | T k | ≥ | K | − δ . Results 1.8 and 1.9 easily imply our main result, namely Theorem 1.1.Finally, combining Theorem 1.8 with a new result of Bors [B2, 1.1.2] weobtain a result which may be regarded as a probabilistic version of the OddOrder Theorem of Feit and Thompson [FT]. For a finite group G denoteby Rad ( G ) the solvable radical of G , namely the maximal solvable normalsubgroup of G . Theorem 1.10. (i)
Let k be an odd integer and let w = x k . Then forany ǫ > there is a number M = M ( k, ǫ ) such that, if G is anyfinite group satisfying p w,G ( g ) ≥ ǫ for some g ∈ G , then | G/ Rad ( G ) | ≤ M. (ii) Let G be a profinite group and suppose that the set of elements of G of finite odd order has positive Haar measure. Then G has aprosolvable open normal subgroup. Therefore profinite groups as in part (ii) above are virtually prosolvable.Part (i) above shows that, for odd k , if the probability that g k = 1 in G isbounded away from zero, then G has a solvable normal subgroup of boundedindex.However, it is not true that if the probability that g ∈ G has odd order isat least ǫ > − ǫ ), then G is solvable-by-bounded. Indeed,simple groups of Lie type in characteristic 2 provide counterexamples (sincemost of their elements are semisimple, hence of odd order). MICHAEL LARSEN AND ANER SHALEV
A result similar to part (i) of Theorem 1.10 with w = [ x , . . . , x d ], a leftnormed commutator, also follows by combining Theorem 1.1.2 of Bors [B2]with Theorem 1.8 above.Let us now discuss the strategy of the proof of Theorem 1.7, which is thebasis of most of our other results. The main idea is, roughly, to convert itto a problem in algebraic geometry. If, to simplify slightly, T = G ( F q ) forsome algebraic group G of bounded rank, the parameter q goes to infinityas | T | → ∞ . Any non-trivial word w defines a non-constant morphism G d → G . The fibers are therefore of dimension ≤ d dim G −
1, and as q goesto infinity and g varies, the “complexity” of the fibers remains bounded, sowe can deduce that | w − ( g ) | = O ( | G ( F q ) | d − / dim G )from standard point counting results for varieties over finite fields. This ideais not new to this paper (see, e.g., [DPSS, LS2]). However, there are techni-cal difficulties in implementing it when we must take outer automorphismsof T into account. For field automorphisms, when F q is large but its charac-teristic p is small, this requires a new idea, namely finding big gaps betweenconsecutive powers of Frobenius appearing in any specified d -tuple of cosetsof T (see the proof of Theorem 1.7, below). The Suzuki and Ree groupsalso pose a technical challenge, since we are no longer counting points onvarieties over finite fields but rather taking fixed points of maps which aresquare roots of ordinary Frobenius maps.In §
2, we develop a simple formalism for making precise the idea ofbounded complexity mentioned above. In §
3, we present upper bounds forcertain point counting problems. There are two main variants, one aimedat proving estimates which are uniform in characteristic and one aimed atdealing with the special difficulties of the Suzuki-Ree case. In § § T k ≤ G ≤ Aut ( T k ) ∼ = Aut ( T ) ≀ S k as in Theorem 1.8, then G induces atransitive permutation group P on the k copies of T , and tools from the the-ory of permutation groups become relevant. Theorem 1.8 is proved, roughly,by using the case k = 1, which has already been established, and by findinga large set of independent equations induced by the given word equation on G .We then prove Proposition 1.9 by bounding the order of P above, anddeduce a finitary version—Theorem 5.2—of Theorem 1.1, which readily im-plies it. 2. Degree Bounds
Definition 2.1.
Let R be a commutative ring. By a generated commutativealgebra over R (GCA for short), we mean a pair ( A, S ) consisting of a
ORDS, HAUSDORFF DIMENSION AND RANDOMLY FREE GROUPS 7 commutative R -algebra A and a finite set S of generators of A over R .An isomorphism ( A, S ) → ( B, T ) of GCAs is an isomorphism A → B of R -algebras which maps S onto T .Thus, every GCA is isomorphic to one of the form( R [ x , . . . , x N ] /I, { x , . . . , x N } ) . For a ∈ A we define deg S a to be the minimum integer m such that a canbe realized as a degree m polynomial in the elements S with coefficients in R . If ( A, S ) and (
B, T ) are GCAs over R and φ : A → B is an R -algebrahomomorphism, we definedeg φ = max s ∈ S deg T φ ( s ) . The following lemma is obvious:
Lemma 2.2.
With notation as above, deg T φ ( a ) ≤ deg φ deg S a. If ( C, U ) is also a GCA over R , and ψ : B → C is an R -algebra homomor-phism, deg ψ ◦ φ ≤ deg φ deg ψ. If R ′ is a commutative R -algebra and ( A ′ , S ′ ) denotes the base change of ( A, S ) to R ′ (i.e., A ′ = A ⊗ R R ′ , and S ′ = { s ⊗ | s ∈ S } ), then deg S ′ a ⊗ ≤ deg S a. The following lemma, asserting that every q -Frobenius map has degree atmost q , is likewise immediate from the definitions: Lemma 2.3. If ( A, S ) is a GCA over R = F q , and φ : A → A denote the q -Frobenius map, then deg φ ≤ q . We say generating sets S and T of A are equivalent if they generatethe same R -submodule of A . In this case, deg S a = deg T a for all a ∈ A .Thus, every surjective R -algebra homomorphism φ : R [ x , . . . , x N ] → A de-termines a well-defined equivalence class of generating sets in A representedby { φ ( x ) , . . . , φ ( x N ) } . Geometrically, a closed immersion of Spec A intoan affine space over R determines a generating set, and two closed immer-sions which are the same up to an affine transformation determine equivalentgenerating sets. Lemma 2.4.
Let R be a field, G = Spec A an adjoint semisimple algebraicgroup over R , α an automorphism of G as algebraic group over R , and ρ : G → A dim G the adjoint representation. Up to equivalence, ρ and ρ ◦ α determine the same equivalence class of generating sets of A .Proof. The homomorphisms ρ and ρ ◦ α give the same representation, whichmeans they are conjugate, so they define the same equivalence class of gen-erating sets. (cid:3) MICHAEL LARSEN AND ANER SHALEV
In general, given an adjoint semisimple algebraic group G over a field F q ,we regard its coordinate ring A as a GCA endowed with a generating set S belonging to this equivalence class. Proposition 2.5.
Let G := Spec A be an adjoint semisimple group over F p with root system Φ . Let F p denote the p -Frobenius endomorphism of G .Let β , . . . , β l be automorphisms of G defined over F p , and let n , . . . , n l ∈{ , . . . , d } . Then, for G as above, the morphism G d → G given by (1) ( g , . . . , g d ) F j q β ( g n ) ± · · · F j l q β l ( g n l ) ± has degree O ( p j ) , where j = max i j i , and the coordinate rings of G and G d have generating sets S and S ` · · · ` S respectively. (The implicit constantdepends on Φ and l but not on p or j .)Proof. As the multiplication and inversion morphisms on G have degreeswhich are bounded independent of q , by Lemma 2.2 and induction on l , itsuffices to prove the claim for g F jp β ( g ). The proposition follows fromLemma 2.2, Lemma 2.3, and Lemma 2.4. (cid:3) Point Bounds
It is well known that a degree s hypersurface in A N has at most sq N − points over F q . We present several variants of this observation. Proposition 3.1.
Let p be prime, q a power of p , X ⊂ A N over F p be theclosed subscheme defined by the ideal ( f , . . . , f n ) , where f i ∈ F p [ x , . . . , x N ] have degrees d , . . . , d n . If d · · · d n ≤ Aq, then | X ( F p ) ∩ A n ( F q ) | ≤ (2 A + 1) N d · · · d n q dim X . Proof.
Let φ i ( x , . . . , x N ) := x d i f i ( x /x , . . . , x N /x ) . The intersection of the hypersurfaces in P N defined by the φ i defines aclosed subscheme X of P N , and the intersection of this subscheme with A N is X . Therefore, the number of irreducible components of X over F p isless than or equal to the number of components in X , which by a suitableversion of B´ezout’s theorem (see [Fu, 12.3.1]), implies that the number ofsuch components is less than or equal to d · · · d n . This implies the theoremfor dim X = 0.We proceed by double induction, first on dim X and then on N . The basecase for the inner induction is N = dim X , and the proposition holds inthis case trivially. For the induction step, we count pairs ( x, H ) consistingof a point x ∈ X ( F p ) ∩ A n ( F q ) and H an F q -rational hyperplane in A N ORDS, HAUSDORFF DIMENSION AND RANDOMLY FREE GROUPS 9 containing x (but not necessarily 0). The number of such pairs for a given x ∈ X ( F p ) ∩ A n ( F q ) is | P N − ( F q ) | , so the total number is(1 + q + · · · + q N − ) | X ( F p ) ∩ A n ( F q ) | . We say that H is bad for X if it contains an irreducible component of X of dimension dim X . The number of hyperplanes containing a given top-dimensional component is at most1 + q + · · · + q N − dim X − , so the number of bad hyperplanes is less than2 q N − dim X − d · · · d n . By the induction hypothesis for N , the number of pairs ( x, H ) where H isbad is less than2 q N − dim X − d · · · d n (2 A +1) N − d · · · d n q dim X ≤ A (2 A +1) N − q N d · · · d n . By the induction hypothesis for dim X , the number of pairs ( x, H ) where H is good is less than( q + q + · · · + q N )(2 A + 1) N − d · · · d n q dim X − . Thus, the total number of pairs is less than( q + q + · · · + q N )(2 A + 1) N d · · · d n q dim X , so | X ( F p ) ∩ A n ( F q ) | ≤ (2 A + 1) N d · · · d n q dim X , and the proposition holds by induction. (cid:3) Proposition 3.2.
Let ( A, S ) be a GCA over Z . Let ( A q , S q ) be a GCA over F q such that there exists an isomorphism ι : A ⊗ F q → A q ⊗ F q F q with respectto which S and S q are equivalent, and let Y = Spec A q . If f ∈ A ⊗ F q isnot a zero divisor, then |{ y ∈ Y ( F q ) | f ( y ) = 0 }| = O ((deg S f ) q dim Y − ) . Here the implicit constant depends on (
A, S ) but not on q . Proof.
Let S = { s , . . . , s M } and S q = { t , . . . , t N } , and consider the homo-morphisms φ : Z [ x , . . . , x N ] → A and φ q : F q [ y , . . . , y N ] → A q mapping x i to s i ⊗ y i to t i ⊗ F q [ x , . . . , x N ] / / φ ⊗ (cid:15) (cid:15) F q [ y , . . . , y N ] φ q ⊗ (cid:15) (cid:15) A ⊗ F q ι / / A q ⊗ F q F q where the top arrow maps each x i to a linear combination of the y j . If f , . . . , f n is a generating set for ker φ . In particular, A q ⊗ F q F q is the quotient of F q [ y , . . . , y N ] by polynomials ι ( f ⊗ n ) , . . . , ι ( f k ⊗
1) of degrees at mostdeg f , . . . , deg f n respectively.The proposition now follows by applying Proposition 3.1 to X = Spec F p [ x , . . . , x N ] / ( ι ( f ⊗ , . . . , ι ( f n ⊗ , f ) . (cid:3) Proposition 3.3.
Let q = p f , ( A, S ) a GCA defined over F p , and X =Spec A . For all a ∈ A , we let X a denote Spec A/ ( a ) regarded as a closedsubscheme of X . Let k be a positive integer prime to f , F q : X → X the q -Frobenius morphism, and F : X → X an endomorphism such that F k = F q .If dim X a < dim X and t is a positive integer, then | X a ( F p ) ∩ X ( F p ) F t | ≤ O ((deg S a ) p ( tf/k )(dim X − ) , where the implicit constant does not depend on t or a .Proof. Let g denote the g.c.d. of t and k . Replacing F by F g and t and k by t/g and k/g respectively, we may assume without loss of generality that t and k are relatively prime.Every endomorphism of a commutative ring maps the nilradical to itself,so without loss of generality, we may assume that X is reduced. We mayalso assume that the irreducible components X , . . . , X h of X are permutedtransitively by powers of F .By induction, we may assume that the proposition holds for all affineschemes of dimension less than dim X . Thus, if U is any F -stable denseopen affine subvariety of X , it suffices to prove the proposition for U . If U is any dense open affine subvariety of X , the orbit of U under h F i isfinite, so we may replace X by an F -stable dense open affine subvarietyof U . Since every quasi-affine variety contains a dense affine subvariety, itsuffices to prove the desired estimate after replacing X by our choice of densequasi-affine U ⊆ X .Applying this observation to the complement of [ ≤ i Let dim X = mk . We claim that there exists a transcendence basis b , . . . , b k , b , . . . , b mk for K in A such that F ∗ b ij = ( b i j +1 if j < k , b qi otherwise.Indeed, by the same reasoning that shows that the transcendence degree of K is a multiple of k , we see that every F ∗ -stable subfield of K has tran-scendence degree a multiple of k . Thus, we can construct the b i iteratively,defining b q +1 1 to be any element of A not algebraic over the (algebraicallyindependent) set { b ij | i ≤ q } and setting b i j +1 = ( F j ) ∗ b i .Endowing A mk = Spec F p [ x ij ] (1 ≤ i ≤ m , 1 ≤ j ≤ k ) with the F -actiongiven by(2) F ∗ x ij = ( x i j +1 if j < k , x qi otherwise,we see that the map x ij b ij is a generically finite F -equivariant morphism X → A mk . Passing to a sufficiently small F -stable affine open subsetSpec A [1 /h ] of the base, we may assume that A [1 /h ] is a finitely generatedfree F p [ x ij ][1 /h ]-module of some rank ρ . We can therefore realize A [1 /h ] as acommutative subring of M ρ ( F p [ x ij ][1 /h ]). In particular, all elements of S canbe realized in this matrix ring, and it follows that any polynomial of degreedeg S a in the elements of S can be realized as a matrix whose entries arepolynomials of degree O (deg S a ) in the x ij and 1 /h . The determinant of thismatrix is a polynomial of degree O (deg S a ) in x ij and 1 /h . Every point in thezero locus X a maps to a point in the zero locus of this polynomial. Since themap Spec A [1 /h ] → Spec F p [ x ij ][1 /h ] is of degree ρ , it suffices to bound thenumber of F t -points on the zero locus of this determinant. Expressing thedeterminant as a rational function in the x ij , the numerator is a polynomialof degree O (deg S a ), and it suffices to bound the number of F t -points of thezero locus of the numerator, viewed as a hypersurface in Spec F p [ x ij ]. Thus,we can reduce the general problem to the case X = Spec F p [ x ij ], with the F -action given by (2).More explicitly, it suffices to prove that if Q is a polynomial in the x ij , |{ ( a ij ) ∈ F mkp | Q ( a ij ) = 0 } F t | = O (deg Q ( t/k ) p mk − ) . We project A mk to A k by π m : ( a ij ) ( a m , . . . , a mk ) . Thus π m is F -equivariant, where F ( c , . . . , c k ) = ( c , c , . . . , c k , c q ) , so F t ( c , . . . , c k ) = (cid:0) c q ⌊ t/k ⌋ t +1 , c q ⌊ ( t +1) /k ⌋ t +2 , . . . , c q ⌊ ( t + k − /k ⌋ t + k − (cid:1) = (cid:0) c q ⌊ t/k ⌋ t +1 , . . . , c q ⌊ t/k ⌋ k , c q ⌊ t/k ⌋ +1 , . . . , c q ⌊ t/k ⌋ +1 t (cid:1) where the c i are numbered cyclically. Since s and k are relatively prime, any F t -fixed ( c , . . . , c k ) is determined by c satisfying c p t = c . The numberof possibilities for c ∈ F q is therefore p t . Any F t -fixed point ( a ij ) projectsunder π m to a F t -fixed point.There are at most deg Q factors of Q over F p [ x ij ] of the form x m − c ,and for any c ∈ F p such that x m − c is not such a factor and ( c , . . . , c k ) isfixed by F t , the fiber π − m ( c , . . . , c k ) is a hypersurface in A ( m − k of degree ≤ deg Q . If ( c , . . . , c m ) is fixed by F t and x m − c divides Q , then thenumber of F t fixed points in π − m ( c , . . . , c m ) is the number of F t fixed pointsof A ( m − k , i.e. q t ( m − . By induction on m , it suffices to consider the basecase, m = 1.For m = 1, if ( c , . . . , c m ) is a fixed point of F t , we have c = c q ⌊ t/k ⌋ t +1 = · · · = c q t −⌈ t/k ⌉ ( k − t +1 , so Q ( c , . . . , c k ) can be expressed as a polynomial in c ( k − t +1 of degree atmost (dim Q ) q t −⌈ t/k ⌉ = (dim Q ) p tf −⌈ t/k ⌉ f ≤ (dim Q ) p tf − tf/k = (dim Q ) p ( tf/k )(dim X − , which proves the proposition. (cid:3) Almost simple groups In this section we bound the size of fibers of word maps for almost simplegroups. Most of the section is devoted to the proof of Theorem 1.7 on almostsimple groups of Lie type of bounded rank.We make essential use of the following result of Nikolov [N2, Corollary 8],ruling out a given coset identity in large almost simple groups. Proposition 4.1. For every word = w ∈ F d there exists c = c ( w ) suchthat if T is a finite simple group of order ≥ c and G is an almost simplegroup with socle T , then for every g , . . . , g d ∈ G we have w ( T g , . . . , T g d ) = { } . This easily yields the following. Corollary 4.2. With notation as above, there exists c = c ( w ) such thatif | T | ≥ c then | w ( T g , . . . , T g d ) | > .Proof. Define v ( x , . . . , x d ) = w ( x , . . . , x d ) w ( x d +1 , . . . , x d ) − .Setting c ( w ) = c ( v ) the result follows. (cid:3) ORDS, HAUSDORFF DIMENSION AND RANDOMLY FREE GROUPS 13 Now let w = x e n · · · x e l n l be a reduced word of length l ≥ F d . Thismeans that n , . . . , n l ∈ { , . . . , d } , e i = ± 1, and n i = n i +1 implies e i = e i +1 .Let G be a group, and let α , . . . , α l ∈ Aut ( G ). The map G d → G givenby ( g , . . . , g d ) α ( g n ) e · · · α l ( g n l ) e l is called a generalized word function (allowing w to be non-reduced) or an automorphic word map on G d based on w (see [Se, § T ⊳ G and consider the word map w : G d → G . Its restriction to T g × · · · × T g d (where g i ∈ G ) can be regarded as a map T d → G whoseimage lies in a T -coset of G . Indeed, more is true, namely: Lemma 4.3. With the above notation, the map ( t , . . . , t d ) w ( g , . . . , g d ) − w ( t g , . . . , t d g d ) is an automorphic word map T d → T based on w . This well known observation is easily proved by induction on l . Lemma 4.4. Given integers j , . . . , j l , there exist nonnegative integers j ′ , . . . , j ′ l ≤ m − ⌈ m/l ⌉ such that j − j ′ ≡ · · · ≡ j l − j ′ l (mod m ) . Proof. Without loss of generality, we may assume 0 ≤ j ≤ · · · ≤ j l < m .Setting j l + i = m + j i for 1 ≤ i ≤ k , we have j i +1 − j i ≥ i = 1 , , . . . , m ,so j r +1 − j r ≥ m/l for some 1 ≤ r ≤ m , so setting j ′ i = ( m − j r +1 + j i if 1 ≤ i ≤ rj i − j r +1 if r + 1 ≤ i ≤ m, the lemma follows. (cid:3) We now prove Theorem 1.7. Proof. For every group T as above, there exists a prime p , an adjoint splitsimple algebraic group G over F p with root system Φ, and a general-ized Frobenius endomorphism F of G such that T is the derived groupof G ( F p ) F . Let S denote a fixed set of generators of the coordinate ring A of G Φ . Let F p denote the p -Frobenius map of G . The condition that F is ageneralized Frobenius endomorphism means that some positive power of F is a positive power of F p .Fix cosets of T g , . . . , T g d of T in Aut ( T ) and let w ∈ F d be as above. ByLemma 4.3 there exist α , . . . , α l ∈ Aut ( T ) such that for ( t , . . . , t d ) ∈ T d the map(3) ( t , . . . , t d ) w ( g , . . . , g d ) − w ( t g , . . . , t d g d ) = α ( t n ) e · · · α l ( t n l ) e l is an automorphic word map T d → T based on w .Each α i can be written F j i p β i , where β i is a product of a diagonal au-tomorphism and a graph automorphism. By Corollary 4.2, the map (3) is not constant, assuming T is large enough given w . We can define (3) asthe restriction to T ⊂ G ( F p ) of a morphism G d → G defined over F p interms of the comultiplication map on the coordinate ring A ⊗ F p of G , theinverse map on A ⊗ F p , and the various endomorphism maps on A ⊗ F p givingrise to the α i . This morphism cannot be constant, and by Proposition 2.5,its degree (with respect to S p ` · · · ` S p ) is O ( p j ), where j = max ≤ i Theorem 4.5. For any non-trivial word w = w ( x , . . . , x d ) there exist N, ǫ > depending only on w such that, if T is any finite simple groupof order ≥ N , then for all g , . . . , g d , g ∈ Aut ( T ) , (cid:12)(cid:12) { ( t , . . . , t d ) ∈ T d | w ( t g , . . . , t d g d ) = g } (cid:12)(cid:12) ≤ | T | d − ǫ . Proof. If T is an alternating group of degree larger than some function f ( w )of w , or a classical group of rank larger than f ( w ), then the conclusionfollows from Theorem 3.1.2 in [B1]. Indeed, the latter result shows that,under our assumption on T , there is ǫ = ǫ ( w ) > T d based on w have size ≤ | T | d − ǫ , which impliesthe required conclusion. ORDS, HAUSDORFF DIMENSION AND RANDOMLY FREE GROUPS 15 Otherwise T is a simple group of Lie type of rank ≤ f ( w ), and the resultfollows by Theorem 1.7. (cid:3) Theorem 4.5 strengthens of Proposition 4.1 of Nikolov: not only there isno fixed coset identity w in large almost simple groups, the probability of w attaining any fixed value g on each subset T g × . . . × T g d tends to zerovery fast as | T | → ∞ .We conclude with the following probabilistic consequence. Corollary 4.6. For any non-trivial word w there exist N, ǫ > dependingonly on w such that, for any almost simple group of order at least N andany element g ∈ G we have p w,G ( g ) ≤ | G | − ǫ . Proof. Since T is generated by two elements we trivially have | G | ≤ | Aut ( T ) | ≤| T | (much better upper bounds hold, of course). Therefore it sufficesto show that, for some N, ǫ > w , if | T | ≥ N / then p w,G ( g ) ≤ | T | − ǫ .This follows from Theorem 4.5 above, by summing up the probabilitiesover all subsets T g × . . . × T g d of G d . (cid:3) From this, we can immediately deduce Theorem 1.8 in the case of almostsimple groups.5. Semisimple groups and proof of main theorem In this section we prove Theorem 1.8 and Proposition 1.9 and then usethem to deduce Theorems 1.1 and 1.10.As in § 4, we let w = x e n · · · x e l n l be a reduced word of length l in F d . Proof of Theorem 1.8. It is well known that Aut ( T k ) = Aut ( T ) ≀ S k . Wefix a d -tuple of cosets of T k in G : T k g , . . . , T k g d and g i = ( h i , . . . , h ik ) .σ i where h ij ∈ Aut ( T ) and σ i ∈ S k .Let w = x e n . . . x e l n l , where 1 ≤ n i ≤ d , e i = ± 1. Note that n i = n i +1 implies e i = e i +1 , since w is reduced. For i = 1 , . . . , d let s i = ( t i , . . . , t ik ) ∈ T k .Fix g ∈ G . If w ( s g , . . . , s d g d ) = g then( s n g n ) e · · · ( s n l g n l ) e l = g, so (( t n h n , . . . , t n k h n k ) σ n ) e · · · (( t n l h n l , . . . , t n l k h n l k ) σ n l ) e l = g. Moving the σ n i terms all the way to the right of the LHS we can expressthis equation in the form(5) τ (( t n h n , . . . , t n k h n k )) e · · · τ l (( t n l h n l , . . . , t n l k h n l k )) e l σ = g, where τ i , σ ∈ S k do not depend on the t ij . Note that τ i (( t n i h n i , . . . , t n i k h n i k ))is simply ( t n i τ i (1) h n i τ i (1) , . . . , t n i τ i ( k ) h n i τ i ( k ) ) . Let H ⊳ G be the kernel of the permutation action of G on the k factorsof T . Then G/H ≤ S k and T k ≤ H ≤ Aut ( T ) k .The equation (5) above has no solutions unless σ is the image of g in G/H . In the latter case it can be written as a system of k equations. Weclaim that each of these is a reduced equation of length l applied to a d -tuple of cosets of T . Indeed, if ( n i , τ i ( j )) = ( n i +1 , τ i +1 ( j )) then n i = n i +1 so e i = e i +1 . Taking all of these equations together, each variable t ij appearsat most l times.We obtain k subsets of { t ij } , each of size at most l , such that each variable t ij occurs at most l times. Therefore there exists at least m := l kl − l +1 m pair-wise disjoint subsets, corresponding to equations in disjoint sets of variables,which are clearly independent.We now apply Corollary 4.6 which implies the case k = 1 of Theorem1.8. We conclude that for some N, δ > w such that,if | T | ≥ N , then any one of the k equations discussed above holds withprobability ≤ | T | − δ . Since our system of equations contains at least m independent equations, we obtain p w,G ( g ) ≤ ( | T | − δ ) m ≤ | T k | − δ/ ( l − l +1) . Setting ǫ = δ/ ( l − l + 1) > (cid:3) The following result, which is Lemma 2.2 of [BCP], will play a key rolein the proof of Proposition 1.9 and Theorem 1.1. Lemma 5.1. Fix c ≥ . A permutation group of degree k without composi-tion factors isomorphic to A i with i > c has order at most c k − .Proof of Proposition 1.9. Fix a positive integer f . Among all chief factors G /G of G (where G i ⊳ G ) corresponding to non-abelian composition factorsof order ≥ f choose one of minimal index (namely | G : G | is minimal).Write G /G = T k for k ≥ T . Let C be thecentralizer in G of G /G and let K = G/C . Then T k ≤ K ≤ Aut ( T k ) =Aut ( T ) ≀ S k . Let H be the kernel of the permutation action of G on the k copies of T . Then G ≤ H ⊳ G and G/H ≤ S k . By the minimality of | G : G | it follows that all non-abelian composition factors of G/H haveorder < f ≤ | T | .Let c ≥ G/H does not have a composition factor A i with i > c . Then | G/H | ≤ c k − by Lemma 5.1. If A i ( i ≥ 5) is acomposition factor of G/H then 2 i < | A i | < f ≤ | T | . This shows (assuming f ≥ 64 as we may) that c ≤ log | T | (where logarithms are to the base 2). ORDS, HAUSDORFF DIMENSION AND RANDOMLY FREE GROUPS 17 It is well known that | Out ( T ) | ≤ log | T | for all finite simple groups T .We conclude that | K | ≤ | Aut ( T ) | k | G/H | ≤ | T | k | Out ( T ) | k (log | T | ) k ≤ | T k | (log | T | ) k . Now, given δ > f = f ( δ ) such that log t ≤ t δ/ for all t ≥ f . Since | T | ≥ f we obtain | K | ≤ | T k | δ . Therefore | T k | ≥ | K | − δ , completing the proof. (cid:3) Combining results 1.9 and 1.8 we obtain the following. Theorem 5.2. For every word = w ∈ F d there exist constants N, ǫ > depending only on w such that, if G is a finite group with a non-abeliancomposition factor S with | S | ≥ N , then G has a quotient K with | K | ≥ | S | such that p w,K ( g ) ≤ | K | − ǫ for all g ∈ K .Proof. Let N ( w ) , ǫ ( w ) be as in Theorem 1.8. Fix ǫ with 0 < ǫ < ǫ ( w ).Define δ = 1 − ǫ/ǫ ( w ) and let f = f ( δ ) be as in Proposition 1.9. Set N = max( f, N ( w )).Now let G, S be as in the theorem. Since | S | ≥ f , Proposition 1.9 (appliedwith | S | in the role of f ) shows that there is a quotient K of G and afinite simple group T with | T | ≥ | S | such that T k ≤ K ≤ Aut ( T k ) and | T k | ≥ | K | − δ .By Theorem 1.8 we have p w,K ( g ) ≤ | T k | − ǫ ( w ) ≤ | K | − (1 − δ ) ǫ ( w ) = | K | − ǫ for every g ∈ G , as required. (cid:3) We now prove Theorem 1.1. Proof. This follows immediately from Theorem 5.2 applied to finite quotientsof Γ with non-abelian composition factors of orders tending to infinity. (cid:3) Proof of Theorem 1.10. We first prove part (i).Theorem 1.1.2 of [B2] shows that, for certain words w , including x k ( k odd) and [ x , . . . , x d ], if G is a finite group, p w,G ( g ) ≥ ǫ > g ∈ G ,and T is a finite simple group, then the multiplicity of T as a compositionfactor of G is bounded above by some function f ( T, w, ǫ ) of T , w and ǫ only.Now, by Theorem 1.8 (applied to a suitable quotient of G ), if p w,G ( g ) ≥ ǫ > T is a non-abelian composition factor of G , then | T | ≤ f ( w, ǫ )for a suitable function f . This implies that the product of the orders of allnon-abelian composition factors of G is bounded above by Y T, | T |≤ f ( w,ǫ ) | T | f ( T,w,ǫ ) ≤ f ( w, ǫ ) , for a suitable function f .Now Soc ( G/ Rad ( G )) has the form Q i T n i i for (non-abelian) simple groups T i , hence | Soc ( G/ Rad ( G )) | ≤ f ( w, ǫ ). Since G/ Rad ( G ) is embedded inAut (Soc ( G/ Rad ( G ))) we obtain | G/ Rad ( G ) | ≤ M, where M = f ( w, ǫ ) = f ( w, ǫ ) log f ( w,ǫ ) . This proves part (i).For part (ii), note that, since the Haar measure is σ -additive, there existsan odd integer k > g of the profinitegroup G satisfying g k = 1 is positive. The required conclusion now followsfrom part (i). (cid:3) References [Ab] M. Ab´ert, Group laws and free subgroups in topological groups, Bull. LondonMath. Soc. (2005), 525–534.[BCP] L. Babai, P.J. Cameron and P.P. P´alfy, On the orders of primitive groups withrestricted nonabelian composition factors, J. Algebra (1982), 161–168.[Bo] A. Borel, On free subgroups of semisimple groups, Enseign. Math. (1983),151–164.[B1] A. Bors, Fibers of automorphic word maps and an application to compositionfactors, arXiv math:1608.00131.[B2] A. Bors, Fibers of word maps and the multiplicities of nonabelian compositionfactors, arXiv math:1703.00408.[DPSS] J.D. Dixon, L. Pyber, ´A. Seress, A. Shalev, Residual properties of free groups andprobabilistic methods, J. reine angew. Math. (Crelle’s) (2003), 159–172.[Ep] D.B.A. Epstein, Almost all subgroups of a Lie group are free, J. Algebra (1971),261–262.[SGA3] M. Demazure M. and A. Grothendieck, Sch´emas en Groupes III , Lecture Notesin Math. , Springer Verlag, Berlin, 1970.[FT] W. Feit and J.G. Thompson, Solvability of groups of odd order, Pacific J. Math. (1963), 775–1029.[Fu] W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebi-ete (3), Springer-Verlag, Berlin, 1984.[GT] R.M. Guralnick and Ph. H. Tiep, Effective results on the Waring problem forfinite simple groups, Amer. J. Math. (2015), no. 5, 1401–1430.[La] M. Larsen, Word maps have large image, Israel J. Math. (2004), 149–156.[LS1] M. Larsen and A. Shalev, Word maps and Waring type problems, J. Amer. Math.Soc. (2009), 437–466.[LS2] M. Larsen and A. Shalev, Fibers of word maps and some applications, J. Algebra (2012), 36–48.[LS3] M. Larsen and A. Shalev, Probabilistic identities and a probabilistic Tits alter-native, Algebra and Number Theory (2016), 1359–1371.[LST] M. Larsen, A. Shalev and Pham Huu Tiep, The Waring problem for finite simplegroups, Annals of Math. (2011), 1885–1950.[LiS] M.W. Liebeck and A. Shalev, Diameters of finite simple groups: sharp boundsand applications, Annals of Math. (2001), 383–406. ORDS, HAUSDORFF DIMENSION AND RANDOMLY FREE GROUPS 19 [LOST] M.W. Liebeck, E.A. O’Brien, A. Shalev, and Pham Huu Tiep, The Ore conjecture, J. Eur. Math. Soc. (2010), 939–1008.[Ma] A. Mann, Finite groups containing many involutions, Proc. Amer. Math. Soc. (1994), 383–385.[Ne] P.M. Neumann, Two combinatorial problems in group theory, Bull. London Math.Soc. (1989), 456–458.[N1] N. Nikolov, Strong approximation methods, Lectures on profinite topics in grouptheory , London Math. Soc. Stud. Texts , Cambridge Univ. Press, Cambridge2011, pp. 63–97.[N2] N. Nikolov, Verbal width in anabelian groups, Israel J. Math. , (2016), 847-876.[Pi] R. Pink, Strong approximation for Zariski dense subgroups over arbitrary globalfields, Comment. Math. Helv. (2000), 608–643.[Se] D. Segal, Words: Notes on Verbal Width in Groups , London Math. Soc. LectureNote Series , Cambridge University Press, Cambridge, 2009.[Sh] A. Shalev, Word maps, conjugacy classes, and a noncommutative Waring-typetheorem, Annals of Math. (2009), 1383–1416.[Sz] B. Szegedy, Almost all finitely generated subgroups of the Nottingham group arefree, Bull. London Math. Soc. (2005), 75–79.[We] B. Weisfeiler, Strong approximation for Zariski-dense subgroups of semisimplealgebraic groups, Annals of Math. (1984), 271–315. E-mail address : [email protected] Department of Mathematics, Indiana University, Bloomington, IN 47405,U.S.A. E-mail address : [email protected]@math.huji.ac.il