Finite axiomatizability for profinite groups
aa r X i v : . [ m a t h . G R ] J un FINITE AXIOMATIZABILITY FOR PROFINITE GROUPS
ANDRE NIES, DAN SEGAL AND KATRIN TENT
Abstract.
A group is finitely axiomatizable (FA) in a class C if it can be de-termined up to isomorphism within C by a sentence in the first-order languageof group theory. We show that profinite groups of various kinds are FA inthe class of profinite groups, as well as in the pro- p groups for some prime p .We develop both algebraic and model-theoretic method to show such results.Reasons why certain groups cannot be FA are also discussed. Contents
1. Introduction 12. Definable subgroups 63. Finite extensions 84. Bi-interpretation of groups and rings 115. Profinite groups of finite rank 216. Special linear groups 337. Some negative results 398. List of formulae 40References 401.
Introduction
Some properties of a group can be expressed by a sentence in the first-orderlanguage L gp of group theory, and some cannot. If the group is assumed to be finite , a lot more can be said about it in first-order language than in the generalcase. We mention examples of these phenomena below.The strongest property of a group G is that of ‘being isomorphic to G ’. If thiscan be expressed by a first-order sentence, G is said to be finitely axiomatizable ,henceforth abbreviated to FA.
It is obvious that every finite group is FA: if | G | = n ,the fact that G has exactly n elements and that they satisfy the multiplicationtable of G is clearly a first-order property. An infinite group cannot be FA by theL¨owenheim-Skolem Theorem ([TZ], Thm 2.3.1); to make the question interestingwe have to limit the universe of groups under consideration. For example, thefirst author in [NSG] called a finitely generated, infinite group G QFA (for quasi-finitely axiomatizable) if some first order sentence determines it up to isomorphismwithin the class of finitely generated groups. He showed that several well-knowngroups, such as the restricted wreath product C p ≀ Z , have this property (here C p denotes the cyclic group of order p ). The QFA nilpotent groups are completelycharacterized by Oger and Sabbagh in [OS]. Further results were obtained byLasserre [L]. Nies [NDG] contains a survey up to 2007. In the present paper, we address the question of relative finite axiomatizability inthe universe of profinite groups . These (unless finite) are necessarily uncountable,so cannot be finitely generated as groups; but from some points of view they behaverather like finite groups. For example, Jarden and Lubotzky show in [JL] that if G is a (topologically) finitely generated profinite group, then the elementary theory of G characterizes G up to isomorphism among all profinite groups (cf. [SW], Thm.4.2.3); in this case one says that G is quasi-axiomatizable . This is very differentfrom the situation in abstract groups: for example, a celebrated theorem of Sela [S](see also [KM]) shows that all finitely generated non-abelian free groups have thesame elementary theory.The elementary theory of G consists of all the sentences satisfied by G . Weconsider the question: which profinite groups can be characterized by a singlesentence? To make this more precise, let us say that a group G is FA (wrt L ) in C if C is a class of groups containing G , L is a language, and there is a sentence σ G of L such that for any group H in C , H | = σ G if and only if H ∼ = G. For instance, QFA means: FA (wrt L gp ) in the class of all f.g. groups. When C is aclass of profinite groups, isomorphisms are required to be topological . Usually, wewill write ‘FA’ to mean ‘FA in the class of all profinite groups.’1.1. Classes of groups and their theories.
It is often the case that a naturalclass of (abstract) groups cannot be axiomatized in the first-order language L gp of group theory. This holds for the class of simple groups (see [WFO]), the 2-generated groups, the finitely generated groups, and classes such as nilpotent orsoluble groups, none of which is closed under the formation of ultraproducts.Since finite groups are FA, every class C of finite groups can be axiomatizedwithin the finite groups: a finite group H is in C if and only if H | = ¬ σ G for everyfinite group G / ∈ C (cf. [WFO], § finitely axiomatizedwithin the finite groups is usually a much subtler question. For example, a theoremof Felgner shows that this holds for the class of non-abelian finite simple groups(see [WFO], Theorem 5.1), and Wilson [WFS] shows that the same is true for theclass of finite soluble groups. On the other hand, Cornulier and Wilson show in[CW] that nilpotency cannot be characterized by a first-order sentence in the classof finite groups.The main object of study in Nies [NSG] was the first-order separation of classesof groups C ⊂ D . Even if the classes are not axiomatizable, can we distinguishthem using first-order logic, by showing that some sentence φ holds in all groups of C but fails in some group in D ? If this holds, one says that C and D are first-orderseparated . One way to establish this is to find a witness for separation: a group G not in C that is FA in D . Then one takes φ to be the negation of a sentencedescribing G within D .Some of our results serve to provide first-order separations of interesting classesof profinite groups: • the finite rank profinite groups are first-order separated from the (topolog-ically) finitely generated profinite groups by Prop. 5.5 • similarly for pro- p groups, also by Prop. 5.5 • the f.g. profinite groups are first-order separated from the class of all profi-nite groups by Cor. 1.5. INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 3
Obstructions to finite axiomatizability.
We know of two obstructions tobeing FA for a profinite group: the centre may ‘stick out too much’, or the groupmay involve too many primes. The first is exemplified by the following result ofOger and Sabbagh, which generalizes work of Wanda Szmielew (see [HMT], ThmA.2.7) for infinite abelian groups; here Z( G ) denotes the centre of G and ∆( G ) /G ′ the torsion subgroup of G/G ′ where G ′ is the derived group: Theorem 1.1. ([OS], Theorem 2).
Let G be a group such that Z( G ) * ∆( G ) . If φ is a sentence such that G | = φ , then G × C p | = φ for almost all primes p . If for example G is a finitely generated profinite group, then G × C p ≇ G for everyprime p , so G cannot be FA.The second obstruction comes from a different direction. Let Z p denote the ringof p -adic integers. In § Proposition 1.2. (T. Scanlon, see [NB])
Let R be the ring Q p ∈ S Z p where S is aninfinite set of primes. Then R is not FA in the class of all profinite rings. Now let UT ( R ) denote the group of upper-unitriangular 3 × R .Using the method of interpretations (see below) one deduces Proposition 1.3.
For R as above, the group UT ( R ) is not FA. Our main results tend to suggest that for a wide range of profinite groups theseare the only obstructions. However, there are two caveats.
One : it is obvious that two groups that are isomorphic (as abstract groups) mustsatisfy the same first-order sentences; it is possible for non-isomorphic profinitegroups to be isomorphic as abstract groups (cf. [K1]), and such groups cannot beFA as profinite groups. In general, there is a strict hierarchy of implications for aprofinite group G : • G is FA = ⇒ G is quasi-axiomatizable = ⇒ G is ‘algebraically rigid’, the third condition meaning: any profinite group abstractly isomorphic to G is topo-logically isomorphic to G .The problem does not arise for groups that are ‘strongly complete’: this meansthat every subgroup of finite index is open . Every group homomorphism fromsuch a group to any profinite group is continuous; in fact these groups are alsoquasi-axiomatizable (see [H]). Every finitely generated profinite group is stronglycomplete (see Theorem 2.1 below). Most of the profinite groups we consider in thispaper are finitely generated (as topological groups), but not all (see Cor. 1.5). Two:
There are only countably many sentences, but uncountably many groups,even among those that avoid the above obstructions. We exhibit in § p groups parametrized by the p -adic integers.There are various ways around this problem. One may restrict attention tothe groups that have a strictly finite presentation : a profinite (or pro- p ) group G has this property if it has a finite presentation as a profinite (or pro- p ) groupin which the relators are finite group words; equivalently, if G is the completionof a finitely presented abstract group. In § L -presentation , which allows for groups like C p b ≀ Z p , the pro- p completionof the aforementioned C p ≀ Z : this is not strictly finitely presentable, but it isfinitely presented within the class of metabelian pro- p groups (cf. [Ha] for abstractmetabelian groups). An L -presentation is like a finite presentation in which theusual relations may be replaced by any sentence in the language L . ANDRE NIES, DAN SEGAL AND KATRIN TENT
Another way is to enlarge the first-order language: given a finite set of primes π ,we take L π to be the language L gp augmented with extra unary function sym-bols P λ , one for each λ ∈ Z π = Q p ∈ π Z p ; for a group element g , P λ ( g ) is interpretedas the profinite power g λ . We shall see that many pro- p groups are indeed FA (wrt L { p } ) within the class of pro- p groups.1.3. Bi-interpretation.
We shall explore two different ways showing that profinitegroups are FA. The first is a model-theoretic procedure known as bi-interpretation ,first used to show that certain groups are finitely axiomatizable by Khelif [Kh].This is defined in [P], Def. 3.1, see also [HMT], Chapter 5; further applications ofbi-intepretation are described in [NDG], § § A is interpreted in B and B is interpreted in A , we have an ‘avatar’ e A of A insome B ( n ) , and an avatar e B of B in some A ( m ) . Composing these procedures pro-duces another avatar ee A of A in A ( mn ) . If now there exists a definable isomorphismfrom A to ee A, then A is said to be bi-interpretable with B . Here we consider a veryparticular case, adapted to deal with profinite groups and profinite rings.We postpone the precise definitions to § , where the following result is estab-lished: • Let R be a profinite ring and G an algebraically rigid profinite group. If G is bi-interpretable with R and R is FA in profinite rings, then G is FA inprofinite groups. As an illustration of the method, we prove
Theorem 1.4.
Let R be a complete, unramified regular local ring with finite residuefield κ . Then each of the profinite groups Af ( R ) , SL ( R ) is FA in the class ofprofinite groups, assuming in the second case that char( κ ) is odd. Here Af ( R ) = ( R, +) ⋊ R ∗ denotes the 1-dimensional affine group over R . Thetheorem combines Theorems 4.5, 4.7 and 4.9, proved below. This result is extendedin [ST] to Chevalley groups of rank at least 2 over a more general class of rings.Although we do not pursue this aspect, it may be of interest to mention thatthe proof of Theorem 1.4 actually shows that the respective groups are uniformlybi-intepretable with the corresponding rings, i.e. the defining formulae are inde-pendent of the ring.In Theorem 1.4 the rings in question are the following: • power series rings in finitely many variables over a finite field • power series rings in finitely many variables over an unramified p -adic ring Z p [ ζ ] ( ζ a ( p f − ( R ) (for these rings R ) are finitely generated as profinitegroups (see Prop. 4.11 below), the groups Af ( R ) are not , in most cases (see theremark following the proof of Prop. 4.8); this shows that a profinite group can bevery far from finitely presented and still be FA. It also establishes Corollary 1.5.
The classes of f.g. profinite groups and all profinite groups arefirst-order separable, with witness group Af ( F p [[ t ]]) . p -adic analytic groups, and more. The other approach to establishing thatcertain groups are FA is purely group-theoretic; as such, it is limited to groups thatare ‘not very big’, in a sense about to be clarified. A pro- p group is an inverse limit INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 5 of finite p -groups, where by convention p always denotes a prime. We observedabove that ‘involving too many primes’ can be an obstruction to being FA. In factall our positive results concern groups that are virtually pro- p (that is, pro- p up tofinite index), or finite products of such groups.The pro- p groups in question are compact p - adic analytic groups . This much-studied class of groups can alternatively be characterized as the virtually pro- p groups of finite rank ; the profinite group G has finite rank r if every closed sub-group can be generated by r elements (‘generated’ will always mean: ‘generatedtopologically’). For all this, see the book [DDMS], in particular Chapter 8.The possibility of showing that (some of) these groups are FA rests on the factthat they have a finite dimension : this can be used rather like the order of a finitegroup, to control when a group has no proper quotients of the same ‘size’.Let π = { p , . . . , p k } be a finite set of primes. A C π group is one of the form G × · · · × G k where G i is a pro- p i group for each i . A C π group of finite rankneed not be strictly finitely presented, but it always has an L π presentation (seeSubection 5.3).The first main result about C π applies in particular to all p -adic analytic pro- p groups, but limits the universe: Theorem 1.6.
Every C π group of finite rank is FA (wrt L π ) in the class C π ; if ithas an L gp -presentation (e.g. if it is strictly finitely presented) then it is FA (wrt L gp ) in the class C π . This will be the key to several theorems showing that groups in certain limitedclasses of C π groups are FA among all profinite groups. The first of these is aprofinite analogue of [OS], Theorem 10. Theorem 1.7.
Let G be a nilpotent C π group, and suppose that G has an L gp -presentation. Then G is FA in the class of all profinite groups if and only if Z( G ) ⊆ ∆( G ) . (The hypothesis implies that G is f.g.; as a product of finitely many nilpotent pro- p groups, G then has finite rank.) Note that by Proposition 1.3, this would fail if π were an infinite set of primes.The Oger-Sabbagh theorem characterizing the nilpotent (abstract) groups thatare QFA has been extended to polycyclic groups by Lasserre [L]: such a group G is QFA iff Z( H ) ⊆ ∆( H ) for each subgroup H of finite index. The analogous classof pro- p groups is the soluble pro- p groups of finite rank, suggesting the Problem.
Let G be a soluble pro- p group of finite rank. Show that the followingare equivalent: a: G is FA in the class of profinite groups b: Z( H ) ⊆ ∆( H ) for each open subgroup H of G .In § C π groups of finite rank that are FA in the class of all pronilpotent groups.The final main result concerns p -adic analytic groups that are far from nilpotent: Theorem 1.8.
Let n ≥ and let p be an odd prime such that p ∤ n . Then each ofthe groups SL n ( Z p ) , SL n ( Z p ) , PSL n ( Z p ) ANDRE NIES, DAN SEGAL AND KATRIN TENT is FA in the class of profinite groups.
Here SL n ( Z p ) denotes the principal congruence subgroup modulo p in SL n ( Z p ).The proof for SL n ( Z p ) uses both Theorem 1.6 and Theorem 1.7, which can beapplied to the upper unitriangular group (when n ≥ n ( Z p )depends on Theorem 3.1, proved in §
3, which establishes some sufficient conditionsfor a finite extension of an FA group to be FA. More general results are obtainedin [ST].1.5.
Organization of the paper.
The next section introduces notation and presentssome general results about definability in profinite groups. Section 3 is devoted toshowing that under certain conditions, a finite extension of an FA group is againFA; this is useful in situations like that of Theorem 1.8, which deal with groups thatare virtually pro- p but not actually pro- p . Section 4 deals with bi-interpretabilityand applications. The material about C π -groups occupies Sections 5 and 6. Somenegative results are collected together in Section 7. The short Section 8 consists ofa list of first-order formulas for lookup.2. Definable subgroups
For a group G and a formula κ ( x ) (possibly with parameters g from G ), we write κ ( G ) = κ ( G ; g ) := { x ∈ G | G | = κ ( g, x ) } . (The notation will also be used, mutatis mutandis , for rings.) A subgroup is defin-able if it is of this form; unless otherwise stated, κ is supposed to be a formula of L gp . Note that κ ( G ) is a subgroup iff G | = s( κ ) wheres( κ ) ≡ ∃ x.κ ( x ) ∧ ∀ x, y. (cid:0) κ ( x ) ∧ κ ( y ) → κ ( x − y ) (cid:1) , and κ ( G ) is a normal subgroup iff G | = s ⊳ ( κ ) wheres ⊳ ( κ ) ≡ s( κ ) ∧ ∀ x, y. (cid:0) κ ( x ) → κ ( y − xy ) (cid:1) . We will say that a subgroup H is definably closed if H = κ ( G ) for a formula κ such that in any profinite group M , the subset κ ( M ) is necessarily closed.Suppose that H = κ ( G ) is a definable subgroup of G . By the usual rela-tivization process, for any formula ϕ ( y , . . . , y k ) there is a ‘restriction’ formulares( κ, ϕ )( y , . . . , y k ) such that for each k -tuple b ∈ H ( k ) we have G | = res( κ, ϕ )( b ) ⇐⇒ H | = ϕ ( b ) . (Note that res( κ, ϕ ) is obtained from ϕ by relativizing the quantifiers of ϕ , i.e.replacing any expression ∀ zψ ( z ) by ∀ z. ( κ ( z ) −→ ψ ( z )) , and any expression ∃ zψ ( z )by ∃ z. ( κ ( z ) ∧ ψ ( z )). Clearly, if ϕ is quantifier-free, then res( κ, ϕ ) is just ϕ . )Similarly, if N = κ ( G ) is a definable normal subgroup, there is a ‘lifted’ formulalift( κ, ϕ ) such that G | = lift( κ, ϕ )( b ) ⇐⇒ G/N | = ϕ ( e b , . . . , e b k ) , where ˜ b denotes the image of b modulo N . To obtain lift( κ, ϕ ) we replace eachatomic formula x = y in ϕ with κ ( x − y ).Suppose that κ ( G ) is a definable subgroup, and let n ∈ N . Then | G : κ ( G ) | ≤ n ⇐⇒ G | = ind( κ ; n ) , | G : κ ( G ) | = n ⇐⇒ G | = ind( κ ; n ) ∧ ¬ ind( κ ; n −
1) := ind ∗ ( κ ; n ) , INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 7 where ind( κ ; n ) ≡ ∃ u , . . . , u n . ∀ x. _ j κ ( x − u j ) . We define the frequently used formulacom( x, y ) := ( xy = yx )For a profinite group G and X ⊆ G, the closure of X is denoted X. (This is notto be confused with x , which stands for a tuple ( x , . . . , x n ) .) We write X ≤ c G, resp. X ≤ o G, for ‘ X is a closed, resp. open, subgroup of G ’.For any group G (abstract or profinite) and Y ⊆ G, the subgroup generated(algebraically) by Y is denoted h Y i . For q ∈ N , G { q } = { g q | g ∈ G } is the set of q -th powers and G q = (cid:10) G { q } (cid:11) . The derived group of G is G ′ = h [ x, y ] | x, y ∈ G i . Note that G ′ G q = G ′ G { q } . The key fact that makes f.g. profinite groups accessible to first-order logic is the definability of open subgroups.
We shall use the following without special mention:
Theorem 2.1. (Nikolov and Segal)
Let G be a f.g. profinite group. (i) Every subgroup of finite index in G is both open and definably closed (withparameters). (ii) Each term γ n ( G ) of the lower central series of G is closed and definable(without parameters). (iii) Every group homomorphism from G to a profinite group is continuous.Proof. The definability of subgroups in a profinite group is related to the topologyof the group through the concept of verbal width. A word w has width f in a group G if every product of w -values or their inverses is equal to such a product of length f . The verbal subgroup w ( G ) generated by all w -values is closed in G if and onlyif w has finite width ([SW], Prop. 4.1.2); in this case it is definable, by the formula κ w,f ( x ) which expresses that(1) x ∈ G w · . . . · G w ( f factors)where w = w ( x , . . . , x k ) has width f and G w = { w ( g ) ± | g ∈ G ( k ) } . Thisformula defines a closed subset in every profinite group, since the verbal mapping G ( k ) → G defined by w is continuous, hence has compact image.In a finitely generated profinite group, each lower-central word and all powerwords have finite width ([NS1], [NS2], [NS]). (ii) follows at once.For (i), suppose H is a subgroup of finite index in G . Then H ≥ G q = h g q | g ∈ G i for some q , and G q is definably closed by the preceding remarks, becausethe word x q has finite width. If G q ≤ N ⊳ o G then G/N is a finite d = d( G )-generator group of exponent dividing q , hence has order bounded by a finite number β ( d, q ) (by the positive solution of the Restricted Burnside Problem [Z1], [Z2]). As G q is the intersection of all such N it follows that G q is open. Now (i) follows bythe lemma below.(iii) is an easy consequence of the fact that every subgroup of finite index open. (cid:3) Lemma 2.2.
Suppose N is a definable subgroup in a group G . If N ≤ H ≤ G and | H : N | is finite then H is definable, by a formula with parameters. If N is ANDRE NIES, DAN SEGAL AND KATRIN TENT definably closed then so is H . If G = N h X i for some subset X, we may choose theparameters in X .Proof. Say N = κ ( G ) and H = N g ∪ . . . ∪ N g n . Then H is defined by W ni =1 κ ( xg − i ).The second claim is clear since the union of finitely many translates of a closed setis closed. For the final claim, we may replace each g i by a suitable word on X . (cid:3) Remark . If every subgroup of finite index in G is open, then every subgroupof finite index contains a definable open subgroup, whether or not G is f.g.: thisfollows from [WS], Theorem 2 in a similar way to the proof of (ii) above; it isimplicit in the proof of [H], Theorem 3.11.The special case of these results where G is a pro- p group is much easier, andsuffices for most of our applications; see e.g. [DDMS], Chapter 1, ex. 19 and [SW], § w ( G ) when w is a word of finite width are definable asin (1) without parameters.When proving that a certain group G is FA in some class C , we often establish astronger property, namely: for some finite (usually generating) tuple g in G , thereis a formula σ G such that for a group H in C and a tuple h in H , H | = σ G ( h ) if andonly if there is an isomorphism from G to H mapping g to h , a situation denoted by( G, g ) ∼ = ( H, h ). In this case we say that (
G, g ) is FA in C . Of course, this impliesthat G is FA in C : indeed, for H in C , we have H ∼ = G if and only if H | = ∃ x.σ G ( x ).3. Finite extensions
If a group G is FA, one would expect that (definable) subgroups of finite indexin G and finite extension groups of G should inherit this property. In this sectionwe establish the latter under some natural hypotheses.Fix a class C of profinite groups, and assume that C is closed under taking opensubgroups. L ⊇ L gp is a language. By ‘FA’ we mean FA (wrt L ) in C .Given a group N and elements h , . . . , h s ∈ N , we say that an element g of N is h -definable in N if there is a formula φ g such that for c ∈ N ,(2) N | = φ g ( h, c ) ⇐⇒ c = g. This holds in particular if g ∈ h h , . . . , h s i . Remarks (i) if θ : N → M is an isomorphism and (2) holds, then gθ is theunique element b of M such that N | = φ g ( hθ, b ) . (ii) If ( N, h ) is FA and g is h -definable in N then ( N, ( h, g )) is FA.(iii) If g is h -definable in N and N = κ ( G ) is a definable subgroup of G, then g is h -definable in G, by the formula κ ( y ) ∧ res( κ, φ g ) . Theorem 3.1.
Let N = h h , . . . , h s i ⊳ o G = h g , . . . , g r i ∈ C , and assume that ( N, h ) is FA. Then G is FA provided one of the following holds: (a) N ∩ h g , . . . , g r i = h h , . . . , h s i , in which case ( G, g ) is FA; or (b) Z( N ) = 1 , { h , . . . , h s } ⊆ h g , . . . , g r i , and h g j i is h -definable in N for each i and j . INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 9
Proof.
Say | G : N | = m . By Theorem 2.1 there is a formula κ such that N = κ ( G ; g ), and such that κ always defines a closed subset in any profinite group.Thus G satisfies Φ ( g ) := s ⊳ ( κ ( g )) ∧ ind ∗ ( κ ( g ) , m ) , which asserts that κ ( G ; g ) is a closed normal subgroup of index m (and is thereforeopen).By hypothesis, there is a formula ψ , where N | = ψ ( h ), such that if k , . . . , k s ∈ M ∈ C and M | = ψ ( k ) then there is an isomorphism N → M sending h to k . Foreach i there is a word w i such that h i = w i ( g ); then G satisfies(3) Φ ( g ) := s ^ i =1 κ ( g, w i ) ∧ res( κ ( g ) , ψ ( w , . . . , w s )) , where for aesthetic reasons w i is written in place of w i ( g ) , a convention we keepthroughout this proof.Since C is closed under taking open subgroups, Φ ( g ) implies that κ ( G ; g ) ∈ C ,and then Φ ( g ) ensures that κ ( G ; g ) ∼ = N . We setΦ := Φ ∧ Φ . To fix the isomorphism type of G, we need also to specify the conjugation actionof G on N , the quotient G/N, and the extension class. These are done in thefollowing manner. To begin with, note that G = N h g , . . . , g r i because N is open;hence there exists a transversal { t i ( g ) | i = 1 , . . . , m } to the cosets of N in G ,where each t i is a word. There is a formula τ ( g ) (depending on κ ) which assertsthat G = S mi =1 N t i ( g ).Now we deal separately with cases (a) and (b). Case (a) : For each i and j we have h g j i = v ij ( h ) for some word v ij . Thus G satisfies conj( g ) := ^ i,j (cid:2) g − j w i g j = v ij ( w , . . . , w s ) (cid:3) . For each i there exist i ∗ and a word u i such that g i = u i ( h ) t i ∗ ( g ). Then G satisfies ρ ( g ) := r ^ i =1 [ g i = u i ( w , . . . , w s ) t i ∗ ( g )]extn( g ) := ^ i,j (cid:2) t i ( g ) t j ( g ) = c ij ( w , . . . , w s ) t s ( i,j ) ( g ) (cid:3) for suitable words c ij ; here ( i, j ) s ( i, j ) describes the multiplication table of G/N, and ( i, j ) c ij ( h ) represents the 2-cocycle defining the extension of N by G/N ; this takes values in h h , . . . , h s i because of hypothesis (a).Now suppose that y , . . . , y r ∈ H ∈ C and that(4) H | = Φ( y ) ∧ τ ( y ) ∧ ρ ( y ) ∧ conj( y ) ∧ extn( y ) . Put M = κ ( H ; y ) and set k i = w i ( y ) for i = 1 , . . . , s .The fact that H | = Φ( y ) implies that each k i ∈ M and that the map sending h to k extends to an isomorphism θ : N → M . Define θ : G → H by( at i ( g )) θ = aθ · t i ( y ) ( a ∈ N, ≤ i ≤ m ) . Then τ ( y ) ensures that θ is a bijection, and using conj( y ) and extn( y ) one verifiesthat θ is a homomorphism; the key point is that conj( g ) determines the conjugationaction of each g i on N because the h j generate N topologically and inner automor-phisms are continuous, and similarly conj( y ) determines the action of each y i on m . This implies that for b ∈ N and each j , t j ( y ) − · bθ · t j ( y ) = (cid:0) t j ( g ) − · b · t j ( g ) (cid:1) θ . Finally, ρ ( y ) implies that g i θ = y i for each i .Thus (4) implies that there is an isomorphism G → H sending g to y . Case (b) : Assume now that Z( N ) = 1 . Given a group N with trivial centre, agroup Q , and a homomorphism γ : Q → Out( N ), there is (up to equivalence) atmost one extension group G of N by Q such that conjugation in G induces themapping γ : Q → Out( N ) ([G], § N, G/N and the action.We fix the multiplication table of
G/N withquot( g ) := ^ i,j κ ( g, t i t j t s ( i,j ) − )(writing t i in place of t i ( g ) throughout). We redefine ρ as follows: ρ ( g ) := r ^ i =1 κ (cid:0) g, t i ∗ g − i (cid:1) . where i ∗ is defined above. To fix the action, we now setconj( g ) := res κ ( g ) , ^ i,j φ v ( i,j ) ( w , . . . , w s , g − j w i g j ) where v ( i, j ) = h g j i , and φ v ( i,j ) defines h g j i in N in terms of h .Now suppose that y , . . . , y r ∈ H ∈ C and that H | = Φ( y ) ∧ τ ( y ) ∧ ρ ( y ) ∧ quot( y ) ∧ conj( y ) . Put M = κ ( H ; y ) and set k i = w i ( y ) for i = 1 , . . . , s . As before we have anisomorphism θ : N → M sending h to k . The map sending t i ( g ) to t i ( y ) for each i induces an isomorphism θ : G/N → H/M.
Thus we have a diagram of groupextensions: 1 → N → G → G/N → ↓ ↓ → N α → H β → G/N → α : N → M ֒ → H and β : H ։ H/M → G/N are induced respectivelyby θ : N → M and θ − : H/M → G/N , and the vertical arrows representidentity maps. Now ρ ( g ) and ρ ( y ) ensure that ( N g i ) θ = M y i for each i . Thenusing conj( g ) and conj( y ) together with Remark (i), we can verify that the twomappings G/N → Out( N ) induced by the top extension and the bottom extensionare identical. Hence there exists a homomorphism θ : G → H making the diagramcommute, and then θ must be an isomorphism since the end maps are bijective.Finally, because G is finitely generated, Theorem 2.1 (iii) ensures that any groupisomorphism G → H is a topological isomorphism. (cid:3) INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 11
Remark
This argument gives the same result for a class C of abstract groups,if we add the hypothesis that N (has finite index and) is definable in G .4. Bi-interpretation of groups and rings
Properly speaking, bi-interpretability should be an equivalence relation betweenstructures. For the sake of simplicity, we will define here a messier, asymmetricrelation, specifically tailored to one purpose: showing that for certain groups builtout of rings, finite axiomatizability of the ring implies that of the group. The preciseresult is Theorem 4.3 in § § Interpreting rings in groups.
All rings are commutative, with identity. L rg is the first-order language of rings. A ring is profinite if it is an inverse limit offinite rings. We also need a slightly weaker version: the ring R is additively profinite if its additive group ( R, +) is profinite as a group.A profinite ring R is FA (resp. strongly FA ) if there is a formula σ of L rg suchthat (i) R | = σ and (ii) if S is a profinite (resp. additively profinite) ring and S | = σ then S is topologically isomorphic to R .Let R be an additively profinite ring. We say that R has an interpretation bya definably closed subgroup in a profinite group G if there are formulae τ, µ and atuple of parameters g in G with the following property: • for every profinite group H and tuple h from H , the set τ ( H ; h ) is a closedsubgroup of H ; • τ ( G ; g ) becomes a ring b τ ( G ) = b τ ( G ; g ) with ring addition given by thegroup operation, and ring multiplication defined by res( µ ( g ) , τ ( g )), in thesense that for x, y, z ∈ τ ( G ; g ) ,x · y = z ⇐⇒ G | = µ ( g, x, y, z ) . • b τ ( G ; g ) is topologically isomorphic to R. (Here b τ stands for ( τ, µ ) , and we will write b τ ( G ) for b τ ( G ; g ) when there is norisk of confusion.)In this situation, there is a formula ρ (depending on τ and µ ) such that (i) G | = ρ ( g ) and (ii) for any profinite group H and tuple h from H , if H | = ρ ( h ) then thesubgroup τ ( H ; h ) is a ring S := b τ ( H ; h ) with operations defined as above. (Theformula ρ expresses the statements that µ defines a binary operation on S and thatthe axioms for a commutative ring with identity are satisfied). This ring S will beadditively profinite, because τ ( H ; h ) is a profinite group.We call such an interpretation strongly topological if it has the following addi-tional property: for any profinite group H and tuple h from H , if H | = ρ ( h ) thenthe ring S = b τ ( H ; h ) is actually a profinite ring: that is, the multiplication mapfrom S × S to S is continuous.For each formula φ of L rg there is a formula φ ∗ of L gp such that b τ ( H, h ) | = φ ⇐⇒ H | = φ ∗ ( h ) , obtained in the obvious way by translating each atomic L rg subformula of φ intoan equivalent L gp formula. Lemma 4.1.
Suppose that the profinite ring R has an interpretation by a definablyclosed subgroup in a profinite group G, and that either R is strongly FA, or theinterpretation is strongly topological. Then there is an L gp formula ψ ( y ) such that G | = ψ ( g ) (where g is as above), and for each profinite group H and tuple h , if H | = ψ ( h ) then b τ ( H ; h ) is a ring topologically isomorphic to R . Indeed, it suffices to set ψ ( y ) = ρ ( y ) ∧ σ ∗ R ( y ). Remark 4.2.
The ring R has the property ‘2 is not a zero divisor’ if and only if b τ ( G ) satisfies a certain formula φ ( g ). In this case, we make the convention that ρ implies φ ∗ . If H as above now satisfies ρ ( h ) then 2 is not a zero divisor in the ring S = b τ ( H ), and then in S the identity(5) 2 xy = ( x + y ) − x − y determines xy . Since addition is continuous, by definition of the topology on S , toestablish continuity of multiplication it will suffice to show that the map x x on S is continuous.Thus if 2 is not a zero divisor in R , for the interpretation be topological it sufficesto have: whenever H as above satisfies ρ ( h ), the squaring map from S = b τ ( H ) to S is continuous.4.2. Interpreting groups in rings.
Let G be a profinite group. We say that G is interpreted in a profinite ring R if, for some d , there are L rg formulae α , α suchthat • for every profinite ring T , the subset α ( T ( d ) ) is closed in T ( d ) ; • α ( R ( d ) ) is a group b α ( R ), with operation defined by a · b = c ⇐⇒ R | = α ( a, b, c ); • G is topologically isomorphic to b α ( R ) (with the subspace topology inducedby α ( R ) ⊆ R ( d ) ).As in the preceding subsection, there is a formula α (depending on α , α ) suchthat (i) R | = α and (ii) for any profinite ring T , if T | = α then α ( T ( d ) ) is a group b α ( T ) with the operation defined as above.For example, if G ≤ SL n is an algebraic group defined over Z , then G ( R ) isinterpreted in R for any ring R ; here d = n , α expresses the defining equationsof G , and α is the formula for matrix multiplication.While first-order language may suffice to determine the algebraic structure ofa group, it cannot say anything about the topology. Recall that the profinitegroup G is algebraically rigid if every profinite group abstractly isomorphic to G istopologically isomorphic to G . This holds in particular if G is strongly complete(i.e. every subgroup of finite index is open), but the conditions are not equivalent;in § Theorem 4.3.
Let G be an algebraically rigid profinite group, interpreted (by α )in a profinite ring R . Suppose that (a) R has an interpretation by a definably closed subgroup τ ( G ; g ) in G , and INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 13 (b) there exists a group isomorphism θ : G → b α ( b τ ( G ; g )) that is definable, in the sense that the (1 + d ) -ary relation R θ = { ( u, v ) ∈ G × G d | uθ = v } is definable (with the parameters g ) in G .If (i) R is FA and the interpretation of R in G is strongly topological, or (ii) R is strongly FA, then G is FA in profinite groups.Proof. Condition (a) says that b α ( b τ ( G ; g )) ∼ = R as topological rings, and Lemma 4.1provides a certain formula ψ ( y ) such that G | = ψ ( g ). The statement that θ is anisomorphism from G onto b α ( b τ ( G )) can be expressed by a certain L gp formula Θ( g ),depending in a straightforward way on R θ , α and τ . Then G | = Σ G ( g ) whereΣ G ( y ) ≡ ψ ( y ) ∧ Θ( y ) . Now let H be a profinite group and suppose that H | = Σ G ( h ) for some tuple h in H . As H | = ψ ( h ), the ring S = b τ ( H ; h ) is topologically isomorphic to R . In par-ticular, S is a profinite ring and S | = α , so b α ( S ) is a group with operation definedby α . As H | = Θ( h ), the formula defining θ establishes a group isomorphism θ ′ : H → b α ( S ) ∼ = b α ( R ) ∼ = b α ( b τ ( G ; g )) . Then θθ ′− is a group isomorphism G → H . As H is a profinite group and G isalgebraically rigid, the groups are topologically isomorphic.Thus ∃ y. Σ G ( y ) determines G as a profinite group. (cid:3) Remark.
For G to be bi-interpetable with R in the usual sense, one needsalso to establish an L rg -definable isomorphism from R to b τ ( b α ( R )). In each of theexamples discussed below such an isomorphism is easy to discern: b α ( R ) will be amatrix group and b τ ( b α ( R )) a certain one-parameter subgroup.4.3. Some profinite rings.
Familiar examples of profinite rings are the completelocal rings with finite residue field: if R is one of these, with (finitely generated)maximal ideal m and finite residue field R/ m ∼ = F q , then R is the inverse limit ofthe finite rings R/ m n ( n ∈ N ). We will keep this notation throughout this section,and set p = char( R/ m ), q = p f .The fundamental structure theorem of I. S. Cohen describes most of these ringsquite explicitly. R is said to be regular if m can be generated by d elements where d = dim R is the Krull dimension of R . Also R is said to be unramified if either pR = 0 or p · R / ∈ m . (For background on regular local rings, see e.g. [E], § o q = Z p [ ζ q − ]where ζ q − is a primitive ( q − f over Q p . According to [ISC], Th. 11,Cor. 2 this is the only complete local domain R of characteristic 0 with maximalideal pR and residue field F q . Theorem 4.4. ([ISC], Theorem 15)
Let R be a regular, unramified complete localring with residue field F q and dimension d ≥ . Then one of the following holds: (a) R ∼ = F q [[ t , . . . , t d ]](b) R ∼ = o q [[ t , . . . , t d − ]] . The point is that R is determined up to isomorphism by its characteristic andthe parameters d, q ; it is then hardly surprising that such a ring is FA. (The samevery likely holds also in the ramified case, when R is an ‘Eisenstein extension’ of thering specified in (b); we shall not go into this here, but it can probably be coveredby suitably extending the arguments below.) Theorem 4.5.
Let R be a regular, unramified complete local ring with finite residuefield . Then R is FA.Proof. Until further notice S denotes an arbitrary ring. Each of the statements ‘ S is an integral domain’, ‘char S = 0’, ‘char S = p ’ is easily expressible by a sentenceof L rg . There are formulae µ, ϕ q and ρ such that • S | = µ ( a , . . . , a d ) if and only if S \ P d a i S consists of units; • S | = ϕ q ( a , . . . , a d ) if and only if (cid:12)(cid:12)(cid:12) S/ P d a i S (cid:12)(cid:12)(cid:12) = q ; • S | = ρ ( a , . . . , a d ) if and only if for each i , the element a i is not a zerodivisor modulo P i − j =1 a j S (the zero ideal when i = 1).Put Σ q ( x ) := µ ( x ) ∧ ϕ q ( x ) ∧ ρ ( x ) ∧ ∀ y, z. ( yz = 0 −→ ( y = 0 ∨ z = 0)) . Now suppose that S satisfies Σ q ( a , . . . , a d ), and set I = P d a i S. Then S is a localdomain with maximal ideal I and residue field S/I ∼ = F q . The sentence ρ ( a , . . . , a d )implies that dim S is at least d , and hence that dim S = d ([ISC], Theorem 14), so S is regular.Now we separate cases. Case (a) : R ∼ = F q [[ t , . . . , t d ]]. Then R satisfiesΣ q,p ( t , . . . , t d ) ≡ Σ q ( t , . . . , t d ) ∧ ∀ y. ( py = 0) . Suppose that S is a profinite ring and that S | = Σ q,p ( s , . . . , s d ) for some s , . . . , s d ∈ S . Put I = P d s i S . Then S is a regular, unramified local domain of dimension d and characteristic p , with maximal ideal I and residue field F q . Case (b) : R ∼ = o q [[ t , . . . , t d − ]] . Note that 1 R is a definable element, by theformula ∀ y. ( xy = y ). Now m = pR + P d − t i R . The fact that p R / ∈ m isexpressible by τ p ( t , . . . , t d − ) := ∀ z ij , y i , x. X i,j t i t j z ij + X i pt i y i + p x = p R . Thus R satisfiesΣ q, ( t , . . . , t d − ) ≡ Σ q ( p R , t , . . . , t d − ) ∧ ∃ y. ( py = 0) ∧ τ p ( t , . . . , t d − ) . Suppose now that S is profinite ring and that S | = Σ q, ( s , . . . , s d − ) for some s , . . . , s d − ∈ S . Put I = pS + P d − s i S . Then again, S is a regular, unramifiedlocal domain of dimension d , with maximal ideal I and residue field F q , and S hascharacteristic 0. INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 15
Conclusion.
Since ring multiplication is continuous, I is compact and thereforeclosed in S ; as it has finite index, I is open. The same argument shows that I n isopen for each n (each I n − /I n is finite because it is finitely generated as a modulefor S/I ). Now S is the inverse limit of finite rings S/J α , where { J a } is a familyof open ideals that form a base for the neighbourhoods of 0. For each α the ring S/J α is finite with Jacobson radical I/J α , so for some n we have I n ⊆ J α . Hencethe system { I n | n ∈ N } is also a base for the neighbourhoods of 0. Thus the givenprofinite topology is the I -adic topology, and as S is complete for the former it iscomplete as a local ring.Now Theorem 4.5 shows that S ∼ = F q [[ t , . . . , t d ]] (Case a) or S ∼ = o q [[ t , . . . , t d − ]](Case b). (cid:3) Remarks (i) If R is one-dimensional and of characteristic 0 , i.e. R ∼ = o q forsome q , then in fact R is strongly FA. Indeed, in Case (b) above we merely need toassume that S is additively profinite, for S satisfies Σ q, with d = 1 , which assertsthat I = pS. As multiplication by p is continuous on the profinite additive group( S, +), the given argument shows that the powers of I define the topology, whichagain implies that S is complete.(ii) We used the well-known fact that in a finite (more generally, Artinian) ring,the Jacobson radical is nilpotent. It is worth stating an immediate consequence. Lemma 4.6.
Let R be a complete local ring with finite residue field. Then everyideal of finite index in R is open. In other words, rings of this kind are ‘strongly complete’ . Some worked examples.
Now we are ready to prove Theorem 1.4: if R is a complete, unramified regular local ring with finite residue field then each ofthe profinite groups Af ( R ), SL ( R ) is FA in the class of profinite groups, assumingin the second case that the residue characteristic is odd. This will follow from Theorem 4.5 once we show that the hypotheses of Theorem4.3 are satisfied.Let R be a complete local domain, with maximal ideal m and finite residue field R/ m . Then R is a profinite ring, a base for the neighbourhoods of 0 being thepowers of m . In particular, R ∗ = R \ m is an open subset, being the union offinitely many additive cosets of m .The semi-direct product ( R, +) ⋊ R ∗ can be identified with the affine groupAf ( R ) = (cid:18) R R ∗ (cid:19) < GL ( R ) . Theorem 4.7. If R is FA then the group Af ( R ) is FA in profinite groups.Proof. We will verify the hypotheses of Theorem 4.3. Write G = Af ( R ) . Definethe following elements of G, where 1 = 1 R and λ ∈ R, ξ ∈ R ∗ : u ( λ ) = (cid:18) λ (cid:19) , h ( ξ ) = (cid:18) ξ (cid:19) , and fix the parameters u := u (1) , h := h ( r ) for some r ∈ R ∗ , r = 1. Then G = H · U where U := u ( R ) = C G ( u ) H := h ( R ∗ ) = C G ( h ) are both definable subgroups. For technical reasons, we want to encode the factthat H is abelian and normalizes U ; to this end, we define κ ( x, u, h ) ≡ ∀ y. ([com( y, h ) → com( y, x )] ∧ [com( y, u ) ←→ com( y x , u )]) ;and note that H = κ ( G ; u, h ).We will frequently use the identity u ( λ ) h ( ξ ) = u ( ξλ ) . All formulae are supposed to involve the parameters u, h , which we will sometimesomit for brevity.
Claim 1.
The ring R has a strongly topological interpretation by U = com( G ; u ).Certainly com( − , u ) defines a closed subgroup in any profinite group, as it definesa centralizer. The map u : R → U is a topological isomorphism from ( R, +) to U .It becomes a topological ring isomorphism if one defines u ( α ) · u ( β ) = u ( αβ ) . We need to provide an L gp formula µ such that for x, y, z ∈ U , x · y = z ⇐⇒ G | = µ ( u, h ; x, y, z ) . Let v = u ( β ) ∈ U. If β ∈ R ∗ then v = u h ( β ) , while if β ∈ m then β + 1 ∈ R ∗ and v = [ u, h ( β + 1)]. Thus v · v = v if and only if there exist x, y ∈ H such that oneof the following holds: v = u x , v = u y , v = u xy , or v = u x , v = [ u, y ] , v = [ u x , y ] , or v = [ u, x ] , v = u y , v = [ u y , x ] , or v = [ u, x ] , v = [ u, y ] , v = [[ u, y ] , x ] . This can be expressed by a first-order formula since H is definable.To say that the interpretation is strongly topological means the following: if aprofinite group e G satisfies the appropriate sentence ρ with parameters e u, e h , whichin particular implies that µ ( e u, e h ; − ) defines a binary operation · on e U = com( e G ; e u ),this operation is continuous.We will write u in place of e u for aesthetic reasons.Let N be an open normal subgroup of e G . If u x ≡ u x ′ (mod N ) and u y ≡ u y ′ (mod N ) with x, x ′ , y, y ′ ∈ e H, then u xy ≡ u x ′ y = u yx ′ ≡ u y ′ x ′ = u x ′ y ′ (mod N )since e H is abelian. Similar congruences hold if u x is replaced by [ u, x ] or u y isreplaced by [ u, y ] . Thus in all cases we see that if v i ≡ v ′ i (mod N ) for i = 1 , v · v = v , v ′ · v ′ = v ′ , then v ≡ v ′ (mod N ). Thus the operation · is continuousas required. Claim 2
There exists a definable map θ : G → Af ( U ) such that θ is a groupisomorphism when U is endowed with the ring operations defined in Claim 1 . INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 17
Let g = (cid:18) λ ξ (cid:19) ∈ G . Then g = e h ( g ) e u ( g ) where e u ( g ) = u ( λ ) and e h ( g ) = h ( ξ ) . Also { e u ( g ) } = Hg ∩ U { e h ( g ) } = gU ∩ H. As H and U are definable subsets of G , the mappings e u : G → G and e h : G → G are both definable. Hence so is the mapping θ : G → Af ( U ) given by gθ = (cid:18) e u ( g )0 u e h ( g ) (cid:19) . The mapping from M ( U ) to M ( R ) induced by u ( λ ) λ is a ring isomorphism(when U is given the ring structure from Claim 1 ), hence restricts to a groupisomorphism ϕ : Af ( U ) → Af ( R ) = G . Now for g as above, gθϕ = (cid:18) u ( λ )0 u h ( ξ ) (cid:19) ϕ = (cid:18) u ( λ )0 u ( ξ ) (cid:19) ϕ = (cid:18) λ ξ (cid:19) = g. It follows that θ = ϕ − is a group isomorphism. Claim 3. G is algebraically rigid . This follows from the stronger result Proposition4.8, below, and completes the proof of Theorem 4.7. (cid:3) Proposition 4.8.
Let R be a complete local domain with finite residue field. Thenevery group isomorphism from Af ( R ) to a profinite group is continuous (and there-fore a topological isomorphism).Proof. G = Af ( R ) = U ⋊ H where U = u ( R ) and H = h ( R ∗ ) (notation as above).A base for the neigbourhoods of 1 in G is the family of subgroups G ( n ) := H ( n ) U ( n ) , n ≥ , where U ( n ) = u ( m n ) , H ( n ) = h (1 + m n ) . Let θ : G → e G be a group isomorphism, where e G is a profinite group. Set e U = U θ and e H = Hθ . As R is an integral domain, H = C G ( H ) , and so e H = C e G ( e H ) is closedin e G . Similarly, e U = C e G ( e U ) is closed. We will show that θ − is continuous (which,for an isomorphism of profinite groups, is equivalent to being a homeomorphism).Suppose that e N ⊳ o e G . Then N := e N θ − is a normal subgroup of finite index in G , so N ∩ U = u ( B ) for some additive subgroup B of finite index in R . If r ∈ R ∗ then u ( Br ) = u ( B ) h ( r ) = u ( B )so B = Br , and as R = R ∗ ∪ ( R ∗ −
1) it follows that B is an ideal of R ; therefore B ⊇ m n for some n , by Lemma 4.6. Thus U ( n ) ⊆ e N θ − .It follows that θ | U : U → e U is a continuous isomorphism, and consequently ahomeomorphism. This in turn implies that each e U ( n ) := U ( n ) θ is open in e U .Now for each n , H ( n ) θ = C H ( U/U ( n )) θ = C e H ( e U / e U ( n ))is open in e H since e U ( n ) is open in e U (here C H ( U/U ( n )) denotes the kernel of theconjugation action of H on the factor U/U ( n ) ). Thus G ( n ) θ = H ( n ) θ.U ( n ) θ is closed in e G, hence open as it has finite index. It follows that θ − is continuous. (cid:3) Remark.
It is not usually the case that G = Af ( R ) is strongly complete. Infact, G is strongly complete if and only if its open subgroup G (1) (the principalcongruence subgroup mod m ) is, and as G (1) is a pro- p group this holds if and onlyif G (1) is finitely generated (e.g. by [WS], Theorem 2: a pro- p group that is not f.g.maps onto an infinite elementary abelian p -group, and so has uncountably manysubgroups of index p ). This in turn holds if and only if the multiplicative subgroup T := 1 + m of R ∗ is finitely generated as a pro- p group. Now T is finitely generatediff T /T p is finite; of the rings listed in Theorem 4.4, only the ones denoted o q havethis property.For the next example, let R be as above, and assume that q = | R/ m | is odd .Every element of R/ m is then a difference of two squares, and as R is completeand q is odd it follows that every element of R is of the form x or x − y with x,y ∈ R ∗ . Fix r ∈ R such that r + m is a generator for ( R/ m ) ∗ and r = 1. Theorem 4.9. If R is FA then the group SL ( R ) is FA in profinite groups.Proof. Put G = SL ( R ) . We will show that all hypotheses of Theorem 4.3 aresatisfied, with a strongly topological interpretation of R in G .Define the following elements of G, where 1 = 1 R and λ ∈ R : u ( λ ) = (cid:18) λ (cid:19) , v ( λ ) = (cid:18) − λ (cid:19) ,h ( λ ) = (cid:18) λ − λ (cid:19) ( λ ∈ R ∗ ) , (6) w = (cid:18) − (cid:19) We take u := u (1) , u ′ := u ( r ) , v := v (1), w and h := h ( r ) as parameters and write u = ( u, u ′ , v, w, h ) . ‘Definable’ will mean definable with these parameters. All formulae below aresupposed to include these parameters, which we mostly omit for brevity. We willuse without special mention the identities u ( λ ) w = v ( λ ) u ( λ ) h ( µ ) = u ( λµ ) . Write U = u ( R ) , V = v ( R ) , H = h ( R ∗ ). Write ± U = h± i · U , etc. Then ± U = C G ( u ) , ± V = C G ( v ) are definable subgroups of G .To show that U is definable, observe first that no element of − U is conjugateto an element of U . On the other hand, we shall see below that each element of U takes one of the forms u x or u x u − y ( x, y ∈ H ). Thus U = ρ ( G ) where ρ ( s ) := com( s, u ) ∧ ∃ x, y. ( s = u x ∨ s = u x u − y ) . Note that ρ will define a closed subset in any profinite group, since centralizersand conjugacy classes are closed. Using Lemma 4.10 below, we now adjust ρ to anew formula ρ ∗ , such that U = ρ ∗ ( G ) , and for any profinite group A the subset ρ ∗ ( A ) is a closed subgroup. INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 19
Then V = U w is also definable, as is the subgroup H = C G ( h ); for technicalreasons, we want to encode the fact that H is abelian and normalizes U ; thus H = η ( G ) where η ( u, x ) ≡ ∀ y. (com( h, y ) → com( x, y ) ∧ ρ ∗ ( y ) ←→ ρ ∗ ( y x )) . Claim 1.
The ring R has a strongly topological interpretation by U = ρ ∗ ( G ).We adapt the method used in [KRT], proof of Theorem 3.2. The map u : R → U is a topological isomorphism from ( R, +) to U . It becomes a topological ringisomorphism if one defines(7) u ( α ) · u ( β ) = u ( αβ ) . Now we need to provide an L gp formula µ such that for v , v , v ∈ U , v · v = v ⇐⇒ G | = µ ( v , v , v ) . If α = ξ (resp. ξ − η ) with ξ, η ∈ R ∗ , then u ( α ) = u x (resp. u x u − y ) where x = h ( ξ ) , y = h ( η ), and u ( α ) = u x , resp. u x u − xy u y . Let sq( v , v ) be the formula asserting that there exist x, y in H such that h v = u x ∧ v = u x i ∨ h v = u − y u x ∧ v = u x u − xy u y i . One verifies easily that this holds if and only if v = v · v in the sense of (7). Nowin view of (5) we can take µ ( v , v , v ) to assert that there exist a and b such thatsq( v , a ) ∧ sq( v , b ) ∧ sq( v v , av b ) . To complete the proof of Claim 1, it remains to establish that the interpretationis strongly topological. In view of Remark 4.2, it will suffice to show that sq( v , v )defines a continuous map – not just on U but on any profinite group arising as ρ ∗ ( G ∗ ; u ∗ ) where G ∗ satisfies the appropriate sentences (which include one assertingthat sq( v , v ) does define a mapping). For simplicity (‘by abuse of notation’) wekeep the notation attached to G , but will not use any special properties of G . Thefact that H = η ( G ; u ) is abelian and normalizes U is now implied by the definitionof η. If N is an open normal subgroup of G, u ∈ U and x, y, x ′ , y ′ ∈ H then u x ≡ u x ′ (mod N ) implies u x ≡ u x ′ x = u xx ′ ≡ u x ′ (mod N ) . Similarly, u − y u x ≡ u − y ′ u x ′ (mod N ) implies u x u − xy u y = ( u − y u x ) x ( u − x u y ) y ≡ ( u − y ′ u x ′ ) x ( u − x ′ u y ′ ) y = ( u x u − y ) x ′ ( u − x u y ) y ′ ≡ ( u x ′ u − y ′ ) x ′ ( u − x ′ u y ′ ) y ′ = u x ′ u − x ′ y ′ u y ′ . Given that sq( v , v ) defines a map, it follows that if v ≡ v ′ (mod N ) and sq( v , v )and sq( v ′ , v ′ ) hold then v ≡ v ′ (mod N ); this map is therefore continuous asrequired.Claim 1 is now established. Claim 2
There exists a definable map θ : G → SL ( U ) such that θ is a groupisomorphism when U is endowed with the ring operations defined in Claim 1 . To begin with, we partition G as G · ∪ G where G = { g ∈ G | g ∈ R ∗ } G = { g ∈ G | g ∈ m } . If g = (cid:18) a bc d (cid:19) ∈ G then g = e v ( g ) e h ( g ) e u ( g ) where e v ( g ) = v ( − a − c ) ∈ V e h ( g ) = h ( a − ) ∈ H e u ( g ) = u ( a − b ) ∈ U. This calculation shows that in fact G = V HU , so G is definable; these threefunctions on G are definable since x = e v ( g ) ⇐⇒ x ∈ V ∩ HU gy = e u ( g ) ⇐⇒ y ∈ U ∩ HV gz = e h ( g ) ⇐⇒ z ∈ H ∩ V gU. If g ∈ G then gw ∈ G since a and b cannot both lie in m , and then g = e v ( gw ) e h ( gw ) e u ( gw ) w − .Define α : V → U by xα = x w . Then v ( λ ) α = u ( λ ).Define β : H → U × U as follows. Let ( x, y , y ) ∈ H × U × U . Then xβ = ( y , y )if and only if ∃ t ∈ H. (cid:0)(cid:0) x = t ∧ y t = u ∧ y = u t (cid:1) ∨ (cid:0) x = t h ∧ y th = u ′ ∧ y = u ′ t (cid:1)(cid:1) . This decodes as h ( λ ) β = ( u ( λ − ) , u ( λ )) . Now we construe M ( U ) as a ring by transferring the ring operations from R to U via Claim 1. Define θ : G → M ( U ) as follows: if g ∈ G then(8) gθ = (cid:18) − e v ( g ) α (cid:19) · e h ( g ) β e h ( g ) β ! · (cid:18) e u ( g )0 1 (cid:19) ;if g ∈ G then gθ = ( gw ) θ · w − . The mapping u ( λ ) λ induces an isomorphism ϕ : SL ( U ) → SL ( R ), and we seethat θϕ is the identity map on G , so θ is an isomorphism as required.This completes the proof of Claim 2. Claim 3 . G is algebraically rigid because it is strongly complete. This is presum-ably well known, but we include a proof in Proposition 4.11 below.The theorem follows. (cid:3) Lemma 4.10.
Let G be a group and ρ a formula such that ρ ( G ) is a subgroup.Then there is a formula ρ ∗ such that ρ ∗ ( G ) = ρ ( G ) , and for any group H the subset ρ ∗ ( H ) is a subgroup. If ρ defines a closed subset in every profinite group then sodoes ρ ∗ .Proof. Put ρ ( x ) = ρ ( x ) ∨ ( x = 1) , and set ρ ∗ ( x ) := ρ ( x ) ∧ ∀ y (cid:0) ρ ( y ) −→ ( ρ ( xy ) ∧ ρ ( x − y )) (cid:1) . One sees easily that this has the required properties. (cid:3)
INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 21
Proposition 4.11.
Let R be a complete local domain, with maximal ideal m andfinite residue field R/ m of odd characteristic. Then the profinite group SL ( R ) isfinitely generated, hence strongly complete.Proof. Let G (1) denote the principal congruence subgroup modulo m in G =SL ( R ). Then G (1) is a so-called R -perfect group : that is, an R -analytic pro- p group whose associated Lie algebra is perfect (see [DDMS], Exercise 13.10 onpage 352). Now Corollary 3.4 of [LS] asserts that such pro- p groups are finitelygenerated (cf. [DDMS] , Proposition 13.29(i)), hence strongly complete ([DDMS],Theorem 1.17).As G (1) is open in G it follows that G has both properties. (cid:3) Remarks. (i) The Lubotzky-Shalev results [LS] hold for SL n ( R ), any n ≥ R/ m ) = 2 provided n ≥ congruence subgroup property : every subgroup of finite index con-tains a principal congruence subgroup modulo m k for some k (this follows from [K],Satz 2). 5. Profinite groups of finite rank
In the context of profinite groups, a ‘generating set’ will always mean a topological generating set. From a first-order point of view, the nicest profinite groups are thefinitely generated pro- p groups, where p is a prime. These have the following specialproperty: a finite generating set can be recognized in a definable finite quotient (aswe shall see below). So for such groups, being generated by d elements is a first-order property. This is the key to most of our main results; it does not hold forf.g. profinite groups in general as we point out in Prop. 5.4, but it does hold forgroups in the larger class C π of pronilpotent pro- π groups , where π is any finite setof primes: a group G is in C π iff it is a direct product G = G p × · · · × G p k where π = { p , . . . , p k } and G p i is a pro- p i group for each i . To save repetition, wemake the convention that π will always denote a finite set of primes .5.1. Some preliminaries, and an example.
For basic facts about profinitegroups, see [DDMS], Chapter 1 and the earlier chapters of [WP]. Besides thelanguage L gp of group theory, we will consider the language • L π : the language L gp augmented by unary function symbols P λ , one foreach λ ∈ Z π = Q p ∈ π Z p ; for g ∈ G , P λ ( g ) is interpreted as the profinitepower g λ .For a profinite group G, • d( G ) is the minimal size of a (topological) generating set for G • rk( G ) = sup { d( H ) | H ≤ c G } This is the rank of G (sometimes called Pr¨ufer rank ). The pro- p groups offinite rank are of particular interest, being just those that are p -adic analytic ; see[DDMS], Chapter 3 (which includes several equivalent definitions of rank). On theface of it, having a particular finite rank is not a first-order property (the definitioninvolves quantifying over subgroups); the following result shows that the rank, iffinite, can be more or less specified by a first-order sentence: Proposition 5.1.
For each positive integer r , there is a sentence ρ r such that fora pro- p group G , rk( G ) ≤ r = ⇒ G | = ρ r = ⇒ rk( G ) ≤ r (2 + log ( r )) . We omit the proof, an application of the techniques described below.We fix the notation q ( π ) = p . . . p k q ′ ( π ) = 2 ε q ( π )where ε = 0 if 2 / ∈ π, ε = 1 if 2 ∈ π. A sharper version of Theorem 2.1(ii) holds in some cases:
Lemma 5.2.
Let G = h a , . . . , a d i be a pronilpotent group. Then (9) G ′ = [ a , G ] . . . [ a d , G ] , a closed subgroup of G .Proof. The set X = [ a , G ] . . . [ a d , G ] is closed in G, so X = T N ⊳ o G XN . If N ⊳ o G then G/N is nilpotent and generated by { a , . . . , a d } , which implies XN/N = G ′ N/N (cf. [DDMS, Lemma 1.23]), so G ′ ≤ XN . Hence G ′ ⊆ X. (cid:3) It is easy to see that (9) can be expressed by a first-order formula. If in a group G every product of d + 1 commutators belongs to the set X defined above, then (byan obvious induction) every product of commutators belongs to X . Hence there isa formula α such that G | = α ( a , . . . , a d ) ⇐⇒ G ′ = [ a , G ] . . . [ a d , G ] . The
Frattini subgroup Φ( G ) of a profinite group G is the intersection of allmaximal open subgroups of G . It follows from the definition that for Y ⊆ G ,(10) h Y i = G ⇐⇒ h Y i Φ( G ) = G. If G is pronilpotent then every maximal open subgroup is normal of prime index;it follows that if G ∈ C π then Φ( G ) = G ′ G q ( π ) . If also G is finitely generated then this subgroup has finite index, so it is open , andthen in (10) we have h Y i Φ( G ) = h Y i Φ( G ) = h Y i G ′ G q ( π ) . Thus if G = h a , . . . , a d i ∈ C π thenΦ( G ) = δ ( G ; a )where δ ( a, x ) ≡ ∃ z, y , . . . , y d . x = [ a , y ] . . . [ a d , y d ] z q ( π ) . The definability of Φ( G ) means that we can define generating sets in L gp : Proposition 5.3.
For each d ≥ there is a formula β d such that for G ∈ C π , G | = β d ( a , . . . , a d ) ⇐⇒ G = h a , . . . , a d i . INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 23
Proof.
Set β d ( u , . . . , u d ) ≡ α ( u , . . . , u d ) ∧ ∀ x. _ s (1) ,...,s ( d ) ∈ S δ ( u, x − u s (1)1 . . . u s ( d ) d )where S = { , , . . . , q − } . We have seen that if G = h a , . . . , a d i ∈ C π then G | = β d ( a , . . . , a d ).Conversely, if G ∈ C π and G | = β d ( a , . . . , a d ) then every element of G belongsto h a , . . . , a d i [ a , G ] . . . [ a d , G ] G q ( π ) ⊆ h a , . . . , a d i Φ( G ) . (cid:3) Now setting e β d ≡ ∃ y , . . . , y d .β d ( y , . . . , y d )we have d( G ) ≤ d ⇐⇒ G | = e β d , d( G ) = d ⇐⇒ G | = e β d ∧ ¬ e β d − := β ∗ d ;thus for groups in C π being d -generated can be expressed by a first-order sentence.We note that the hypothesis that the group G be in C π is necessary: Proposition 5.4.
Within profinite groups, being d -generated cannot be expressedby a single first order sentence.Proof. According to Proposition 1.1, if φ is a sentence in the language of groupsand a non-periodic abelian group G satisfies φ then G × C q satisfies φ for almostall primes q .If φ expresses being d -generated, then b Z d | = φ . Let q be a prime as above, thenalso b Z d × C q | = φ . But this group needs d + 1 generators. (cid:3) The same argument works for the category of abstract groups, using Z d in placeof b Z d . A slightly more elaborate argument shows that being finitely generated, forprofinite groups, also cannot be expressed by a single first order sentence φ . Oneworks with b Z × ( C q ) ℵ , and uses the fact that φ can be expressed as a Booleancombination of Szmielew invariant sentences: see [HMT, Thm. A.2.7].To conclude this introductory section, we discuss a ‘small’ f.g. pro- p group ofinfinite rank, G := C p b ≀ Z p = lim ←− n →∞ C p ≀ C p n . This group is the semidirect product of the ‘base group’ M by a procyclic group T ∼ = Z p ; here M ∼ = F p [[ T ]]as a T -module, where F p [[ T ]] is the completed group algebra of T (see [DDMS, § F p [[ T ]] is a 1-dimensional complete local ring with residue field F p ,whose non-zero closed ideals are just the powers of the maximal ideal, and thereforehave finite index. Proposition 5.5.
The pro- p group C p b ≀ Z p is FA within profinite groups. Proof.
Let a be a generator of the T -module M, and let f be a generator of T .Then G has the pro- p presentation h a, f ; a p = [ w, a ] = [ u, w ] = 1 ( u, w ∈ F ′ ) i where F denotes the free group on { a, f } .Let σ ( a, f ) be the formula saying for a pro- p group G that • the elements a, f generate G , i.e. β ( a, f ) holds; • the element a has order p and commutes with every commutator; • all commutators commute; • the centre of G is trivial.Then σ ( a, f ) holds in G .Suppose to begin with that H is a pro- p group and that H | = σ ( b, h ) for some b, h ∈ H . Then the map sending a to b and f to h extends to an epimorphism θ : G → H . Let N = ker θ . We aim to show that N = 1.Now G/N ∼ = H is a non-trivial pro- p group with trivial centre, so it is infinite(as every non-trivial finite p -group has non-trivial centre). Also N ∩ M correspondsto a closed ideal of F p [[ T ]] , so if N ∩ M = 1 then M/ ( N ∩ M ) is finite. As G/M is procyclic this implies that the centre of
G/N has finite index (if f n centralizes M/ ( N ∩ M ) then [ G, M h f n i ] ≤ N ). This contradicts Z( H ) = 1, and we concludethat N ∩ M = 1. But then N ≤ C G ( M ) ∩ N = M ∩ N = 1.It follows that H ∼ = G . Thus ( G ; a, f ) is FA in pro- p groups.To deal with the general case of profinite groups, we need a way to identify theprime p . Now we have G ′ = [ M, f ] = { [ x, y ] | x, y ∈ G } M = G ′ h a i = G ′ ∪ G ′ a ∪ . . . ∪ G ′ a p − so M is definable by a formula µ ( a, f ) say. Since G is pro- p , C G ( M ) = M and M p = 1 , the following holds:[ M, x ] ⊆ [ M, x p ] = ⇒ [ M, x ] = 1 = ⇒ x ∈ M = ⇒ x p = 1 . So G satisfies a formula τ ( a, f ) that expresses M ⊳ G and [ M, M ] = M p = 1 and [ M, x ] ⊆ [ M, x p ] = ⇒ x p = 1 . Suppose now that H is a profinite group and that H | = σ ( b, h ) ∧ τ ( b, h ) . Let N = µ ( b, h ; H ) , so N is an abelian normal subgroup of H, of exponent p . Suppose x belongs to a Sylow pro- q subgroup of H where q = p . Then x = x pλ for some λ ∈ Z q , and then [ N, x ] = [
N, x pλ ] ⊆ [ N, x p ]. As H | = τ ( b, h ) this implies that x p = 1 , and hence that x = 1. It follows that H is a pro- p group, and then H | = σ ( b, h ) implies that H ∼ = G . (cid:3) Corollary 5.6.
The classes of profinite, respectively pro- p , groups of finite rankand of f.g. profinite, respectively pro- p , groups are first-order separable, with witnessgroup C p b ≀ Z p . Powerful pro- p groups. Next we discuss a special class of pro- p groups,where p always denotes a prime. Fix ε = 0 if p = 2 , ε = 1 if p = 2 . INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 25
A pro- p group G is powerful if G/G p is abelian (replace p by 4 when p = 2). If G is also finitely generated, then G p n = G p n = G { p n } for each n ≥
1. Thus for a f.g. pro- p group G, G is powerful iff G | = ∀ x, y ∃ z. ([ x, y ] = z p )(replace p by 4 when p = 2). In this case we haveΦ( G ) = G p (resp. G if p = 2).For all this, see [DDMS, Chapter 3]. A key result of Lazard [Lz] characterizesthe compact p -adic analytic groups as the f.g. profinite groups that are virtuallypowerful (cf. [DDMS, Chapter 8]).The definition of a uniform pro- p group is given in [DDMS, Chapter 4]. Ratherthan repeating it here, we use the simple characterization ( loc. cit. Theorem 4.5):a pro- p group is uniform iff it is f.g., powerful and torsion-free .The following theorem summarizes key facts established in Chapters 3 and 4 of[DDMS]. We set λ ( r ) = ⌈ log r ⌉ + ε. Theorem 5.7. (i)
Let G be a pro- p group of finite rank r . Put m = λ ( r ) . Then G has open normal subgroups W ≥ W , with W ≥ Φ m G ≥ G p m , such that every open normal subgroup of G contained in W is powerful, and everyopen normal subgroup of G contained in W is uniform. (ii) If G is f.g. and powerful then rk( G ) = d( G ) . (iii) If G = h a , . . . , a d i is powerful then G = { a µ . . . a µ d d | µ , . . . , µ d ∈ Z p } . (iv) If G is f.g. and powerful then G has a uniform open normal subgroup U ,and d( V ) = rk( V ) = rk( U ) = d( U ) for every uniform open subgroup V of G . The common rank of open uniform subgroups of such a group G is denoteddim( G ); this is the dimension of G as a p -adic analytic group. Lemma 5.8.
Let G be a uniform pro- p group and let N ⊳ c G . If G/N is uniformthen N is uniform and dim( G ) = dim( N ) + dim( G/N ) . Proof.
As explained in [DDMS, Chapter 4], G has the structure of a Z p -Lie algebra L ( G ), additively isomorphic to Z dim( G ) p . Proposition 4.31 of [DDMS] says that N isuniform, L ( N ) is an ideal of L ( G ), and the quotient mapping G → G/N induces anepimorphism L ( G ) → L ( G/N ). The claim follows from the additivity of dimensionfor free Z p -modules. (cid:3) Corollary 5.9.
Let G be a pro- p group of finite rank and let N ⊳ c G . Then dim( G ) = dim( N ) + dim( G/N ) . Proof.
Let H be a uniform open normal subgroup of G . Then H/ ( H ∩ N ) ispowerful, hence has a finite normal subgroup M/N such that
H/M is uniformby [DDMS, Theorem 4.20]. The claim follows on replacing G by H and N by M . (cid:3) These results can be applied to C π groups of finite rank. Let G ∈ C π . Then G = G × · · · × G k where each G t is a pro- p t group, the Sylow pro- p t subgroupof G . If H is a closed subgroup H of G then H = H × · · ·× H k where H t = H ∩ G t ,notation we keep for the remainder of this subsection.If G has finite rank, we defineDim( G ) = dim G + · · · + dim G k . If p t ∤ m then every element of H t is an m th power in H ; thus if q = p e . . . p e k k then H { q } = H { p e } × · · · × H { p ekk } k H q = H p e × · · · × H p ekk k . We call H semi-powerful if each H t is a powerful pro- p t group. If H ∈ C π is finitelygenerated, then H is semi-powerful if and only if H/H q ′ ( π ) is abelian. This holdsiff H | = pow ≡ ∀ x, y ∃ z. (cid:16) [ x, y ] = z q ′ ( π ) (cid:17) .H is semi-uniform if each H t is uniform. In this case, the dimension of H is the k -tuple dim H = (dim H , . . . , dim H k ) . Lemma 5.10.
Let H and K be semi-uniform C π groups, and θ : H → K anepimorphism. If dim H = dim K then θ is an isomorphism.Proof. Restricting to Sylow subgroups, we may suppose that H and K are uniformpro- p groups of the same dimension. Then Lemma 5.8 shows that ker θ is a uniformgroup of dimension 0, i.e. the trivial group. (cid:3) Corollary 5.11.
Let G ∈ C π have finite rank. If N ⊳ c G and Dim(
G/N ) =Dim( G ) then N is finite.Proof. This follows likewise from Corollary 5.9. (cid:3)
For q, f ∈ N set µ f,q ( x ) ≡ ∃ y , . . . , y f . ( x = y q . . . y qf ) . As before, we see that the word x q has width f in a group H, that is, H q = ( H { q } ) ∗ f := n h q . . . h qf | h i ∈ H o if and only if H satisfiesm f,q ≡ ∀ x. ( µ f +1 ,q ( x ) → µ f,q ( x )) . Of course, this holds iff H | = s( µ f,q ); we can use either formulation. INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 27
Proposition 5.12.
Let G be a f.g. profinite group and let q ∈ N . There exists f ∈ N such that G | = m f,q and G q = ( G { q } ) ∗ f = µ f,q ( G ) is a definable open normal subgroup of G . Both f and | G : G q | can be bounded interms of q and d( G ) . This is part of Theorem 2.1; the second claim was not made explicit in the statementbut is included in the proof.If H is semi-uniform of dimension ( d , . . . , d k ), we haveΦ( H ) = H q ( π ) = H { q ( π ) } = µ ,q ( π ) ( H ) , | H : Φ( H ) | = p d . . . p d k k . Thus for semi-uniform H , the dimension is determined by H | = ∂ d ,...,d k ≡ ind( µ ,q ( π ) ; p d . . . p d k k ) . Presentations.
In the context of profinite groups, a ‘finite presentation’ mayinvolve relators that are ‘profinite words’, i.e. limits of a convergent sequence ofgroup words. For present purposes we need to consider concepts of finite presenta-tion that are both more and less restrictive.Let C be a class of groups and L ⊇ L gp a language. For a group G ∈ C and aformula ψ ( x , . . . , x r ) of L , we say that ψ is an L -presentation of G in C if G hasa generating set { g , . . . , g r } such that(i) G | = ψ ( g , . . . , g r ), and(ii) if h , . . . , h r ∈ H ∈ C and H | = ψ ( h , . . . , h r ) then there is an epimorphism θ : G → H with g i θ = h i for each i .In this case, we say that ψ is an L -presentation on { g , . . . , g r } .The concept of L -presentation generalizes the familiar idea of a finite presentationin group theory. We mention two particular cases. Proposition 5.13.
A group G ∈ C π has an L -presentation in C π in each of thefollowing cases: (i) L = L gp , and G is strictly f.p. in C π ; that is, G has a finite presentation asa C π -group in which the relators are finite group words, or equivalently, G is the C π -completion of a finitely presented (abstract) group. (ii) L = L π , and G has finite rank.Proof. (i) We have an epimorphism φ : F → G where F is the free C π -group on afinite generating set X = { x , . . . , x r } and ker φ is the closed normal subgroup of F generated by a finite set R of ordinary group words on X . Set ψ ( x ) := β r ( x ) ∧ ^ w ∈ R w ( x ) = 1(recall that G ∈ C π satisfies β r ( a ) iff { a , . . . , a r } generates G ).Now put g i = x i φ for each i . Then G | = ψ ( g ). Suppose that h , . . . , h r ∈ H ∈ C π and H | = ψ ( h ). Then h , . . . , h r generate H , so the homomorphism µ : F → H sending x to h is onto. Also for each w ∈ R we have w ( x ) µ = w ( h ) = 1, soker φ ≤ ker µ . It follows that µ factors through an epimorphism θ : G → H with g i θ = h i for each i . Thus ψ is an L gp presentation for G in C π . (cid:3) Before proving (ii) we need yet another definition: • Let G ∈ C π . Then h X ; R i is a Z π -finite C π presentation for G if G ∼ = F/N where F = F ( X ) is the free C π -group on a finite generating set X and N is the closed normal subgroup of F generated by a finite set R of elementsof the form(11) w µ . . . w µ n n where each w i is a group word on X and µ , . . . , µ n ∈ Z π .Note that the free C π -group on a set X is the direct product of its Sylow pro- p t subgroups, which themselves are free pro- p t groups on the p t -components of X .Expressions like (11) will be called π -words. For a subset Y of G , one says that h X ; R i is a presentation on Y if the implied epimorphism F ( X ) → G maps X to Y . Lemma 5.14.
Let G = h Y i ∈ C π have finite rank, where Y is finite. Then G hasa Z π -finite C π presentation on Y .Proof. Suppose U is a uniform pro- p group with generating set X = { x , . . . , x d } .Then dim U ≤ d and U has a pro- p presentation on X with relators(12) [ x i , x j ] x λ ( i,j )1 . . . x λ d ( i,j ) d , ≤ i < j ≤ d, each λ l ( i, j ) ∈ p Z p . If d = dim U this is [DDMS, Prop. 4.32]; when d > dim U, apply this to a minimal generating subset of X to obtain relators (12) for somepairs ( i, j ), and add (redundant) relators for the remaining pairs using Theorem5.7(iii). Now consider a semi-uniform C π group V = U × · · · × U k where each U t is a uniform pro- p t group of dimension d t ≤ d . A generating set X = { x , . . . , x d } for V projects to a generating set X t = { x ( t )1 , . . . , x ( t ) d } for U t ; then U t has a pro- p t presentation on X t like (12), with exponents λ ( t ) l ( i, j ) ∈ Z p t .Let λ l ( i, j ) ∈ Z π have Z p t -component λ ( t ) l ( i, j ) for each t . Then (12) gives apresentation for V on X .Finally, we have G = G ×· · ·× G k = h y , . . . , y m i where each G t is a pro- p t groupof finite rank and Y = { y , . . . , y m } . Let V = U × · · · × U k be a semi-uniform opennormal subgroup of G . Using the Schreier process we obtain a finite generating set X = { x , . . . , x d } for V, each element of X being equal to a finite word on y , say x i = w i ( y ). Substitute w i ( y ) for x i in (12) to obtain a set of relators R on y . Notethat R consists of π -words.By Theorem 5.7(iii), each element of V is a finite product of Z π -powers of el-ements of X . The conjugation action of G on V is determined by specifying, for j = 1 , . . . , m and for each x i ∈ X,y − j x i y j = W ij ( X )where each W ij ( X ) is a finite product of Z π - powers (for clarity, we keep w forfinite group words and write W for π -words).Let S := (cid:8) y − j w − i y j .W ij ( w , . . . , w d ) | j = 1 , . . . , m, i = 1 , . . . , d (cid:9) . A standard argument (see for example [PG, Chapter 8, Lemma 10]) now showsthat h Y ; R ∪ S i is a C π presentation for G . (cid:3) INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 29
Now we can give the
Proof of Proposition 5.13 (ii). Let G = h Y i be as in the preceding lemma. Set ρ ( y ) := ^ w ∈ R ∪ S w ( y ) = 1where R and S are given above. As these are finite sets of π -words, ρ is a formulaof L π . Now put ψ ( y ) := β m ( y ) ∧ ρ ( y ) . If h , . . . , h m ∈ H ∈ C π and H | = ψ ( h ) then h , . . . , h m generate H and satisfy therelations R ∪ S = 1; as h Y ; R ∪ S i is a C π presentation for G it follows that themap sending h to y extends to an epimorphism from G to H . Thus ψ is an L π presentation for G in C π .5.4. Finite axiomatizability in C π . Until further notice, L stands for one of L gp , L π . Theorem 1.6 is included in Theorem 5.15.
Suppose that G ∈ C π has finite rank, and that G has an L presen-tation on the generating tuple ( a , . . . , a r ) . Then ( G, a ) is FA in C π . Note that when L = L π , the existence of an L presentation is guaranteed byProposition 5.13(ii). Proof.
We have G = G × · · · × G k where each G t is a pro- p t group of finite rankand dimension d t . There is a formula ψ of L such that (i) G | = ψ ( a ) and (ii) if h , . . . , h r ∈ H ∈ C π and H | = ψ ( h ) then there is an epimorphism θ : G → H sending a to h .It follows from Theorem 5.7 that G has an open normal subgroup W such thatevery open normal subgroup of G contained in W is semi-uniform. Then W ≥ G q for some π -number q . Now Proposition 5.12 shows that for some f , G satisfies m f,q and G q = µ f,q ( G )is open in G, hence semi-uniform. Settf ≡ ∀ x ( x q ( π ) = 1 → x = 1) . As G q is semi-powerful and torsion-free, G satisfiesres( µ f,q , pow ∧ tf)(see § | G : G q | = m and dim( G q ) = ( d , . . . , d k ) . Then G also satisfiesind ∗ ( µ f,q ; m ) ∧ res( µ f,q , ∂ d ,...,d k ) . We have established that G satisfies σ G ( a ) ≡ ψ ( a ) ∧ m f,q ∧ res( µ f,q , pow ∧ tf) ∧ ind ∗ ( µ f,q ; m ) ∧ res( µ f,q , ∂ d ,...,d k ) . Now suppose H ∈ C π satisfies σ G ( h ). Let θ : G → H be the epimorphismspecified above. To complete the proof it will suffice to show that ker θ = 1 .σ G ( h ) implies that H q = µ f,q ( H ) is semi-uniform, that | H : H q | = m = | G : G q | ,and that dim( H q ) = dim( G q ). Applying Lemma 5.10 to θ | G q we infer that ker θ ∩ G q = 1. As | H : H q | = | G : G q | is finite, it follows that θ induces an isomorphism G/H q → H/H q , whence ker θ ≤ G q . Thus ker θ = 1 as required. (cid:3) Finite axiomatizability in profinite groups.
The first case of Theorem1.7 to be established was for the specific group G = UT ( Z p ) [NB, § Z p from the group structureof G : specifically, the commutator map in the group carries enough informationto reconstruct multiplication in the ring. With Theorem 5.15 at our disposal, weshall see that it will suffice merely to identify the prime p , and this in turn isquite easy provided there are ‘enough’ commutators. The appropriate conditionwas identified by Oger and Sabbagh in [OS], in the context of abstract nilpotentgroups; fortunately for us it transfers perfectly to the profinite context.We will say that a group G satisfies the O-S condition if Z( G ) / ( G ′ ∩ Z( G )) isperiodic. Theorem 1.7 is included in Theorem 5.16.
Let L be either L gp or L π . Suppose that G ∈ C π is nilpotent, andthat G has an L presentation as a C π group on the generating tuple ( a , . . . , a r ) .Then the following are equivalent: (a) ( G, a ) is FA (wrt L ) in the class of all profinite groups; (b) G is FA (wrt L ) in the class of all f.g. nilpotent virtually pro- π groups; (c) G satisfies the O-S condition. For the proof, we write Z ( x ) ≡ ∀ u. ( xu = ux ) Z ( x ) ≡ ∀ v.Z ([ x, v ]);these define the centre and second centre in a group. Let S p ( x ) be a formulaasserting, for x ∈ G , that [ Z ( G ) , x ] ⊆ [ Z ( G ) , x p ] . Let ψ π be a sentence asserting that V p ∈ π S p ( x ) → [ Z ( G ) , x ] = 1 for each x . Lemma 5.17.
Let G be a nilpotent profinite group. (i) If Z( G ) is pro- π then G | = ψ π . (ii) If G | = ψ π then G/ Z( G ) is a pro- π group.Proof. Write Z = Z( G ) and Z = Z ( G ). We use the facts that for each x ∈ G the map y [ x, y ] is a continuous homomorphism from Z to Z , with kernelcontaining Z , and that ( x, y ) [ x, y ] induces a bilinear map from G ab × Z /Z into Z .(i) Suppose that Z is a pro- π group. Let x ∈ G . If S p ( x ) holds for each p ∈ π then [ x, Z ] = [ x p , Z ] = [ x, Z ] p for each p ∈ π. But [ x, Z ] is a closed subgroup of the abelian pro- π group Z , andso [ x, Z ] = 1. Thus G | = ψ π .(ii) Let q / ∈ π be a prime, and let Q be a Sylow pro- q subgroup of G . Let x ∈ Q and p ∈ π. Then x = x λp where λ ∈ Z q satisfies λp = 1. Then for any u ∈ Z wehave [ x, u ] = [ x p , u λ ] . Thus S p ( x ) holds.Now suppose that G | = ψ p . Then [ x, Z ] = 1 holds for each x ∈ Q , so Q ∩ Z ≤ Z( Q ) . Now Q ∩ Z = Z ( Q ), so Z ( Q ) = Z( Q ). As Q is nilpotent this forces Q = Z( Q ) ≤ Z .It follows that G/Z is a pro- π group. (cid:3) INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 31
To prove Theorem 5.16, we have to show that the following are equivalent: a: ( G, a ) is FA w.r.t. L in the class of all profinite groups; b: G is FA w.r.t. L in the class of all f.g. nilpotent virtually pro- π groups; c: G satisfies the O-S condition.Note to begin with that G has finite rank (a familiar property of f.g. nilpotentgroups).Proposition 1.1 shows that if G does not satisfy the O-S condition then G cannotbe f.a. in the class of groups { G × C q , q prime } . Thus (a) = ⇒ (b) = ⇒ (c) , since( b ) is formally weaker than ( a ).Now suppose that G does satisfy the O-S condition. Given the hypotheses,Theorem 5.15 gives us a formula σ G of L such that for b , . . . , b r ∈ H ∈ C π , we have H | = σ G ( b ) ⇐⇒ ( H, b ) ∼ = ( G, a ) . Also Z( G ) / ( G ′ ∩ Z( G )) is a periodic pro- π group of finite rank, so it is finite, ofexponent q say; here q is a π -number. Recall (Lemma 5.2) that every element of G ′ is a product of d commutators, where d = d( G ). Therefore G satisfies(13) θ q ≡ ∀ y. Z ( y ) −→ ∃ u , v , . . . , u d , v d . y q = d Y i =1 [ u i , v i ] !! . Say G is nilpotent of class c . This is expressed by a sentence Γ c (all simplecommutators of weight c + 1 are equal to 1). Now defineΣ G ≡ ψ π ∧ σ G ∧ θ q ∧ Γ c . Then G satisfies Σ G ( a ) . Suppose H is a profinite group and that H | = Σ G ( b ) forsome b ∈ H ( r ) . Then H is nilpotent, so by Proposition 5.17(ii) H/ Z( H ) is a pro- π group. Also Z( H ) H ′ /H ′ has exponent dividing q , so H/H ′ is a pro- π group. As H is nilpotent this implies that H is a pro- π group (to see this, note that each finitecontinuous quotient e H of H is the direct product of its Sylow subgroups, and itsabelianization is the direct product of their respective abelianizations. So if e H/ e H ′ is a π -group then the Sylow q -subgroups of e H for q / ∈ π have trivial abelianization,and as they are nilpotent this means that they are trivial. Therefore e H is a π -group). Thus H ∈ C π .As H | = σ G ( b ) it follows that ( H, b ) ∼ = ( G, a ). Thus (c) = ⇒ (a) .5.6. Finite axiomatizability in pronilpotent groups.
The nilpotency hypoth-esis in Theorem 5.16 is very restrictive. Without it, we can prove a weaker result,giving finite axiomatizability in the class of all pronilpotent groups ; this is strictlyintermediate between C π and the class of all profinite groups, so the following results‘interpolate’ the two preceding theorems: Theorem 5.18.
Let G ∈ C π have finite rank, and assume that G has an L -presentation in C π . Then the following are equivalent: (a) G is FA (wrt L ) in the class of all pronilpotent groups; (b) G is FA (wrt L ) in the class of all pronilpotent virtually pro- π groups of finiterank; (c) G satisfies the O-S condition. Theorem 5.19.
Let G ∈ C π have finite rank, and assume that G has an L -presentation in C π . If G/γ m ( G ) satisfies the O-S condition for some m ≥ then G is FA (wrt L ) in the class of all pronilpotent groups. Here, γ m ( G ) denotes the m th term of the lower central series (a closed normalsubgroup when G is a f.g. profinite group). Note that when L is L π , Proposition5.13(ii) makes the assumption of an L -presentation redundant.We will use the fact that for each d and m there exists f = f ( d, m ) such that inany d -generator pronilpotent group G we have γ m ( G ) = X ∗ f := { x . . . x f | x , . . . , x f ∈ X } where X = { [ y , . . . , y m ] | y , . . . , y m ∈ G } ([SW], Lemma 4.3.1). It follows that γ m ( G ) is definable by a formula Γ d,m . Lemma 5.20.
Let H be a f.g. pronilpotent group. Put H n = γ n ( H ) and Z n /H n =Z( H/H n ) for each n . Suppose that H qs ≤ H s +1 for some s ≥ , where q is a π -number. Then Z s / Z( H ) is a pro- π group.Proof. We begin with some properties of the series ( H n ). (i) H qn ≤ H n +1 for each n ≥ s. Proof by induction on n . Let n ≥ s and supposethat H qn ≤ H n +1 . Now H n +1 is generated by elements [ x, h ] with x ∈ H n and h ∈ H . These satisfy [ x, h ] q ≡ [ x q , h ] ≡ H n +2 ) . As H n +1 /H n +2 is abelian it follows that H qn +1 ≤ H n +2 . (ii) Z qn ≤ Z n +1 for each n ≥ s. To see this, let z ∈ Z n and h ∈ H . Then[ z q , h ] ≡ [ z, h ] q ≡ H n +1 )by (i) so z q ∈ Z n +1 . (iii) T ∞ n = s Z n = Z( H ) . This is immediate from the fact that T ∞ n = s H n = 1,which holds because H is pronilpotent.To conclude the proof, observe that the subgroups H n , and therefore also Z n , are closed in H . Let n > s . Then H/Z n is a f.g. nilpotent profinite group and Z s /Z n has exponent dividing q n − s , so Z s /Z n is a finite π -group, and Z n is open in Z s . The claim now follows from (iii). (cid:3) Lemma 5.21.
Let G ∈ C π have finite rank, and let ( G n ) n ∈ N be a descending chainof closed normal subgroups of G . Then there exists s such that G n /G n +1 is finitefor each n ≥ s .Proof. The sequence Dim(
G/G n ) is non-decreasing and bounded by Dim( G ), so itbecomes stationary at some point n = s . Then G s /G n is finite for all n ≥ s, byCorollary 5.9. (cid:3) Now let π be a finite set of primes, let G ∈ C π have finite rank, and assumethat G has an L -presentation in C π . For Theorem 5.18 we have to establish theequivalence of a: G is FA (wrt L ) in the class of all pronilpotent groups; b: G is FA (wrt L ) in the class of all pronilpotent virtually pro- π groups offinite rank; c: G satisfies the O-S condition.Theorem 5.19 asserts that these follow from d: G/γ m ( G ) satisfies the O-S condition for some m ≥ . INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 33
The proof that (a) = ⇒ (b) = ⇒ (c) is the same as in the preceding subsection(proof of Theorem 5.16).By Theorem 5.15, there is a sentence σ G such that for any H ∈ C π , H | = σ G iff H ∼ = G ; we may assume that σ G → e β d , where d( G ) = d (recall: e β d asserts that a C π group can be generated by d elements, cf. § (d) holds. We have γ m ( G ) = Γ d,m ( G ). By Theorem 5.16, there isa sentence Σ such that a profinite group L satisfies Σ iff L ∼ = G/γ m ( G ).Let ξ ≡ lift(Γ d,m , Σ) ∧ σ G . Now suppose that H is a pronilpotent group and that H | = ξ . Then d( H ) ≤ d , so γ m ( H ) = Γ d,m ( H ), and so H/γ m ( H ) | = Σ. It follows that H/γ m ( H ) ∼ = G/γ m ( G ).Thus H/γ m ( H ) is a pro- π group. As m ≥ H is a pro- π group (as in the proof of Theorem 5.16, above). Thus H ∈ C π and so H ∼ = G . Thus (a) holds.Suppose now that (c) holds. According to Lemma 5.21, there exists s ≥ γ s ( G ) /γ s +1 ( G ) is finite. Then γ s ( G ) q ≤ γ s +1 ( G ) for some π -number q . Thus G satisfies η ≡ ∀ x. (Γ d,s ( x ) → Γ d,s +1 ( x q )) . Condition (c) implies that G satisfies θ q ′ , defined in (13), for some π -number q ′ .Let ψ π be as in Lemma 5.17. Then G/γ s ( G ) | = ψ π so G | = lift(Γ d,s , ψ π ).Now put Σ ≡ η ∧ θ q ′ ∧ lift(Γ d,s , ψ π ) ∧ σ G . Let H be a pronilpotent group and define Z n ≥ H n as in Lemma 5.20. Supposethat H satisfies Σ. As above, d( H ) ≤ d , so H n = γ n ( H ) = Γ d,n ( H ) for each n . Inparticular, H/H s | = ψ π . It follows by Lemma 5.17 that H/Z s is a pro- π group.As H | = η we have H qs ≤ H s +1 , so Z s / Z( H ) is a pro- π group, by Lemma 5.20.As H | = θ q ′ , we have Z( H ) q ′ ≤ H ′ . It follows that H/H ′ is a pro- π group, andhence (as before) that H is pro- π . As H | = σ G it follows that H ∼ = G .Thus (c) implies (a) . 6. Special linear groups
We assume that p ∤ n , and consider the groupsΓ = SL n ( Z p ) ,G = SL n ( Z p ) = ker (Γ → SL n ( F p )) . We write T p = 1 + p Z p for the group of 1-units in Z p , and recall that this is a procyclic pro- p group iso-morphic to ( Z p , +) (via the mapping log : T p → p Z p , see e.g. [DDMS] 6.25, 6.36).In particular, T pp = 1 + p Z p , so an element η ∈ T p generates T p iff η p ).We fix η ∈ T p r T pp such that η p = (1 + p ) − ; this exists because x x p is abijection T p → T pp .In this section we only consider the language L = L gp . The congruence subgroup.
Note that G is a uniform pro- p group, by[DDMS] , Theorem 5.2 and Lemma 5.8 above (this will also be clear from thestructural information below).Define n × n matrices for 1 ≤ i < j ≤ n : u ij ( λ ) = 1 + λe ij , v ij ( µ ) = 1 − µe ji u ij = u ij ( p ) , v ij = v ij ( p ) . Let w = e + · · · + e n − ,n ± e n, be the permutation matrix for the n -cycle (12 . . . n ) , adjusted to have determinantequal to 1 (here e ij denotes the matrix with just one non-zero entry 1 in the ( i, j )place, not the usual elementary matrix). Thus u wij = u i +1 ,j +1 , v wij = v i +1 ,j +1 ( i < j < n )(14) u win = v ± ,i +1 , v win = u ± ,i +1 . (15)For i = 1 , . . . , n set h i = ζ i n · diag( η − , . . . , η − , η, . . . , η )where the last η − occurs in the i th place and the first η in the ( i + 1)th place, η ∈ T p \ T pp satisfies η p = (1 + p ) − , and ζ i ∈ T p satisfies ζ n η n − i = 1, to ensurethat det( h i ) = 1. Note that ζ i exists because p ∤ n . Note also that h n = 1. Forconvenience we also define h = 1. Then(16) h wi = h − i h i +1 h wi − (1 ≤ i ≤ n − . For i = j define U ij = u ij ( p Z p ) = h u ij i V ij = v ij ( p Z p ) = h v ij i ( i > j )and denote by H the group of all diagonal matrices in G .We make the convention that a list indexed by pairs ( i, j ) with i < j is orderedlexicographically w.r.t. ( j − i, i ) , i.e. as in(1 , , (2 , , . . . , ( n − , n ) , (1 , , (2 , , . . . , ( n − , n ) , . . . , (1 , n ) . Define subsets of
G U = Y ≤ i Experts will observe that the U ij and V ij are the root groups if G is construed as a Chevalley group of type A n − . The following proposition is aversion of the Steinberg presentation and some if its consequences; see for example[DDMS, Chapter 13, Ex. 11], which exhibits a Chevalley group of arbitrary type(with suitable points in Z p ) as a uniform pro- p group. These groups, over a widerrange of rings, are considered in [ST].We summarize some basic structural features of G that will be required; theseare all well known and can be verified by calculation. INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 35 Proposition 6.1. (i) U, V are the subgroups of all upper, respectively lower uni-triangular matrices in G , and (17) the multiplication map: Y ≤ i 3. The case n = 2 is sketched below.For brevity, we will say ‘formula’ to mean ‘formula of L gp with parameters u, v, h ’,except where parameters are explicitly mentioned. We will establish the followingclaims.( ) H is definable, in fact H = χ ( G ; h ) for some formula χ ; moreover, χ alwaysdefines a closed subgroup in any profinite group.( ) U and V are definable, in fact U = ϕ ( G ; u, h ) , V = ϕ ( G ; v, h ) for someformulae ϕ , ϕ ; moreover, these always define closed subsets in any profi-nite group. ( ) U has an L gp presentation on u as a pro- p group; V has an L gp presentationon v as a pro- p group.( ) G has an L gp presentation on ( u, v, h ) as a pro- p group.Given these claims, the proof is concluded as follows.Given Claims and , we can construct a formula Φ( x, y, z ) such that Φ( u, v, h )expresses the conjunction of the facts: (a) multiplication maps V × H × U bijectively to G , (b) [ V, H ] ≤ V p and C H ( V ) = 1, (c) [ U, H ] ≤ U p and C H ( U ) = 1 , (d) H is abelian and has no p -torsion.Both U and V are nilpotent pro- p groups, and they satisfy the O-S conditionby Proposition 6.1 (v). Theorem 5.16 now provides formulae σ U ( x ) , σ V ( x ) thatdetermine ( U, u ) and ( V, v ) among all profinite groups. Set(24)Ψ( x, y, z ) = s( ϕ ( x, z )) ∧ s( ϕ ( y, z )) ∧ res( ϕ ( x, z ) , σ U ( x )) ∧ res( ϕ ( y, z ) , σ V ( y ))(recall that G | = s( ϕ ) means: the subset ϕ ( G ) is a subgroup).Now suppose that e G is a profinite group and u ∼ , v ∼ , h ∼ are tuples in e G of theappropriate lengths such that e G | = Φ( u ∼ , v ∼ , h ∼ ) ∧ Ψ( u ∼ , v ∼ , h ∼ ) . Let e U , e V , e H be the subsets of e G defined by ϕ ( u ∼ , h ∼ ) , ϕ ( v ∼ , h ∼ ) , χ ( h ∼ ).Then Ψ ensures that e U ∼ = U and e V ∼ = V are pro- p groups, generated respectivelyby u ∼ , v ∼ . Also Φ ensures that e H is closed, normalizes e V , acting faithfully byconjugation, and that [ e V , e H ] ⊆ e V p ; this now implies that e H is a pro- p group, andhence that e V · e H is a pro- p group. Φ also ensures that e G = e V · e H · e U . Since aproduct of two pro- p subgroups is again pro- p , it now follows that e G is a pro- p group.Finally, Claim with Theorem 5.15 provides a formula σ G ( x, y, z ) that deter-mines ( G, u, v, h ) among pro- p groups. It follows thatΦ( x, y, z ) ∧ Ψ( x, y, z ) ∧ σ G ( x, y, z )determines ( G, u, v, h ) among all profinite groups. Proof of Claim (17), (21) and (22) imply that H = C G ( H ), so we may take χ ( x, y ) := n − ^ i =1 com( x i , y )(recall that com( x, y ) ≡ ( xy = yx )). This defines a closed subgroup in any profinitegroup because it defines a centralizer. Proof of Claim Proposition 6.1(vi) shows that the U ij are definable, asrepeated centralizers. It is then clear how to define U = Q U ij . The same will holdfor V by symmetry.To establish the final part of Claim 2, note that the formulae defining U ij and V ij always define a closed subgroup in any profinite group (a double centralizer),and the result follows since the product of finitely many closed subsets is closed. INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 37 Proof of Claim . (17) implies that u is a basis for the uniform pro p group U . It follows by [DDMS], Proposition 4.32, that the relations (18) (with µ = p )provide a pro- p presentation for U on that basis. Similarly, for V, v . Proof of Claim Similarly, the relations (18) – (22) (with µ = p ) provide apro- p presentation for the uniform pro- p group G . However, some of them cannotbe expressed in L gp as they involve non-integral powers.Using (21) and (22), (20) can be re-written as(25) [ v ij , u ij ] = h pi − h − pi v − p ij u p ij h − pj − h pj , so it is harmless.Now fix i ≤ j < k and set u = u ij , v = v ij , h = h k . Then (21) (with µ = p )says that u h = u η , whence u h − p = u η − p = u p .Similarly, v h p = v p . Thus (21) (with µ = p ) and (22) imply u h − pk ij = u p ij ( i ≤ k < j )(26) v h pk ij = v p ij ( i ≤ k < j ) . (27)Now we have shown above that given Claims 1, 2 and 3, there are formulae Φand Ψ such that for any profinite group e G , if e G | = Φ( u ∼ , v ∼ , h ∼ ) ∧ Ψ( u ∼ , v ∼ , h ∼ )then e H is abelian, has no p -torsion, and acts faithfully on e U . In this situation, theaction of h ∈ e H on e U is determined by the action of h p , and similarly for the actionon e V . This now implies that (26), (27) are equivalent to (the middle lines of) (21)(with µ = p ), (22).Let ∆( u, v, h ) be a formula that expresses the relations (26), (27), (25), (18),(19) (with µ = p ), and the parts of (21) (with µ = p ) and (22) regarding k < i and k ≥ j . The preceding argument shows that ∆ ∧ Φ ∧ Ψ is equivalent to theconjunction of Φ ∧ Ψ with the original set of relations (18) – (22) (with µ = p ).As the latter give a pro- p presentation of G , it follows that ∆ ∧ Φ ∧ Ψ is an L gp presentation of G as a pro- p group. (cid:3) The case n = 2. We only sketch this. In the above argument, the hypothesis n ≥ U and V satisfy the O-S condition, which inturn is only used to establish that e U ∼ = U and e V ∼ = V are pro- p groups. If n = 2, thishas to be established by a different route. The idea is to show that the ring Z p canbe interpreted in G by the definable subgroup U , and then use the fact that Z p isFA in the class of rings whose additive group is profinite (see the proof of Theorem ?? , and Theorem 4.5). Together, these allow us to express the fact that U ∼ = Z p bya suitable formula with parameters u, h. The same applies with V, v in place of U, u .With these formulae in place of res( ϕ ( x, z ) , σ U ( x )) and res( ϕ ( y, z ) , σ V ( y )) in thedefinition of Ψ( x, y, z ) (see (24)), one finds that e U ∼ = e V ∼ = Z p , and the argumentthen proceeds as before.6.2. SL n ( Z p ) and PSL n ( Z p ) . In this subsection, n can be any integer ≥ 2, but wekeep the assumption that p ∤ n . We continue with the notation of the precedingsubsection, and begin with two lemmas.We set u = u (1) and recall that u ij = u ij (1) p , v ij = v ij (1) p . Lemma 6.3. Let m ∈ Z with m ≡ p ) . There is a formula χ m ( y, z, x ) suchthat, G | = χ m ( v, h, x ) ⇐⇒ x = h ( m − ) := diag( m, m − , , . . . , . Proof. Set χ m ( v, h, x ) := χ ( h, x ) ∧ (cid:16) v x = v m (cid:17) ∧ (( v x ) m = v ) ∧ ^ j> (cid:0) v xj − ,j = v j − ,j (cid:1) , where H = G χ ( h ) . If this holds for x ∈ G then x ∈ H and x acts like h ( m − ) on V ∗ = h v i,i +1 | ≤ i < n i . It follows that x = h ( m − ) because C H ( V ∗ ) = C H ( V ) =1 , since every element of V has some power in V ∗ and extraction of roots is uniquein the torsion-free nilpotent group V . (cid:3) The next lemma is a simple calculation: Lemma 6.4. Put m = 1 + p . Then u − v u = v m − h ( m − ) u m − , ( u − h u ) p = u p h p . Now we can deduce Theorem 6.5. The groups SL n ( Z p ) and PSL n ( Z p ) are FA in the class of all profi-nite groups.Proof. Write ˜ : Γ = SL n ( Z p ) → e Γ = PSL n ( Z p ) for the quotient map. As ˜ restrictsto an injective map on G , we may consider both Γ and e Γ as finite extensions of G .As Z( G ) = 1 we may apply Theorem (ii). The following argument deals withΓ; the same argument with ˜ applied to everything will give the result for e Γ.For convenience, we shall allow u, v, h to denote the sets { u ij . . . } etc. listed in(23), as well ordered tuples. By conjugating with w and forming commutators andinverses we can obtain every u ij (1) and v ij (1) from u . It follows that G = (cid:10) u, v, h (cid:11) ⊳ Γ = (cid:10) u , w, h (cid:11) , (cid:10) u, v, h (cid:11) ⊆ (cid:10) u , w, h (cid:11) . Thus it will suffice to verify that x y is ( u, v, h )-definable in G for each x ∈ u ∪ v ∪ h and y ∈ { u , w, h } . This is obvious for y ∈ h and follows from (16) for y = w .The relations (21), (18) and (19) with µ = 1 show that u commutes with every u ij and every h k for k ≥ 2, and conjugates each v ij with ( i, j ) = (1 , 2) into thegroup h u, v i . So it remains to deal with v u and h u .Now we use the two preceding lemmas, and keep their notation. Set m = 1 + p,α ( u, v, h, x, y, z ) := ( x m = v ) ∧ χ m ( v, h, y ) ∧ ( z m = u ) ,ϕ ( u, v, h, t ) := ∃ x, y, z. (cid:0) t = xyz ∧ α ( u, v, h, x, y, z ) (cid:1) . Then G | = ϕ ( u, v, h, v u ), and this defines v u in G because of the uniquenessproperty (17).Put ψ ( u, h, t ) := ( t p = u p h p ) . Then ψ ( u, h, t ) defines h u in G, because extraction of p th roots is unique in theuniform pro- p group G .The result follows from Theorem 3.1(ii). (cid:3) INITE AXIOMATIZABILITY FOR PROFINITE GROUPS 39 Some negative results Infinitely many primes. All rings are supposed to be commutative, withidentity. Proposition 1.2, due to Scanlon, states the following: Let R be the ring Q p ∈ S Z p where S is an infinite set of primes. Then R is not FAin the class of all profinite rings .The proof uses the Feferman-Vaught theorem from model theory (see [HMT], § φ of L rg there exist finitely many sentences ψ , . . . , ψ n of L rg and a formula θ ( x , . . . , x n ) in the language of Boolean algebrassuch that for any family of rings { A i | i ∈ I } , setting X j = { i ∈ I | A i | = ψ j } wehave Y i ∈ I A i | = φ ⇐⇒ P ( I ) | = θ ( X , . . . , X n )( P ( I ) denotes the power set of I ).Suppose now that φ is a sentence that determines R among profinite rings. Bythe pigeonhole principle, we can find distinct primes r = q in S such that for every j ≤ n , Z r | = ψ j ⇐⇒ Z q | = ψ j . Define R ′ = Z q × Y r = p ∈ S Z p . Then r is invertible in R ′ but not in R , so R ′ ≇ R . But R ′ | = φ by the Feferman-Vaught theorem.Now let G ( R ) = UT ( R ) denote the Heisenberg group over the ring R . Proposition 7.1. Let R S = Q p ∈ S Z p . Then the group G ( R S ) is FA among profi-nite groups if and only if S is finite.Proof. The group G ( R S ) satisfies the O-S condition. If S is finite, G ( R S ) is anilpotent C S group, and has the strictly finite C S presentation h x, y ; [ x, y, x ] , [ x, y, y ] i . So it is FA by Theorem 5.16.Assume now that S is infinite. The group G ( R ) can be interpreted in R by acollection of first-order formulae independent of R . It follows that for each sentence θ of L gp there is a sentence e θ of L rg such that for any ring R , G ( R ) | = θ ⇐⇒ R | = e θ. Suppose θ is a sentence that determines G ( R S ) among profinite groups. Take R = R S and define R ′ as above. Then R ′ | = e θ and so G ( R ′ ) | = θ . But Z( G ( R )) ∼ =( R, +) is not divisible by the prime r , while Z( G ( R ′ )) ∼ = ( R ′ , +) is r -divisible, so G ( R ′ ) ≇ G ( R ), contradiction. Thus G ( R S ) is not FA. (cid:3) Uncountably many pro- p groups. For λ ∈ Z p let T ( λ ) be the class-2 nilpo-tent pro- p group with pro- p presentation on generators x , . . . , x , y , . . . , y , e, f and relations [ x i , x j ] = [ y i , y j ] = 1 (all i, j )[ x i , y j ] = 1 (all i = j )[ x , y ] = e, [ x , y ] = ef − [ x , y ] = f, [ x , y ] = ef − λ e, f centralThis is clearly a pro- p group of rank 10 (with centre Z p and central quotient Z p )and so contains an open normal uniform subgroup T ∗ ( λ ) (for example the subgroupgenerated by e, f and the p th powers of the x i and y j ). It is proved in [GS, § T ( λ ) are pairwise non-commensurable. It follows thatthe groups T ∗ ( λ ) are pairwise non-isomorphic.Note that T ( λ ) is strictly f.p. when λ is a rational p -adic integer a/b ( a, b ∈ Z , p ∤ b ): the relation involving λ is equivalent to[ y , x ] b e b = f a , [ x , y ] central(because we have unique extraction of b th roots).8. 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