aa r X i v : . [ m a t h . G R ] J un On the structure of just infinite profinite groups
Colin ReidSchool of Mathematical SciencesQueen Mary, University of LondonMile End Road, London E1 [email protected] 4, 2018
Abstract
A profinite group G is just infinite if every closed normal subgroup of G is of finite index. We prove that an infinite profinite group is just infiniteif and only if, for every open subgroup H of G , there are only finitelymany open normal subgroups of G not contained in H . This extendsa result recently established by Barnea, Gavioli, Jaikin-Zapirain, Montiand Scoppola in [1], who proved the same characterisation in the case ofpro- p groups. We also use this result to establish a number of features ofthe general structure of profinite groups with regard to the just infiniteproperty. Keywords : Group theory; profinite groups; just infinite groups NB Throughout this paper, we will be concerned with profinite groups astopological groups. As such, it will be tacitly assumed that subgroups are re-quired to be closed, that homomorphisms are required to be continuous, andthat generation refers to topological generation. When we wish to suppress topo-logical considerations, the word ‘abstract’ will be used, for instance ‘abstractsubgroup’.
Definitions
A profinite group G is just infinite if it is infinite, and every non-trivial normal subgroup of G is of finite index. Say G is hereditarily just infinite if every open subgroup of G is just infinite, including G itself.The main theorem of this paper is the following: Theorem A (Generalised obliquity theorem) . Let G be an infinite profinitegroup. Then the following are equivalent:(i) G is just infinite;(ii) The set K H = { K E o G | K H } is finite for every open subgroup H of G ;(iii) there exists a family F of open subgroups of G with trivial intersection,such that K H is a finite set for every H ∈ F . G is a pro- p group.It follows from the theorem that just infinite profinite groups have finitelymany open subgroups of given index. This is trivial in the case of just infinitevirtually pro- p groups, since they are always finitely generated; but a just infi-nite profinite group need not be finitely generated in general.To prove results about just infinite profinite groups, we will make use ofnormal Frattini subgroups, as defined below. Similar methods are used byZalesskii in [9] to give conditions under which a profinite group has a just infiniteimage, but the idea is also useful for studying just infinite profinite groupsthemselves. Definitions
The normal Frattini subgroup Φ ⊳ ( G ) of a profinite group G isthe intersection of all maximal closed normal subgroups, or equivalently theintersection of the open normal subgroups N such that G/N is simple. Notethat if G is non-trivial then Φ ⊳ ( G ) is necessarily a proper subgroup. We say G is a Φ ⊳ -group if Φ ⊳ ( G ) has finite index in G ; say G is a hereditary Φ ⊳ -group ifevery open subgroup of G is a Φ ⊳ -group.In this paper we will give a description of profinite groups G for whichΦ ⊳ ( G ) = 1 (Lemma 2.2 (ii)), from which it will follow that every just infiniteprofinite group is a hereditary Φ ⊳ -group. Definitions
Given a profinite group G , define Φ ⊳ n ( G ) by Φ ⊳ ( G ) = G andthereafter Φ ⊳ ( n +1) ( G ) = Φ ⊳ (Φ ⊳ n ( G )). Define the Φ ⊳ -height of a finite groupto be the least n such that Φ ⊳ n ( G ) = 1.Let X be a class of finite groups. The X -residual O X ( G ) of a profinite group G is the intersection of all open normal subgroups N such that G/N ∈ X . Say G is residually- X if O X ( G ) = 1. Given a hereditary Φ ⊳ -group G , note thatΦ ⊳ n ( G ) = O X ( G ), where X is the class of finite groups of Φ ⊳ -height at most n , and that O X ( G ) is therefore an open subgroup of G . The X -radical O X ( G )is the subgroup generated by all subnormal X -subgroups. A radical of G is asubgroup which is the X -radical of G for some class X .Given a profinite group G and subgroup H , we define the oblique core Ob G ( H ) and strong oblique core Ob ∗ G ( H ) of H in G as follows:Ob G ( H ) := H ∩ \ { K E o G | K H } Ob ∗ G ( H ) := H ∩ \ { K ≤ o G | H ≤ N G ( K ) , K H } Note that Ob G ( H ) and Ob ∗ G ( H ) have finite index in H if and only if the relevantintersections are finite.As motivation for the term ‘generalised obliquity’, we recall the followingdefinition: 2 efinition [Klaas, Leedham-Green, Plesken [5]] Let G be a pro- p group forwhich each lower central subgroup is an open subgroup. Then the i -th obliquity of G is given as follows (with the obvious convention that log p ( ∞ ) = ∞ ): o i ( G ) := log p ( | γ i +1 ( G ) : Ob G ( γ i +1 ( G )) | ) . The obliquity of G is given by o ( G ) := sup i ∈ N o i ( G ).It is an immediate consequence of Theorem A that if G is a pro- p group suchthat each lower central subgroup is an open subgroup, then G is just infinite ifand only if o i ( G ) is finite for every i .Theorem A also gives a characterisation of the hereditarily just infinite prop-erty: an infinite profinite group G is hereditarily just infinite if and only ifOb ∗ G ( H ) has finite index for every open subgroup H of G . Definitions
Given a profinite group G , let I ⊳ n ( G ) denote the intersection ofall open normal subgroups of G of index at most n . Now define OI n ( G ) to beOb G (I ⊳ n ( G )), and OI ∗ n ( G ) to be Ob ∗ G (I ⊳ n ( G )). We thus obtain functions ob G and ob ∗ G from N to N ∪ {∞} defined byob G ( n ) := | G : OI n ( G ) | ; ob ∗ G ( n ) := | G : OI ∗ n ( G ) | . These are respectively the generalised obliquity function or ob -function and the strong generalised obliquity function or ob ∗ -function of G . Given a function η from N to N , let O η denote the class of profinite groups for which ob G ( n ) ≤ η ( n )for every n ∈ N , and let O ∗ η denote the class of profinite groups for whichob ∗ G ( n ) ≤ η ( n ) for every n ∈ N .These functions give characterisations of the just infinite property and thehereditarily just infinite property in terms of finite images, as described in thefollowing: Theorem B.
Let G be a profinite group. Let C η indicate either O η or O ∗ η .Then the following are equivalent:(i) G is finite or a J -group, where J is the class of just infinite groups if C η = O η , or the class of hereditarily just infinite groups if C η = O ∗ η ;(ii) There is some η for which G is a C η -group;(iii) There is some η for which every image of G is a C η -group;(iv) There is some η , and some family of normal subgroups { N i | i ∈ I } ,such that each G/N i is a C η -group and such that G ∼ = lim ←− G/N i .Moreover, (ii), (iii) and (iv) are equivalent for any specified η . Remark
In more specific contexts, the subgroups I ⊳ n ( G ) in the definition of(strong) generalised obliquity functions can be replaced with various other char-acteristic open series, and Theorem B would remain valid, with essentially thesame proof. For instance, in case of pro- p groups, one could use the lower centralexponent- p series, and in the case of prosoluble groups with no infinite solubleimages, one could use the derived series.3he definitions of ob-functions and ob ∗ -functions lead to the following gen-eral question: Question 1.
Which functions from N to N can occur as ob -functions or ob ∗ -functions for (hereditarily) just infinite profinite groups? What growth rates arepossible? As a straightforward example, consider the significance of linear growth ofthe ob-function, given a pro- p group G . Proposition.
Let G be a pro- p group. The following are equivalent:(i) G is either Z p or has finite obliquity;(ii) there is a constant k such that ob G ( n ) ≤ kn for all n . Another example is that of self-reproducing branch groups, in the sense of[4]; in particular, such a group G has an open subgroup that is isomorphic toa direct product of copies of G . Here, self-similarity properties can be used toobtain a bound on the growth rate of the obliquity function. Proposition.
Let G be a just infinite profinite branch group that is self-reproducingat some vertex (see later). Then there is a constant c such that ob G ( n ) ≤ c n for all n . We also consider how the ob-function and ob ∗ -function of a just infiniteprofinite group G relate to those of its open normal subgroups. Theorem C.
Let G be a just infinite profinite group, and let H be a subgroupof G of index h .(i) The following inequality holds for sufficiently large n : h − ob G ( hn ) ≤ ob H ( n ) . (ii) Let t = | G : Core G ( H ) | . The following inequality holds for all n : ob ∗ H ( n ) ≤ h − ob ∗ G ( tn h ) . (iii) For a given n , let I n be the set of subgroups of G containing I ⊳ n ( G ) .The following inequality holds: ob ∗ G ( n ) ≤ Y L ∈I n | G : L | ob L ( n ) . Another consequence of generalised obliquity concerns non-abelian normalsections of a just infinite profinite group G . In sharp contrast to the case ofabelian normal sections, a given isomorphism type of non-abelian finite groupcan occur only finitely many times as a normal section of G . In fact, more canbe said here. Theorem D.
Let G be a just infinite profinite group.(i) Let F be a class of non-abelian finite groups, and let A be the class ofgroups A satisfying Inn( F ) ≤ A ≤ Aut( F ) for some F ∈ F . Suppose there areinfinitely many pairs ( M, N ) of normal subgroups of G such that N ≤ M and M/N ∈ F . Then either G is residually- A , or it has an open normal subgroup hat is residually- F . In particular, at least one of A and F contains groups ofarbitrarily large Φ ⊳ -height, and hence F must contain infinitely many isomor-phism classes.(ii) Suppose G is not virtually abelian, and let H be a proper open nor-mal subgroup of G . Then G has only finitely many normal subgroups K suchthat K/ Φ ⊳ ( K ′ ) ∼ = H/ Φ ⊳ ( H ′ ) . In particular, G has only finitely many normalsubgroups that are isomorphic to H . Definition
We say a profinite group G is index-unstable if it has a pair ofisomorphic open subgroups of different indices, and index-stable otherwise.Any just infinite virtually abelian profinite group G is virtually a free abelianpro- p group for some p , and hence G is index-unstable. On the other hand, givenpart (ii) of Theorem D it seems to be difficult to construct just infinite profinitegroups which are index-unstable but not virtually abelian. We consider thefollowing question: Question 2.
Let G be a (hereditarily) just infinite profinite group which isindex-unstable. Is G necessarily virtually abelian? This question is also motivated by the study of commensurators of profinitegroups, in the sense of Barnea, Ershov and Weigel ([2]). The relevant definitionswill be recalled briefly later, but the reader should consult [2] for a detailedaccount. Although Question 2 remains open, we do obtain some results in thisdirection. In particular, the following will be shown:
Theorem E.
Let G be a just infinite profinite group.(i) Suppose there is a proper open subgroup H of G isomorphic to G itself.Then G is virtually abelian.(ii) Suppose G has infinitely many distinct radicals. Then G is index-stable. Remarks If G is just infinite profinite group that is virtually abelian, it caneasily be shown that G is isomorphic to a proper open subgroup of itself if andonly if G is a split extension of a free abelian pro- p group for some p .Part (ii) of Theorem E is not vacuous, as there are certainly just infiniteprofinite groups that have infinitely many distinct radicals. For instance, Ershov([3]) has proved that for p >
3, the Nottingham group J p over the field of p elements satisfies Comm( J p ) ∼ = Aut( J p ); since J p is hereditarily just infinite, itfollows that every characteristic subgroup of J p is a radical. Definitions
The finite radical
Fin( G ) of a profinite group G is the union ofall finite normal subgroups of G . This is an abstract subgroup of G , though itneed not be closed. 5 emma 2.1 (Wilson, Corollary 3.8 (a) of [8]) . Let G be a just infinite profinitegroup. Then G has no non-trivial finite subnormal subgroups. Lemma 2.2. (i) Let G be a profinite group, and let H E G . Then Φ ⊳ ( H ) ≤ Φ ⊳ ( G ) .(ii) Let G be a profinite group such that Φ ⊳ ( G ) is trivial. Then G is aCartesian product of elementary abelian groups and non-abelian finite simplegroups. In particular, Fin( G ) is dense in G .Proof. (i)It suffices to show that Φ ⊳ ( H ) is contained in every normal subgroup N of G such that G/N is simple; let N be such a subgroup. Then HN/N isa normal subgroup of
G/N , so either
HN/N = G/N or HN/N = 1. In theformer case, H ∩ N is a normal subgroup of H such that H/ ( H ∩ N ) ∼ = HN/N is simple, so that Φ ⊳ ( H ) ≤ H ∩ N ; in the latter case, H ≤ N .(ii) Let A be the set of open normal subgroups N such that G/N is non-abelian simple, and let A = T A . Let B be the set of open normal sub-groups N such that G/N is cyclic of prime order, and let B = T B . Then A ∩ B = Φ ⊳ ( G ) = 1. Also G = BN whenever N ∈ A , as G/BN is an imageof both an abelian group
G/B and a perfect group
G/N . Hence G = AB bycompactness, and so G ∼ = A × B .It follows that A ∼ = G/B is abelian, and hence is a Cartesian product of itsSylow subgroups. Every finite image of
G/B has squarefree exponent, so itsSylow subgroups are all elementary abelian.It also follows that B ∼ = G/A , and so every finite image of B is a subdirectproduct of non-abelian simple groups; but every finite group that is a subdirectproduct of non-abelian simple groups is isomorphic to a direct product of non-abelian simple groups. Hence every finite image of B is isomorphic to a directproduct of non-abelian simple groups. By a standard inverse limit argument,it follows that B is isomorphic to a Cartesian product of non-abelian simplegroups. Corollary 2.3. (i) Let G be a profinite group, such that every open normalsubgroup of G is a Φ ⊳ -group. Then every open subgroup of G is a Φ ⊳ -group.(ii) Let G be a just infinite profinite group. Then G is a hereditary Φ ⊳ -group.Proof. (i) Let H be an open subgroup of G , and let K be the core of H in G .Then K is an open normal subgroup of G , so Φ ⊳ ( K ) has finite index in K andhence in G . Now Φ ⊳ ( K ) ≤ Φ ⊳ ( H ) by the lemma, so Φ ⊳ ( H ) has finite index in H . (ii) By part (i) it suffices to consider an open normal subgroup H of G . Thisensures Fin( H ) = 1, so Φ ⊳ ( H ) > ⊳ ( H ) isof finite index in H , since it is characteristic in H and hence normal in G .Say a set N of open normal subgroups of a profinite group G is upward-closed if, given any open normal subgroups N and N of G such that N ∈ N and N ≤ N , then N ∈ N . 6 emma 2.4. Let G be a hereditary Φ ⊳ -group, and let K be an infinite upward-closed set of normal subgroups of G . Then there is an infinite descending chain K > K > . . . of open normal subgroups of G , such that K i ∈ K for all i .Proof. Every closed normal subgroup of infinite index is contained in infinitelymany open normal subgroups, and so we may assume K consists of open normalsubgroups. Define a directed graph Γ with vertex set K as follows: place an ar-row from K to K if K < K , and there is no K ∈ K such that K < K < K .Let K ∈ K . If there is an arrow from K to another vertex K , then K/K is characteristic-simple by the maximality property of K in K , so K containsΦ ⊳ ( K ). So given K , there are finitely many possibilities for K , correspondingto some of the sections of the finite group K/ Φ ⊳ ( K ). Hence each vertex of Γhas finite outdegree, and clearly any vertex can be reached from the vertex G ,so Γ has an infinite directed path by K˝onig’s lemma; this gives the requireddescending chain. Lemma 2.5.
Let G be a compact topological group, and let O be an open neigh-bourhood of in G . Let K > K > . . . be a descending chain of closed normalsubgroups of G such that K i O for every i ∈ I . Let K be the intersection ofthe K i . Then K O ; in particular, K is non-trivial.Proof. Let C i = K i ∩ ( G \ O ). Then each C i is closed and non-empty, and hencethe intersection of finitely many C i is non-empty, since the C i form a descendingchain. Since G is compact, it follows that the intersection K ∩ ( G \ O ) of all the C i is non-empty. Hence K O . Proof of Theorem A.
Suppose K H is infinite for some H ≤ o G . Then K H isupward-closed, so by Lemma 2.4 there is an infinite descending chain K >K > . . . of open normal subgroups occurring in K H for which K i H . Nowapply Lemma 2.5, to conclude that the intersection K of these K i is a non-trivialnormal subgroup of infinite index. Hence G is not just infinite, demonstratingthat (i) implies (ii).Clearly (ii) implies (iii), so it now suffices to show (iii) implies (i). Assume(iii), and let K be a non-trivial closed normal subgroup of G . Then thereis an element H of F which does not contain K . It follows that K , being theintersection of the open normal subgroups of G containing K , is the intersectionof some open normal subgroups not contained in H . All such subgroups containOb G ( H ), which is of finite index, since it is the intersection of a finite set ofopen normal subgroups of G . Hence K ≥ Ob G ( H ), and hence K is open in G ,proving (i). Corollary 2.6.
Let G be a just infinite profinite group.(i) Let n be an integer. Then G has finitely many open subgroups of index n .(ii) Let H be an infinite profinite group such that every finite image of H isisomorphic to some image of G . Then G ∼ = H .Proof. (i) Since a subgroup of index n has a core of index at most n !, and a nor-mal subgroup of finite index can only be contained in finitely many subgroups, it7uffices to consider normal subgroups. Suppose G has an open normal subgroup K of index n . Then K does not contain any open normal subgroup of G index n , other than itself. Hence by the theorem, the set of such subgroups is finite.(ii) By part (i), G has finitely many open normal subgroups of any givenindex. It is shown in [7] that in this situation, given any profinite group H such that every finite image of H is isomorphic to an image of G , then H isisomorphic to an image of G . Hence there is some N ⊳ G such that G/N ∼ = H ;since H is infinite and G is just infinite, N = 1. Corollary 2.7.
Let G be an infinite profinite group. Then G is hereditarily justinfinite if and only if Ob ∗ G ( H ) has finite index for every open subgroup H of G .Proof. Suppose G has an open subgroup H which is not just infinite. Then bythe theorem, there is an open subgroup R of H which fails to contain infinitelymany normal subgroups of H , and so Ob ∗ G ( R ) has infinite index.Conversely, suppose G is hereditarily just infinite. Let H be an open sub-group of G , and let H be the set of subgroups of G containing H ; then H isfinite. Let K be a subgroup of G such that H ≤ N G ( K ) but K H . Let L = HK . Then L is just infinite and K is a normal subgroup of L not contain-ing H . Hence Ob ∗ G ( H ) contains T M ∈H Ob M ( H ); by the theorem each Ob M ( H )has finite index, and hence Ob ∗ G ( H ) has finite index. Lemma 3.1.
Let G be a profinite group, and let n be a positive integer.(i) Let N be a normal subgroup of G . Then: I ⊳ n ( G ) N/N ≤ I ⊳ n ( G/N );OI n ( G ) N/N ≤ OI n ( G/N );OI ∗ n ( G ) N/N ≤ OI ∗ n ( G/N ) . (ii) Let I be a directed set, and let N = { N i | i ∈ I } be a family of normalsubgroups of G with trivial intersection, such that N i < N j whenever i > j . Let π i be the quotient map from G to G/N i . Then: I ⊳ n ( G ) = \ i ∈ I π − i (I ⊳ n ( G/N i ));OI n ( G ) = \ i ∈ I π − i (OI n ( G/N i ));OI ∗ n ( G ) = \ i ∈ I π − i (OI ∗ n ( G/N i )) . roof. Let L = I ⊳ n ( G ), let M/N = I ⊳ n ( G/N ), and let M i /N i = I ⊳ n ( G/N i ).(i) If H/N is a normal subgroup of index at most n in G/N , then H also hasindex at most n in G . This proves the first inequality, in other words L ≤ M .If H/N is a normal subgroup of
G/N not contained in
M/N , then H is alsonot contained in M and hence not in L . This proves the second inequality.If H/N is a subgroup of
G/N that is normalised by
M/N but not containedin it, then H is also normalised by but not contained in M , and hence alsonormalised by but not contained in L . This proves the third inequality.(ii) Given part (i), it suffices to show for each equation that the left-handside contains the right-hand side.If H is a normal subgroup of G index at most n , then there is some N i contained in H , which means that M i is contained in H , since H/N i has indexat most n in G/N i . This proves the first equation, in other words L = T i ∈ I M i .If H is a normal subgroup of G not contained in L , then there is some M i that does not contain H , by the first equation. This proves the second equation.Let H be an open subgroup of G that is normalised by L but not containedin it. Then HL is an open subgroup of G which contains T i ∈ I M i . By Lemma2.5, this means that there is some M i contained in HL , which implies that this M i normalises H . By the first equation, there is some M j not containing H .Now take M k ≤ M i ∩ M j , and note that Ob ∗ G/N k ( M k /N k ) is contained in H .This proves the third equation. Proof of Theorem B.
We give the proof only for C η = O η , as the proof for C η = O ∗ η is entirely analogous, with ob ∗ in place of ob and Ob ∗ in place of Ob.Clearly (iii) implies both (ii) and (iv). It is clear from the lemma that (ii)implies (iii), and that (ii) and (iv) are equivalent. These implications hold forany specified η . So it remains to show that (i) and (ii) are equivalent.Suppose (i) holds. Then G has finitely many normal subgroups of index n for any integer n , so I ⊳ n ( G ) has finite index. It follows by Corollary 2.7 thatOb G (I ⊳ n ( G )) also has finite index, so ob G ( n ) is finite. This implies (ii) by taking η = ob G .Suppose (i) is false. Then by Corollary 2.7, there is an open subgroup H of G such that Ob G ( H ) has infinite index in G . Now H has index h say, sothat I ⊳ h ( G ) ≤ H . It follows that OI h ( G ) must be contained in Ob G ( H ), and soob G ( h ) = | G : OI h ( G ) | = ∞ . This implies that (ii) is also false.As an illustration, consider a pro- p group G of finite obliquity o . As men-tioned in [1], this also implies that there is some constant w such that | γ i ( G ) : γ i +1 ( G ) | ≤ w . It is proved in [1] that the condition of finite obliquity is equiva-lent to the following:There exists a constant c such that for every normal subgroup N of G , andfor every normal subgroup M not contained in N , we have | N : N ∩ M | ≤ p c .Lower and upper bounds for ob G can easily be derived in terms of these9nvariants, from which follows a characterisation of the pro- p groups G for whichob G is bounded by a linear function. Proposition 3.2. (i) The ob -function of Z p is given by ob Z p ( n ) = p k , where k is the largest integer such that p k ≤ n . In particular ob Z p ( n ) ≤ n for all n .(ii) Let G be a pro- p group of finite obliquity, with invariants as describedabove. Then ob G ( p n ) ≤ p n + c + w + o − for all n . In particular, there is a constant k such that ob G ( n ) ≤ kn for all n .(iii) Let G be a pro- p group for which there is a constant k such that ob G ( n ) ≤ kn for all n . Then either G ∼ = Z p , or G has obliquity at most log p ( k ) .Proof. (i) This is immediate from the definitions.(ii) Let N be a normal subgroup of G of index at most p n . Then N is notproperly contained in any lower central subgroup that has index at least that of N ; the first such, say γ r ( G ), has index at most p n + w − . Hence I ⊳ p n ( G ) containsOb G ( γ r ( G )), which has index at most p n + w + o − .Now let M be a normal subgroup of G not contained in I ⊳ p n ( G ). Then M is not contained in some normal subgroup K of index at most p n . Hence M properly contains a normal subgroup M ∩ K of G of index at most p n + c . Inparticular, M is of index at most p n + c − , so contains I ⊳ p n + c − ( G ). Thus OI p n ( G )contains I ⊳ p n + c − ( G ), a subgroup of index at most p n + c + w + o − .(iii) By Theorem B, G is finite or just infinite. We may assume G is not Z p , which ensures that all lower central subgroups are open (see [1]). Let H bea lower central subgroup, of index h say. Then H contains I ⊳ h ( G ), so Ob G ( H )contains OI h ( G ), which in turn is a subgroup of G of index at most kh . Hence | H : Ob G ( H ) | is at most k .We now consider profinite branch groups. The definitions given here aremostly based on those of Grigorchuk in [4], which should be consulted for a moredetailed account, and for constructions of such groups (including the group nowgenerally known as the profinite Grigorchuk group). Definitions A rooted tree T is a tree with a distinguished vertex, labelled ∅ .We require each vertex to have finite degree, though the tree itself will be infinitein general. The norm | u | of a vertex u is the distance from ∅ to u ; the n -th layer is the set of vertices of norm n . Denote by T [ n ] the subtree of T induced by thevertices of norm at most n ; by our assumptions, T [ n ] is finite for every n . WriteAut( T ) for the (abstract) group of graph automorphisms of T that fix ∅ . ThenAut( T ) also preserves the norm, and so there are natural homomorphisms fromAut( T ) to Aut( T [ n ] ), with kernel denoted St Aut( T ) ( n ), the n -th level stabiliser .Declare the level stabilisers to be open; this generates a topology on Aut( T ),turning Aut( T ) into a profinite group.10 efinitions Let G be a subgroup of Aut( T ). Then G is said to act sphericallytransitively if it acts transitively on each layer. Given a vertex v , write T v forthe rooted tree with root v induced by the vertices descending from v in T .Define U Gv to be the group of automorphisms of T v induced by the stabiliser of v in G , and define L Gv to be the subgroup of G that fixes v and every vertex of T outside T v . Note that if G acts spherically transitively, the isomorphism typesof U Gv and L Gv depend only on the norm of v ; also, there are natural embeddings L Gv × · · · × L Gv k ≤ St G ( n ) ≤ U G [ n ] := U Gv × · · · × U Gv k , where v , . . . , v k are all the vertices at a given level. Now G is a branch group if G acts spherically transitively and | U G [ n ] : L Gv × · · · × L Gv k | is finite for all n . Say G is self-reproducing at v if there is an isomorphism from T to T v that inducesan isomorphism from G to U Gv . (The definition of self-reproducing given in [4]is that this should hold at every vertex.) Proposition 3.3.
Let G be a just infinite profinite branch group acting on therooted tree T , such that G is self-reproducing at some vertex v . Then there is aconstant c such that ob G ( n ) ≤ c n for all n .Proof. Since G is a just infinite profinite group, and the subgroups St G ( n ) areall open in G , we can define a function f from N to N by the property thatOb G (St G ( n )) contains St G ( n + f ( n )) but not St G ( n + f ( n ) − n .Suppose | v | = k , and consider a normal subgroup K of G not contained inSt G ( k + n ). If K is not contained in St G ( k ), then it contains St G ( k + f ( k )).Otherwise, there is some vertex u of norm k such that K acts non-trivially on( T u ) n ; since G is spherically transitive, we may take u = v . This means that K/ St K ( T v ) contains Ob V (St V ( n )), where V = U Gv ; since V ∼ = G as groups oftree automorphisms, we have in turn Ob V (St V ( n )) ≥ St V ( n + f ( n )). Since K isnormal in G , it follows that K induces all automorphisms of T occurring in G that fix the layers up to k + n + f ( n ), and hence K contains St G ( k + n + f ( n )).Thus Ob G (St G ( k + n )) contains St G ( k + f ( k )) ∩ St G ( k + n + f ( n )), which meansthat f ( k + n ) ≤ max { f ( n ) , f ( k ) } . By induction on n , this implies f ( n ) ≤ r forall n , where r = max ≤ i ≤ k f ( i ).Let N be a normal subgroup of index at most n , where n ≥
2. Let l ( n ) bethe greatest integer such that St G ( l ( n )) has index less than n . Then N is notproperly contained in St G ( l ( n ) + 1), so it contains St G ( l ( n ) + 1 + f ( l ( n ) + 1)),and hence Ob G ( N ) contains St G ( l ( n ) + 1 + 2 f ( l ( n ) + 1)), which in particularcontains St G ( l ( n ) + 2 r + 1). Hence OI n ( G ) contains St G ( l ( n ) + 2 r + 1). Thismeans that ob G ( n ) ≤ | G : St G ( l ( n )) || St G ( l ( n )) : St G ( l ( n ) + 2 r + 1) |≤ n | St G ( l ( n )) : St G ( l ( n ) + 2 r + 1) | . By applying the self-reproducing property of G repeatedly, we obtain an em-bedding St G ( l ( n ))St G ( l ( n ) + 2 r + 1) ֒ → St G ( t )St G ( t + 2 r + 1) × · · · × St G ( t )St G ( t + 2 r + 1) , where t is the integer in the interval (0 , k ] such that l ( n ) ≡ t modulo k , andthe direct factors on the right correspond to the vertices of T of norm l ( n )11escending from a given vertex of norm t . Since G is spherically transitive,there are less than n vertices of T of norm l ( n ), so thatob G ( n ) ≤ n ( max Let G be a profinite group, and let H be a subgroup of G of index h . Then:(i) I ⊳ n ( G ) ≥ I ⊳ n ( H ) ≥ I ⊳ tn h ( G ) , where t = | G : Core G ( H ) | ;(ii) If G is just infinite, then I ⊳ hn ( G ) ≥ I ⊳ n ( H ) for sufficiently large n .Proof. (i) If K is a normal subgroup of G of index at most n , then H ∩ K isa normal subgroup of H of index at most n . On the other hand, let L be anormal subgroup of H of index at most n . Then M = L ∩ Core G ( H ) has indexat most n in Core G ( H ), and M has at most h conjugates in G , all of which arecontained in Core G ( H ), so that Core G ( M ) has index at most n h in Core G ( H ),and hence index at most tn h in G . Thus every normal subgroup of H of indexat most n contains a normal subgroup of G of index at most tn h .(ii) If G is just infinite, there is some integer m such that H contains everynormal subgroup of G of index at least hm ; furthermore, there is some m such that any normal subgroup of G of index less than hm contains a normalsubgroup of G of index at least hm , but at most hm . The claimed equalityholds for any n ≥ m . Proof of Theorem C. For part (i), we may assume that n is large enough thatOb G ( H ) ≥ I ⊳ hn ( G ) ≥ I ⊳ n ( H ). The claimed inequalities are demonstrated by therelationships between subgroups given below:OI hn ( G ) = I ⊳ hn ( G ) ∩ \ { N E o G | N I ⊳ hn ( G ) }≥ I ⊳ hn ( G ) ∩ \ { N E o H | N I ⊳ hn ( G ) } ∩ Ob G ( H ) ≥ I ⊳ n ( H ) ∩ \ { N E o H | N I ⊳ n ( H ) } = OI n ( H ) . OI ∗ n ( H ) = I ⊳ n ( H ) ∩ \ { L ≤ o H | I ⊳ n ( H ) ≤ N H ( L ) , L I ⊳ n ( H ) }≥ I ⊳ tn h ( G ) ∩ \ { L ≤ o G | I ⊳ tn h ( G ) ≤ N G ( L ) , L I ⊳ tn h ( G ) } = OI ∗ tn h ( G ) . OI ∗ n ( G ) = I ⊳ n ( G ) ∩ \ { H ≤ o G | I ⊳ n ( G ) ≤ N G ( H ) , H I ⊳ n ( G ) }≥ \ L ∈I n I ⊳ n ( L ) ∩ \ L ∈I n \ { H E o L | H I ⊳ n ( L ) } = \ L ∈I n OI ∗ n ( L ) . Isomorphism types of normal sections and opensubgroups Proposition 4.1. Let G be a just infinite profinite group. Let M and N beopen normal subgroups of G such that N ≤ M , and let H be an open subgroupof G , with Core G ( H ) of index h . Then at least one of the following holds:(i) M/N is abelian;(ii) H contains both M and the centraliser of M/N ;(iii) M contains the open subgroup Ob G (Ob G ( H )) , and so | G : M | ≤ ob G (ob G ( h )) .Proof. Assume (i) and (ii) are false. Since M is a normal subgroup of G ,to demonstrate (iii) it suffices to prove that M is not properly contained inOb G ( H ). Let K be the centraliser of M/N in G ; note that since (i) is false, K does not contain M . If H does not contain M , then M contains Ob G ( H ), so wemay assume H contains M . It now follows that H does not contain K , by theassumption that (ii) is false. Since K is normal in G , it must contain Ob G ( H ),and hence Ob G ( H ) cannot contain M . Proof of Theorem D. (i) We may assume that G is not a residually- A group, so O A ( G ) has finite index. Let M and N be normal subgroups such that M/N is a F -group, and let H = C G ( M/N ). Then G/H is a A -group, and hencean image of G/O A ( G ). On the other hand, H does not contain M . By theabove proposition, this means that | G : M | is bounded by a function of G and | G/O A ( G ) | , and hence there are only finitely many possibilities for M . Thismeans that for some open normal subgroup M , there must be infinitely manyimages of M that are F -groups. Hence O F ( M ) is a normal subgroup of G ofinfinite index, and hence trivial, so that M is residually- F .(ii) Since G is not virtually abelian, H ′ is an open normal subgroup of G .Since G is a hereditary Φ ⊳ -group, it follows that Φ ⊳ ( H ′ ) is a proper normalsubgroup of H ′ of finite index, so H/ Φ ⊳ ( H ′ ) is finite and non-abelian. Theresult follows by part (i) applied to F = [ H/ Φ ⊳ ( H ′ )]. Remark Part (i) of the above proposition does not extend to abelian sections.Indeed, given any positive integer n and a prime p , then any just infinite pro- p group has infinitely many abelian normal sections of order p n ; this is clearfor Z p , and for any non-nilpotent pro- p group G one can take suitable sectionsinside γ k ( G ) /γ k ( G ) for any k ≥ n . Corollary 4.2. Let G be a just infinite profinite group. Then G has infinitelymany isomorphism types of open normal subgroup if and only if G is not virtuallyabelian.Proof. Suppose G is virtually abelian. Then G has an open normal subgroup V which is a free abelian pro- p group. Every open subgroup of G contained in V is isomorphic to V , and by Theorem A, all but finitely many open normalsubgroups of G are contained in V . Hence G has only finitely many isomorphism13ypes of open normal subgroup. The converse follows from part (ii) of TheoremD. Recall the definition of index-stability given in the introduction. This prop-erty, or the absence of it, is important for understanding the commensuratorsof just infinite profinite groups, in the sense of [2].A virtual automorphism of a profinite group G is an isomorphism betweentwo open subgroups. Two virtual automorphisms are regarded as equivalent ifthey agree on an open subgroup of G . Composition of virtual automorphismsis defined up to equivalence by composing suitable equivalence class represen-tatives. Under this composition, the set of equivalence classes of virtual auto-morphisms of G forms an abstract group, the commensurator Comm( G ) of G .The commensurator depends only on the commensurability class of G .Let H and K be isomorphic open subgroups of G . Given an isomorphism θ from H to K , write ι ( θ ) for | G : H | / | G : K | . This is clearly invariant underequivalence, so ι ( φ ) is defined for φ ∈ Comm( G ) as ι ( θ ) for any θ representing φ . This defines a function ι from Comm( G ) to the multiplicative group Q × + ofpositive rationals, which we call the index ratio . Let V Z ( G ) be the union of thefinite conjugacy classes of G . In [2], a topology (called the strong topology) isdefined on Comm( G ) so that it becomes a topological group Comm( G ) S , and itis shown that if V Z ( G ) = 1, then Comm( G ) S is locally compact, and the indexratio is in fact the modular function for this topology. For just infinite profinitegroups, it can easily be seen that V Z ( G ) = 1 if and only if G is not virtuallyabelian.So Question 2 from the introduction is equivalent to the following question:Let G be a (hereditarily) just infinite profinite group which is not virtuallyabelian. Is the index ratio of Comm( G ) (or equivalently the modular functionof Comm( G ) S ) necessarily trivial?This appears to be a difficult question in general, in either form. First, somegeneral comments about the index ratio. Lemma 4.3. Let G be a profinite group. Then the index ratio ι is a homomor-phism of abstract groups from Comm( G ) to Q × + . In particular, if G is index-unstable, then for all k ∈ N there exists φ ∈ Comm( G ) such that ι ( φ ) > k .Proof. Let φ, ψ ∈ Comm( G ), and let φ ′ and ψ ′ be representatives of φ and ψ respectively such that the composition φ ′ ψ ′ is defined. Let H be the domain of φ ′ . Then ι ( φψ ) = | G : H || G : H φ ′ ψ ′ | = | G : H || G : H φ ′ | | G : H φ ′ || G : H φ ′ ψ ′ | = ι ( φ ) ι ( ψ ) . The conclusions are now clear.Write H E o G to indicate that H is an open subnormal subgroup of G ofdefect at most 2. 14 emma 4.4. Let G be a profinite group. Let H and K be open subgroups of G , and suppose θ is an isomorphism from H to K . Then there are subgroups H ∗ ≤ H and K ∗ ≤ K , with H ∗ E o G and K ∗ E o G , such that the restriction of θ to H ∗ induces an isomorphism from H ∗ to K ∗ .Proof. Let H be the core of H in G , and let K be its image under θ . Now let K ∗ be the core of K in G , and let H ∗ be its preimage under θ . By construction, K ∗ is normal in G , and hence normal in K . Since θ maps H isomorphicallyto K , this means that H ∗ must be the corresponding normal subgroup of H .But H is normal in G , so H ∗ E o G . Definition Let G be a just infinite profinite group that is not virtually abelian,and let N be an open normal subgroup of G . We define the following invariantof G : j N ( G ) = inf {| G : M | | M ∼ = N, M E o G } inf {| G : M | | M ∼ = N, M E o G } . Clearly, if G is index-stable then j N ( G ) = 1 for all N E o G . In fact, there isa strong converse to this statement. Proposition 4.5. Let G be a just infinite profinite group. Suppose that thereare infinitely many isomorphism types of open normal subgroup N of G forwhich j N ( G ) ≤ k , for some constant k . Then G is index-stable.Proof. By Corollary 4.2, G is not virtually abelian. Suppose G is index-unstable.Then by Lemmas 4.3 and 4.4, there are isomorphic subgroups H and K of G such that | G : H | / | G : K | > k , and such that H E o G and K E o G . Now H contains all but finitely many normal subgroups of G , so all but finitely manyisomorphism types of open normal subgroups of G occur only as subgroups of H . This means that there is a normal subgroup N of G such that j N ( G ) ≤ k ,and such that all normal subgroups of G isomorphic to N are subgroups of H ;take N to be of least possible index. Then N θ E o G , where θ is any isomorphismfrom H to K , and N θ is isomorphic to N . Since N was chosen to be of leastpossible index, it follows that j N ( G ) ≥ ι ( θ ) > k , a contradiction.We now have enough information to prove Theorem E. Proof of Theorem E. (i) Suppose G is not virtually abelian. Then there is anopen subgroup N of H such that N is normal in G but N θ is not normal in G ,as shown in the proof of Proposition 4.5. But then N is normal in H , and so N θ ⊳ H θ = G , a contradiction.(ii) Let N = O X ( G ) for some class of groups X , and suppose N is non-trivial.Let M be a subnormal subgroup of G isomorphic to N . Then by definition, M is generated by its subnormal X -subgroups. But these are then subnormal in G ,and so contained in N . Hence M ≤ N , demonstrating that j N ( G ) = 1. Hencethe non-trivial radicals of G form an infinite set of pairwise non-isomorphic opennormal subgroups N satisfying j N ( G ) = 1. By Proposition 4.5, this ensures that G is index-stable. 15 Acknowledgments This paper is based on results obtained by the author while under the super-vision of Robert Wilson at Queen Mary, University of London (QMUL). Theauthor would also like to thank Charles Leedham-Green for introducing the au-thor to the study of profinite groups, for his continuing advice and guidancein general and for his detailed feedback and corrections concerning this paper;Yiftach Barnea for his advice on the presentation of some of the proofs and forinforming the author about the recent papers [1] and [2]; and Pavel Zalesskii fornotifying the author of [9]. The author acknowledges financial support providedby EPSRC and QMUL for the duration of his doctoral studies. References [1] Y. Barnea, N. Gavioli, A. Jaikin-Zapirain, V. Monti, C.M. Scoppola, Pro- p groups with few normal subgroups, Journal of Algebra 321 (2009), 429–449.[2] Y. Barnea, M. Ershov, T. Weigel, Abstract commensurators of profinitegroups, preprint, http://arxiv.org/abs/0810.2060v1 .[3] M. Ershov, On the commensurator of the Nottingham group, submitted2008.[4] R.I. Grigorchuk, Just Infinite Branch Groups, ch. 4 of New Horizons in Pro- p groups, editors M. du Sautoy, D. Segal, A. Shalev, Birkh¨auser, 2000.[5] G. Klaas, C.R. Leedham-Green, W. Plesken, Linear pro- p groups of finitewidth, Springer-Verlag Berlin Heidelberg, 1997.[6] N. Nikolov, D. Segal, On finitely generated profinite groups. I. Strong com-pleteness and uniform bounds, Ann. of Math. (2) 165 (2007), no. 1, 171–238;On finitely generated profinite groups. II. Products in quasisimple groups,ibid. 239–273.[7] J.S. Wilson, Profinite groups, Clarendon Press, Oxford, 1998.[8] J.S. Wilson, On Just Infinite Abstract and Profinite Groups, ch. 5 of NewHorizons in Pro- pp