aa r X i v : . [ m a t h . GN ] O c t The Chabauty space of Q × p Antoine Bourquin and Alain ValetteOctober 30, 2019
Abstract
Let C ( G ) denote the Chabauty space of closed subgroups of the locallycompact group G . In this paper, we first prove that C ( Q × p ) is a propercompactification of N , identified with the set N of open subgroups withfinite index. Then we identify the space C ( Q × p ) r N up to homeomorphism:e.g. for p = 2 , it is the Cantor space on which 2 copies of N (the 1-pointcompactification of N ) are glued. In 1950, Chabauty [Cha] introduced a topology on the set F ( X ) of closed subsetsof a locally compact space X , turning F ( X ) into a compact space, see Definition1 below; for X discrete, this is nothing but the product topology on 2 X . When G is a locally compact group, the set C ( G ) of closed subgroups of G is a closedsubset of F ( G ), so C ( G ) is a compact set canonically associated with G : we callit the Chabauty space of G . Definition 1.1.
For a locally compact space X , the Chabauty topology on F ( X )has as open sets finite intersections of subsets of the form O K = { F ∈ F ( X ) : F ∩ K = ∅} with K compact in X , and O ′ U = { F ∈ F ( X ) : F ∩ U = ∅} with U open in X .Let us give some examples of Chabauty spaces, first for additive groups ofsome locally compact fields: 1 xample 1.2.
1. For G = R , the Chabauty space C ( R ) is homeomorphic toa closed interval, say [0 , + ∞ ] , with the subgroup λ Z (with λ > ) beingmapped to λ , the subgroup { } being mapped to + ∞ , and the subgroup R being mapped to 0.2. For G = C , the situation is already much more subtle, and it was provedby Hubbard and Pourezza [HP] that C ( C ) is homeomorphic to the 4-sphere.3. For G = Q p , the field of p -adic numbers, every non-trivial closed subgroupis of the form p k Z p for some k ∈ Z , so C ( Q p ) is homeomorphic to the2-point compactification of Z , namely Z ∪ {±∞} , with the subgroup p k Z p being mapped to k ∈ Z , the subgroup { } being mapped to + ∞ and thesubgroup Q p being mapped to −∞ . Let us turn to multiplicative groups of some locally compact fields:
Example 1.3.
1. For G = R × , since G ≃ R × Z / Z , we get 3 types of closedsubgroups: • Closed subgroups of R , contributing a copy of [0 , + ∞ ] . • Subgroups which are products H × Z / Z , with H a closed subgroup of R . They contribute another copy of [0 , + ∞ ] , with origin the subgroup R × Z / Z and extremity the subgroup { } × Z / Z . • Infinite cyclic subgroups which are not contained in R ; those are of theform h ( λ, i , for some λ > . So they contribute a copy of ]0 , + ∞ [ .As h ( λ, i converges to R × Z / Z for λ → , and to { (0 , } for λ → + ∞ ,we see that subgroups of the 3rd type “connect” subgroups of the first andthe second type, so that C ( R × ) is homeomorphic to a closed interval.2. For G = C × ≃ R × T , the structure of C ( G ) was elucidated by Haettel[Hae]: it is path connected but not locally connected, and with uncountablefundamental group. In this paper, we deal with the multiplicative group G = Q × p of the field Q p . We enjoy the general results by Y. Cornulier [Cor] on C ( H ) for H a locallycompact abelian group. Applied to G = Q × p , they yield that C ( G ) is totallydisconnected and uncountable (Theorems 1.5 and 1.6 in [Cor]), that { } is notisolated in C ( G ) while { G } is isolated in C ( G ) (Lemmas 4.1 and 4.2 in [Cor]).More generally, a closed subgroup H defines an isolated point in C ( G ) if and onlyif H is open with finite index in G (Theorem 1.7 in [Cor]).2o describe our main result, recall that a compact metric space Y is a propercompactification of N if Y contains an open, countable, dense, discrete subset N . It is known (see e.g. Propositions 2.1 and 2.3 in [Tsa]) that every non-emptycompact metric space can be written as Y r N , with Y and N as above, in aunique way up to homeomorphism.We will use the following notations: N = N ∪ {∞} is the one-point compact-ification of N , C is the Cantor space, [ k ] is the set { , , ..., k } , and d ( n ) is thenumber of divisors of n . Theorem 1.4.
Let p be a prime. For G = Q × p :1. C ( G ) is a proper compactification of N , viewed as the set N of open sub-groups of finite index.2. For p odd, the space C ( G ) r N is homeomorphic to the space obtained byglueing [ d ( p − × N on C , with the d ( p − accumulation points of [ d ( p − × N being identified to d ( p − pairwise distinct points of C .3. For p = 2 , the space C ( G ) r N is homeomorphic to the space obtainedby glueing [2] × N on C , with the 2 accumulation points of [2] × N beingidentified to 2 pairwise distinct points of C . In the above picture, the isolated points of C ( G ) r N correspond to the closedinfinite subgroups of Z × p , the invertible group of the ring Z p of p -adic integers; and(for p odd) the glueing points are the finite subgroups of Q × p , which are exactlythe cyclic groups C d of d -roots of unity, for d dividing p −
1. It follows from theresult that every non-isolated point of C ( G ) r N is a condensation point, and theCantor-Bendixson rank of C ( G ) r N is 1.To prove Theorem 1.4, we use the decomposition as topological groups Q × p ≃ Z p × C p − × Z (for p odd), and Q × ≃ Z × C × Z (for all this, see section 3in Chapter II of [Ser]). For a locally compact abelian group H with Pontryagindual ˆ H = hom( H, T ), the space C ( H ) identifies with C ( ˆ H ) for H : see Proposition2.5 below for the precise statement of this result of Cornulier [Cor]. So, actuallywe work with the Pontryagin dual of Q × p , which for p odd identifies with C p ∞ × C p − × T , where C p ∞ denotes the p -Pr¨ufer group.At the end of the Introduction in [Cor], Cornulier mentions as a non-trivialquestion to determine the homeomorphism type of C ( G ), when G is a totallydisconnected locally compact abelian group that fits into a short exact sequence0 → A × P → G → D → , A is finitely generated abelian, D is Artinian (i.e. it has no infinite de-creasing chain of subgroups), and P is a finite direct product of finite groupsand groups isomorphic to Z p (for various p ’s, multiplicities allowed). Since Q × p is of that form, Theorem 1.4 and its extension Theorem 4.7 can be seen as acontribution to Cornulier’s question.The structure of the paper is as follows: in section 2, we gather some pre-liminary material about the Chabauty space C ( G ). In section 3, we determinethe Chabauty space of C ( C n × Z ): the Chabauty space of any finitely generatedabelian group has been determined in Theorem C of [CGP2], but we give a directproof for completeness in the case of C n × Z ; this will be needed in the proofof our main result. Although we do not need it, we see that the determinationof the Chabauty space of Z follows easily from the one of C n × Z . Finally, insection 4 we prove Theorem 1.4 along the lines sketched above. Acknowledgements:
We thank Y. Cornulier and P. de la Harpe for useful com-ments on a first draft of this paper.
In this subsection G will denote a locally compact second countable group. First,we recall a result on convergence in C ( G ) that we will use throughout the paperwithout further reference (see [Pau] for a proof) : Proposition 2.1.
A sequence ( H n ) n> in C ( G ) converges to H if and only if1. ∀ h ∈ H , ∃ h n ∈ H n such that h = lim h n .2. For every sequence ( h n k ) k> in G with h n k ∈ H n k , n k +1 > n k , converging tosome h ∈ G , we have h ∈ H . (cid:3) The following is lemma 1.3(1) in [CGP1].
Lemma 2.2.
Let
N ⊳ G be a closed normal subgroup, let p : G → G/N denotethe quotient map. The map p ∗ : C ( G/N ) → C ( G ) : H p − ( H ) is a homeomorphism onto its image. (cid:3) The proof of the next lemma is an easy exercise.4 emma 2.3.
Let
N ⊳ G be a compact normal subgroup. If ( H k ) k> converges to H in C ( G/N ) , then ( H k N ) k> converges to HN in C ( G ) . (cid:3) If L is a closed subgroup of G , the map C ( G ) → C ( L ) : H H ∩ L is ingeneral not continuous (take e.g. G = R , and L a 1-dimensional subspace). Wesingle out a case where it is continuous. Lemma 2.4.
Let G = D × L be a locally compact group, with D discrete. Identify L with { } × L . The map C ( G ) → C ( L ) : H H ∩ L is continuous. Proof:
Assume that the sequence ( H n ) n> converges to H in C ( G ), let uscheck that ( H n ∩ L ) n> converges to H ∩ L in C ( L ). • If (1 , l ) is in H ∩ L , we can write (1 , l ) = lim n →∞ ( d n , l n ), with ( d n , l n ) ∈ H n for every n >
0. Since D is discrete, we have d n = 1 for n large enough, sothat we have also (1 , l ) = lim n →∞ (1 , l n ), with (1 , l n ) ∈ H n ∩ L for every n . • If a sequence ((1 , l n k )) k> , with (1 , l n k ) ∈ H n k ∩ L , converges to (1 , l ) ∈ L ,then by assumption we have (1 , l ) ∈ H , i.e. (1 , l ) ∈ H ∩ L . (cid:3) If G is a locally compact abelian group, recall that ˆ G = hom( G, T ) denote itsPontryagin dual. For H a closed subgroup of G , denote by H ⊥ the orthogonal of H : H ⊥ = { χ ∈ ˆ G : χ | H = 1 } . The following result is due to Cornulier (Theorem 1.1 in [Cor]):
Proposition 2.5.
The orthogonal map C ( G ) → C ( ˆ G ) : H H ⊥ is an inclusion-reversing homeomorphism. (cid:3) The symbol k ≫ for k large enough ”. In this section, G denotes adiscrete group. Let ( P k ) k> , P denote subsets of G . We say that P is locallycontained in ( P k ) k> if, for every finite subset S ⊂ P , we have S ⊂ P k for k ≫ P is locally contained in ( P k ) k> if every finite subset of P iscontained in all but finitely many P k ’s. Lemma 2.6.
Let ( H k ) k> , H be subgroups of G .1. The sequence ( H k ) k> converges to H in C ( G ) if and only if H is locallycontained in ( H k ) k> and G r H is locally contained in ( G r H k ) k> .2. If ( H k ) k> converges in C ( G ) to H and H is finitely generated, then H ⊂ H k for k ≫ . . The sequence ( H k ) k> converges to the trivial subgroup { e } of G , if and onlyif for every finite subset F ⊂ G r { e } , we have F ∩ H k = ∅ for k ≫ . Proof:
1. Assume that ( H k ) k> converges to H . If S is a finite subset of H , then forany g ∈ S we find a sequence ( h k ) k> in G , with h k ∈ H k for every k , suchthat g = lim k →∞ h k . As G is discrete, this means g = h k for k ≫
0, i.e. g ∈ H k for k ≫
0. As S is finite, we have S ⊂ H k for k ≫
0. Similarly, if T is finite and disjoint from H , we must show that T is disjoint from all butfinitely many H k ’s. Suppose not, i.e. T intersects infinitely many H k ’s. As T is finite, we find g ∈ T and a subsequence ( k i ) i> such that g ∈ H k i forevery i >
0. Setting g k i := g , and writing g = lim i →∞ g k i , because ( H k ) k> converges to H this implies g ∈ H , which contradicts T ∩ H = ∅ .Conversely, if H is locally contained in ( H k ) k> and G r H is locally con-tained in ( G r H k ) k> , let us check that ( H k ) k> converges to H . For g ∈ H ,set h k = g so that h k ∈ H k for k ≫
0: so we write g as a limit of elementsin the ( H k ) ′ s . Now, let ( k i ) i> be a subsequence and h k i ∈ H k i such that( h k i ) i> converges to g ∈ G . This means g = h k i for i ≫
0, so that g belongsto infinitely many H k ’s. As G r H is locally contained in ( G r H k ) k> , thisforces g ∈ H , hence lim k →∞ H k = H .2. Apply the previous point to a finite generating S of H : it is contained in H k for k ≫
0, so the same holds for H .3. The condition is equivalent to G r { e } being locally contained in ( G r H k ) k> . (cid:3) Lemma 2.7.
Let G be a finitely generated group.1. Any finite index subgroup defines an isolated point in C ( G ) .2. The index of a subgroup is continuous on C ( G ) . More precisely, the map C ( G ) → N : H [ G : H ] is continuous. Proof:
1. Let H be a finite index subgroup of G and let ( H k ) k> be a sequence in C ( G )converging to H . Let us show that eventually H k = H . As H is finitelygenerated, by Lemma 2.6 we already know that H ⊂ H k for k ≫
0. Butthere are finitely many distinct subgroups containing H , as H has finiteindex. Passing to a subsequence we may assume that H k = K for k ≫ H k ) k> converges both to H and K , so H = K .6. Let ( H k ) k> ⊂ C ( G ) be a sequence converging to H ∈ C ( G ); let us provethat [ G : H k ] → [ G : H ]. We have to consider two cases : • Let assume that [ G : H ] = n < ∞ . By the first part, H is isolated in C ( G ), so H k = H for k ≫
0, and the result follows. • Now assume that [ G : H ] = ∞ . If [ G : H k ] = ∞ for k ≫
0, the resultfollows. Else, we may assume that [ G : H k ] = n k < ∞ . If n k
6→ ∞ ,passing to a subsequence, we may assume that n k = n for all k . As G is finitely generated, the number of subgroups of index n is finite. Bythe pigeonhole principle, we may assume that the sequence ( H k ) k> isconstant. Hence, [ G : H ] < ∞ and this is a contradiction. (cid:3) Let IC ( G ) denote the set of infinite cyclic subgroups of a group G . Lemma 2.8.
Let G = Z d ⊕ F be a finitely generated abelian group, with F afinite abelian group, written additively.1. The closure of IC ( G ) in C ( G ) is IC ( G ) ∪ { (0 , } .2. Let ( g k ) k> be a sequence in G . The sequence h g k i k> converges to { (0 , } if and only if lim k →∞ g k = ∞ in G .3. The topology induced by C ( G ) on IC ( G ) is discrete. Proof:
1. Assume that the sequence h g k i k> in C ( G ) converges to a subgroup H . Weknow that every subgroup of G is finitely generated and by Lemma 2.6, H ⊂ h g k i for k ≫
0. So H is a subgroup of the cyclic infinite group h g k i .So H is either trivial or infinite cyclic.2. This follows immediately from the last part of lemma 2.6.3. Let ( h g k i ) k> ⊂ IC ( G ) be a sequence converging to h g i ∈ IC ( G ). Let usshow that h g k i = h g i for k ≫
0. By Lemma 2.6, we may assume that h g i ⊂ h g k i for k >
0. So, there exists non-zero integers m k ∈ Z suchthat g = g k m k . We may assume that m k > g k by − g k . If m k → ∞ , then g is divisible by infinitely many integers, a contradiction. Sothe m k ’s are bounded and, passing to a subsequence, we may assume that m k = m for all k >
0, i.e. g = mg k . Since the equation g = mx has finitelymany solutions in G , by the pigeonhole principle we may assume that thesequence ( g k ) k> is constant. Because of the convergence of ( h g k i ) k> to h g i , this implies g k ∈ h g i , i.e. m = 1, and g k = g . (cid:3) The case Z × Z /n Z and Z × Z /n Z In this subsection, we set G = Z × Z /n Z . Recalling that the rank of a finitelygenerated group is the minimal number of generators, by viewing G as a quotientof Z we see that every subgroup of G has rank at most 2. Now G has threetypes of subgroups: • Finite subgroups, i.e. subgroups of { } × Z /n Z : there are exactly d ( n ) ofthem, where d ( n ) is the number of divisors of n ; • Infinite cyclic subgroups; • Infinite subgroups of rank 2.We describe the structure of infinite subgroups more precisely.
Lemma 3.1.
Let H be an infinite subgroup of G . Set F := H ∩ ( { } × Z /n Z ) .There exists ( a, b ) ∈ H , with a > , such that H = F ⊕ h ( a, b ) i . In particular, H is cyclic if F is trivial and H has rank 2 if F is non-trivial. Proof:
Let p : G → Z be the projection onto the first factor. As p ( H )is infinite, it is of the form p ( H ) = a Z for some a >
0. Let ( a, b ) ∈ H beany element such that p ( a, b ) = a . Then clearly F ∩ h ( a, b ) i = { (0 , } , and for( x, y ) ∈ H , writing x = ma for some integer m , we get ( x, y ) = m ( a, b )+(0 , y − mb )so that H = F ⊕ h ( a, b ) i . (cid:3) The group G has the feature that every infinite subgroup has finite index, sodefines an isolated point in C ( G ), by Lemma 2.7. By Lemma 2.8, the only accu-mulation point of infinite cyclic groups is the trivial subgroup. So it remains tostudy accumulation points of rank 2 subgroups, which are necessarily finite sub-groups. Every non-trivial finite subgroup H of Z /n Z is the limit of the sequence( k Z × H ) k> of rank 2 subgroups. The converse is provided by: Proposition 3.2.
Let m be a divisor of n . Consider a sequence ( H k ) k> ofinfinite subgroups of rank 2 of G . It converges in C ( G ) to { } × h m i if and onlyif, for k ≫ , there exists g k ∈ G with lim k →∞ g k = ∞ , such that H k is generatedby (0 , m ) and g k . Proof:
For the sufficient condition: if H k = h (0 , m ) , g k i = h (0 , m ) i ⊕ h g k i , aslim k →∞ h g k i = { (0 , } by Lemma 2.8, we have lim k →∞ H k = h (0 , m ) i by Lemma2.3. 8or the necessary condition: assume ( H k ) k> converges to { } × h m i . By Lemma3.1, we have H k = F k ⊕ h g k i with F k = H k ∩ ( { } × Z /n Z ) and g k = ( a k , b k ) with a k >
0. Because of the assumed convergence, we have F k = h (0 , m ) i for k ≫ k →∞ g k = ∞ . (cid:3) Recall that we denote by [ k ] the set { , , ..., k } , and by d ( n ) the number ofdivisors of n . From Proposition 3.2, we get immediately the following special caseof Theorem C in [CGP1]: Corollary 3.3.
The Chabauty space C ( Z × Z /n Z ) is homeomorphic to N × [ d ( n )] ,the accumulation points corresponding to the subgroups { } × h m i , with m adivisor of n . (cid:3) The list of subgroups of G = Z is as follows: • The trivial subgroup { (0 , } ; • Subgroups of rank 1, i.e. infinite cyclic subgroups; • Subgroups of rank 2, i.e. finite index subgroups in G (which define isolatedpoints in C ( G ), by Lemma 2.7).In each infinite subgroup H of G , pick a minimal vector m H (so m H hasminimal norm among all non-zero vectors in H ). From the 3rd part of Lemma2.6, we immediately get: Proposition 3.4.
Let ( H k ) k> be a sequence of infinite subgroups of G . Thissequence converges to { (0 , } in C ( G ) if and only if lim k →∞ k m H k k = + ∞ . (cid:3) It remains to see how a rank 1 subgroup h h i can be a limit in C ( G ). Proposition 3.5.
A sequence ( H k ) k> in C ( G ) converges to the rank 1 subgroup H = h h i if and only if, for k ≫ , there exists g k ∈ G such that H k = h h, g k i andthe sequence ( g k + H ) k> goes to infinity in G/H . Proof:
Write h = ( a, b ), let n > a and b , set p = ( an , bn ),so that p is a primitive vector proportional to h . Let c, d ∈ Z be integers suchthat ad − bc = n , so that, with q = ( c, d ), the set { p, q } is a basis of G , and everyvector in G may be written uniquely αp + βq , for α, β ∈ Z . The map π : Z → ( Z /n Z ) × Z : αp + βq ( α (mod n ) , β )9s then a surjective homomorphism with kernel H .If ( H k ) k> is a sequence of subgroups converging to H , we have H ⊂ H k for k ≫
0, by Lemma 2.6. So to study convergence to H , we may as wellassume that H ⊂ H k for every k >
0. By Lemma 2.2, such a sequence ( H k ) k> converges to H if and only if the sequence ( π ( H k )) k> converges to the trivialsubgroup in C (( Z /n Z ) × Z ). By Proposition 3.2, this happens if and only if,for k ≫
0, the subgroup π ( H k ) is infinite cyclic, say π ( H k ) = π ( h g k i ) for some g k ∈ π − π ( H k ) = H k , with the property that lim k →∞ π ( g k ) = ∞ . This concludesthe proof. (cid:3) As a consequence, we get another special case of Theorem C in [CGP1]:
Corollary 3.6.
The Chabauty space C ( Z ) is homeomorphic to N . (cid:3) Let p be a prime. Recall that C k denotes the cyclic group of order k (viewed asthe group of k -th roots of 1 in T ). It is classical (see [Ser]) that Q × p ≃ Z p × C p − × Z ( p odd); Q × ≃ Z × C × Z . Hence by Pontryagin duality c Q × p ≃ C p ∞ × C p − × T ( p odd); c Q × ≃ C ∞ × C × T where C p ∞ = ∪ ∞ ℓ =1 C p ℓ denotes the Pr¨ufer p -group. By Proposition 2.5, Q × p and c Q × p have canonically isomorphic Chabauty spaces. We choose to work with c Q × p as it is a 1-dimensional Lie group. More generally, for k > G p,k := C p ∞ × C k × T and we aim to determine C ( G p,k ).We will need some notation. We will denote by π , π , π the projections of G p,k onto C p ∞ (resp. C k , resp. T ). Set also π = ( π , π ) : G p,k → C p ∞ × C k .We first give a list of closed subgroups of G p,k : as it is a 1-dimensional Liegroup, closed subgroups are either discrete, or 1-dimensional. Lemma 4.1.
Every closed 1-dimensional subgroup of G p,k is of the form H (1) D := D × T , where D is any subgroup of C p ∞ × C k . The set of 1-dimensional subgroupsis closed in C ( G p,k ) and identifies with C ( C p ∞ × C k ) via π ∗ (defined as in Lemma2.2). roof: A 1-dimensional subgroup of G p,k has the same connected compo-nent of identity as G p,k , namely { } × { } × T . The second statement followsimmediately from Lemma 2.2. (cid:3) Remark 4.2.
Note that C ( C p ∞ × C k ) is homeomorphic to N × [ d ( k )], with ac-cumulation points corresponding to the subgroups C p ∞ × C d , for d a divisor of k :this follows from Proposition 3.2 by dualizing.We now turn to infinite discrete subgroups of G p,k . For F a finite subgroup of C k × T , we denote by q F : G p,k → G p,k / ( { } × F ) the quotient map. As we willneed homomorphisms C p ∞ → ( C k × T ) /F in Proposition 4.4 below, we start bydescribing those homomorphisms. We denote by T the connected component ofidentity of the 1-dimensional compact abelian Lie group ( C k × T ) /F . We choosesome identification T ≃ T , and we denote by ι : T → ( C k × T ) /F the inclusion. Lemma 4.3.
The map d C ∞ p → hom( C p ∞ , ( C k × T ) /F ) : f ι ◦ f is an isomorphism of compact groups. Proof:
It is enough to see that any homomorphism f : C p ∞ → ( C k × T ) /F takes values in T . First observe that, being a 1-dimensional compact abeliangroup, ( C k × T ) /F is isomorphic to A × T , for some finite abelian group A .Second, C p ∞ is a divisible group, and so is every homomorphic image of C p ∞ .This applies in particular to the projection of f ( C p ∞ ) to the first factor A . Buta finite divisible group must be trivial, hence f ( C p ∞ ) ⊂ T . (cid:3) Proposition 4.4.
For a finite subgroup F of C k × T and a homomorphism f : C p ∞ → ( C k × T ) /F , set H F,f := q − F ( Graph ( f )) . Then H F,f is an infinite discretesubgroup of G p,k and every infinite discrete subgroup is of that form. Proof:
It is clear that H F,f is an infinite discrete subgroup of G p,k . Con-versely, let H be an infinite discrete subgroup. We first claim that π ( H ) = C p ∞ :otherwise we would have that π ( H ) = C p ℓ for some ℓ >
0, so that H appears asa discrete subgroup of the closed subgroup π − ( C p ℓ ) of G p,k : as this is a compactsubgroup, this forces H to be finite, a contradiction.Set then F := ker( π | H ) = H ∩ ( { } × C k × T ): as a discrete subgroup in acompact group, F is finite. We consider two cases. Special case : F is trivial. Then π | H is injective, hence f := ( π , π ) ◦ ( π | H ) − is a homomorphism C p ∞ → C k × T , and H is exactly the graph of f .11 eneral case : Let F be arbitrary. We will reduce to our special case. Weconsider the infinite discrete subgroup q F ( H ) in G p,k / ( { } × F ). By the previousremark, for some divisor d of k , we may identify ( C k × T ) /F with C d × T , hencealso G p,k / ( { } × F ) with G p,d . By construction, the kernel ker( π | q F ( H ) ) is trivial,so we are back to the special case: therefore there exists a homomorphism f : C p ∞ → ( C k × T ) /F such that q F ( H ) = Graph ( f ). Then H = q − F ( q F ( H )) = H F,f as claimed. (cid:3)
Example 4.5. (with k = 1 ) Let F = C p viewed as a subgroup of T . The quotientmap q F identifies with the map T → T : z z p . Let ι denote the inclusion of C p ∞ into T . Then H F,ι = { ( w, z ) ∈ C p ∞ × T : w = z p } . Observe that ι does not lift, i.e. there is no ˜ ι : C p ∞ → T such that ι = q F ◦ ˜ ι . To summarize, from Proposition 4.4 and Lemma 4.1, we have now the com-plete list of closed subgroups of G p,k : • Finite groups; • Discrete infinite subgroups H F,f , as described by Proposition 4.4; • H (1) D , as described by Lemma 4.1.The first part of Theorem 1.4 follows from: Proposition 4.6. C ( G p,k ) is a proper compactification of N , identified with theset N of finite subgroups of G p,k . Proof:
By Theorem 1.7 in [Cor], every finite subgroup defines an isolatedpoint in C ( G p,k ). So N is open and discrete in C ( G p,k ), it remains to show thatit is dense. This follows from: • H (1) D is the limit of the sequence ( D × C n ) n> of finite subgroups. • H F,f is the limit of the sequence ( q − F ( Graph ( f | C pℓ ))) ℓ> , also consisting infinite subgroups. (cid:3) Our aim is now to determine the homeomorphism type of C ( G p,k ) r N . Let C (1) be the set of one-dimensional subgroup, and let C dis be the set of infinite discretesubgroups, so that C ( G p,k ) r N is the disjoint union of C (1) and C dis . By lemma 4.1and the remark following it, C (1) is homeomorphic to N × [ d ( k )], the accumulationpoints corresponding to the subgroups C p ∞ × C d × T , with d a divisor of k . Note12hat the latter subgroup is also the limit of the sequence ( H C d × C n , d,n ) n> where1 d,n denotes the trivial homomorphism C p ∞ → ( C k × T ) / ( C d × C n ).We denote by C fin ( C k × T ) the set of finite subgroups of C k × T . By the dualversion of Corollary 3.3, this is a discrete set in C ( C k × T ). Theorem 4.7.
1. The map α : C fin ( C k × T ) × d C p ∞ → C dis : ( F, f ) H F,f is a homeomorphism.2. The closure of C dis in C ( G p,k ) is the union of C dis with the d ( k ) subgroups C p ∞ × C d × T , with d a divisor of k . Proof: • α is onto, by combining lemma 4.3 with Proposition 4.4. • To show that α is injective, we observe that H F,f determines both F and f : first, F is obtained as the intersection of H F,f with { }× C k × T ;second, q F ( H F,f ) is the graph of f , which of course determines f . • α is continuous. As C fin ( C k × T ) is discrete, it is enough to showthat, for each F ∈ C fin ( C k × T ), the map d C p ∞ → C dis : f H F,f iscontinuous.Let fix F ∈ C fin ( C k × T ) and let ( f n ) n> ⊂ d C p ∞ be a sequence con-verging to f ∈ d C p ∞ . We want to show that H F,f n → H F,f . – Let z ∈ H F,f . We have q F ( z ) = ( a, f ( a )) for a certain a ∈ C p ∞ . As f n → f , we have that ( a, f n ( a )) → ( a, f ( a )). So q − F ( a, f n ( a )) → q − F ( a, f ( a )) in the Chabauty topology of closed subsets. More-over, | F | = | q − F ( a, f n ( a )) | = | q − F ( a, f ( a )) | < ∞ , so we can find z n ∈ H F,f n such that z n → z . – Let z n i ∈ H F,f ni such that z n i → z ∈ G p,k and let us show that z ∈ H F f . We have q F ( z n i ) = ( a n i , f ( a n i )) for a certain a n i ∈ C p ∞ .By discreteness of C p ∞ , convergence of ( z n i ) and continuity of q F ,we may assume that a n i = a for all i >
0. As f n → f , q F ( z n i ) → ( a, f ( a )). This implies that z ∈ q − F ( a, f ( a )) and z ∈ H F,f . • α is open. To show this, it is enough to show that, for every F ∈C fin ( C k × T ), the set { H F,f : f ∈ d C p ∞ } is open. This is in turnimplied by the continuity of the map C dis → C fin ( C k × T ) : H F,f F .Since F can be expressed as H F,f ∩ ( { } × C k × T ), the result followsfrom lemma 2.4. 13. Let ( H F n ,f n ) n> be a sequence in C dis converging to H ∈ C ( G p,k ). If theorders of the F n ’s remain bounded, passing to a sub-sequence we may as-sume that F n = F for n ≫
0. By compactness of d C p ∞ , again passing toa subsequence we may assume that ( f n ) n> converges to f ∈ d C p ∞ . Then( H F n ,f n ) n> converges to H F,f . Assume now that the orders of the F n ’s areunbounded. Passing to a subsequence, we may assume that ( F n ) n> con-verges in C ( C k × T ) to C d × T , for some divisor d of k . By the dualversion of Proposition 3.2, F n contains C d × { } for n ≫
0. For ev-ery ( λ n , z n ) ∈ C d × T , the sequence of cosets (( λ n , z n ) F n ) n> convergesto C d × T in the Chabauty topology of closed subsets of C d × T . Sofor ( w, λ, z ) ∈ C p ∞ × C d × T , choose ( λ n , z n ) in q − F n ( f n ( w )), then choose( λ ′ n , z ′ n ) ∈ F n such that ( λ n λ ′ n , z n z ′ n ) converges to ( λ, z ). So we have ex-pressed ( w, λ, z ) as the limit of the ( w, λ n λ ′ n , z n z ′ n )’s in H F n ,f n . Conversely,if ( w i , λ i , z i ) ∈ H F ni ,f ni converges to ( w, λ, z ) ∈ G p,k , then λ i = λ for i ≫ λ ∈ C d , and ( w, λ, z ) ∈ C p ∞ × C d × T . (cid:3) The second part of Theorem 1.4 is then a special case of :
Corollary 4.8. C ( G p,k ) r N is homeomorphic to the space obtained by glueing [ d ( k )] × N on the Cantor space C , with the d ( k ) accumulation points of [ d ( k )] × N being identified to d ( k ) pairwise distinct points of C . Proof:
By Theorem 4.7, the closure of C dis is a metrizable compact spacewhich is totally disconnected and perfect (no isolated point), so it is homeo-morphic to the Cantor space C . On the other hand we already observed afterProposition 4.6 that C (1) is homeomorphic to N × [ d ( k )], and the d ( k ) accumula-tion points are identified with d ( k ) pairwise distinct points of C . (cid:3) References [Cha] Claude
Chabauty , Limite d’ensembles et g´eom´etrie des nombres .Bull. Soc. Math. France 78 (1950), 143-151.[Cor] Yves
Cornulier , On the Chabauty space of locally compact abeliangroups , Algebr. Geom. Topol. 11 (2011) 2007-2035.[CGP1] Yves
Cornulier , Luc
Guyot and Wolfgang
Pitsch , On the isolatedpoints in the space of groups , J. Algebra 307(1) (2007), 254-277.[CGP2] Yves
Cornulier , Luc
Guyot and Wolfgang
Pitsch , The space ofsubgroups of an abelian group , J. London Math. Soc. 81(3) (2010) 727-746. 14Hae] Thomas
Haettel . L’espace des sous-groupes ferm´es de R × Z . Algebr.Geom. Topol. 10 (2010) 1395-1415.[HP] J. Hubbard , I.
Pourezza . The space of closed subgroups of R ,Topology 18 (1979), no 2, 143-146.[Pau] Fr´ed´eric Paulin , De la g´eom´etrie et de la dynamique de SL n ( R ) etde SL n ( Z ), pp. 47-110 in Sur la dynamique des groupes de matriceset applications arithm´etiques , edited by N. Berline, A. Plagne and C.Sabbah, ´Ed. de l’ ´Ecole Polytechnique (2007).[Ser] Jean-Pierre
Serre , Cours d’arithm´etique , Presses universitaires deFrance (1970).[Tsa] Todor
Tsankov
Compactifications of N and Polishable subgroups of S ∞∞