Contraction groups and the big cell for endomorphisms of Lie groups over local fields
aa r X i v : . [ m a t h . G R ] J a n Contraction groups and the big cell forendomorphisms of Lie groups over local fields
Helge Gl¨ockner Abstract If G is a Lie group over a totally disconnected local field and α : G → G an analytic endomorphism, let con( α ) be the set of all x ∈ G such that α n ( x ) → e as n → ∞ . Call sequence ( x − n ) n ∈ N in G an α -regressivetrajectory for x if α ( x − n ) = x − n +1 for all n ∈ N and x = x . Letcon − ( α ) be the set of all x ∈ G admitting an α -regressive trajectory( x − n ) n ∈ N such that x − n → e as n → ∞ . Let lev( α ) be the setof all x ∈ G whose α -orbit is relatively compact, and such that x admits an α -regressive trajectory ( x − n ) n ∈ N such that { x − n : n ∈ N } is relatively compact. The big cell associated to α is the subset Ω :=con( α ) lev( α ) con − ( α ). We show: Ω is open in G and the product mapcon( α ) × lev( α ) × con − ( α ) → Ω, ( x, y, z ) xyz is ´etale for suitableimmersed Lie subgroup structures on con( α ), lev( α ), and con − ( α ).Moreover, we study group-theoretic properties of con( α ) and con − ( α ). Classification : 22E20 (primary); 22D05, 22E25, 22E35, 32P05, 37B05, 37C05, 37C86,37D10 (secondary).
Key words : endomorphism, automorphism, Lie group, p -adic Lie group, local field, contrac-tion group, anti-contraction group, Levi subgroup, big cell, parabolic subgroup, nilpotentgroup, iterated kernel, HNN extension. Let G be a totally disconnected, locally compact topological group, with neu-tral element e , and α : G → G be a continuous endomorphism. Then various α -invariant subgroups of G can be associated with α , which are importantfor the structure theory of totally disconnected, locally compact groups andits applications (see [27], [5], [4]; for automorphisms, see [1], [13], [18]; cf. Supported by Deutsche Forschungsgemeinschaft, project GL 357/10-1. con-traction subgroup of α is the set con( α ) of all x ∈ G such that α n ( x ) → e as n → ∞ . The anti-contraction subgroup of α is the set of all x ∈ G admitting an α -regressive trajectory ( x − n ) n ∈ N (as recalled in the abstract)such that lim n →∞ x − n = e . Neither of the subgroups con( α ) and con − ( α )need to be closed in G ; if α is a (bicontinuous) automorphism of G , thencon − ( α ) = con( α − ). The parabolic subgroup of α is the set par( α ) of all x ∈ G such that { α n ( x ) : n ∈ N } is relatively compact in G ; the anti-parabolic subgroup of α is the set par − ( α ) of all x ∈ G admitting an α -regressive trajectory ( x − n ) n ∈ N such that { x − n : n ∈ N } is relatively compactin G . Then par( α ) and par − ( α ) are closed subgroups of G , whence also the Levi subgroup lev( α ) := par( α ) ∩ par − ( α ) is closed; moreover, α (con( α )) ⊆ con( α ) , α (con − ( α )) = con − ( α ) , α (par( α )) ⊆ par( α ) ,α (par − ( α )) = par − ( α ) , and α (lev( α )) = lev( α );see [27] (notably Proposition 19). We are interested in the subsetΩ := con( α ) lev( α ) con − ( α )of G , the so-called big cell . If con( α ) is closed in G , then also con − ( α ) isclosed and the product map π : con( α ) × lev( α ) × con − ( α ) → Ωis a homeomorphism, see [4, Theorems D and F] (cf. already [21] for the caseof automorphisms of p -adic Lie groups). If α is a K -analytic endomorphismof a Lie group G over a totally disconnected local field K and con( α ) isclosed, then con( α ), lev( α ), and con − ( α ) are Lie subgroups of G and π is a K -analytic diffeomorphism, see [11, Theorem 8.15]. If K has characteristic 0,then con( α ) is always closed (see [11, Corollary 6.7]). Our goal is twofold:First, we strive for additional information concerning the subgroups justdiscussed, e.g., group-theoretic properties like nilpotency. Second, we wishto explore what can be said when con( α ) fails to be closed. While our focusis on Lie groups, we start with a general observation: Proposition 1.1
Let G be a totally disconnected, locally compact group and α : G → G be a continuous endomorphism. Then Ω := con( α ) lev( α ) con − ( α ) is an open e -neighbourhood in G such that α (Ω) ⊆ Ω and the map par( α ) × par − ( α ) → Ω , ( x, y ) xy is continuous, surjective, and open. K be a totally disconnected local field. For the study of a K -analyticendomorphism α of a K -analytic Lie group G , it is useful to consider ( G, α )as a K -analytic dynamical system with fixed point e . We shall use local in-variant manifolds for analytic dynamical systems as introduced in [8] and[9], stimulated by classical constructions in the theory of smooth dynamicalsystems over R (cf. [14] and [23]). See [10] and [11] for previous applicationsof such tools to invariant subgroups for analytic automorphisms and endo-morphisms. We recall the concepts of local stable manifolds, local unstablemanifolds and related notions in Section 2. Theorem 1.2
Let α : G → G be a K -analytic endomorphism of a Lie group G over a totally disconnected local field K . Then the following holds: (a) There exists a unique K -analytic manifold structure on con( α ) makingit a K -analytic Lie group con ∗ ( α ) , such that con ∗ ( α ) has an open subsetwhich is a local stable manifold for α around e . (b) There exists a unique K -analytic manifold structure on con − ( α ) makingit a K -analytic Lie group con −∗ ( α ) , such that con −∗ ( α ) has an open subsetwhich is a local unstable manifold for α around e . (c) There exists a unique K -analytic manifold structure on lev( α ) makingit a K -analytic Lie group lev ∗ ( α ) , such that lev ∗ ( α ) has an open subsetwhich is a centre manifold for α around e . (d) There exists a unique K -analytic manifold structure on par( α ) makingit a K -analytic Lie group par ∗ ( α ) , such that par ∗ ( α ) has an open subsetwhich is a centre-stable manifold for α around e . (e) There exists a unique K -analytic manifold structure on par − ( α ) makingit a K -analytic Lie group par −∗ ( α ) , such that par −∗ ( α ) has an open subsetwhich is a local centre-unstable manifold for α around e . In the following three theorems, we retain the situation of Theorem 1.2.
Theorem 1.3
The manifolds just constructed have the following properties. (a)
Each of con ∗ ( α ) , con −∗ ( α ) , lev ∗ ( α ) , par ∗ ( α ) , and par −∗ ( α ) is an im-mersed Lie subgroup of G and α restricts to a K -analytic endomorphism α s , α u , α c , α cs , and α cu thereon, respectively. We have con( α s ) = con ∗ ( α ) and con − ( α u ) = con −∗ ( α ) . (1) Moreover, lev( α c ) , par( α cs ) , and par − ( α cu ) are open subgroups of lev ∗ ( α ) , par ∗ ( α ) , and par −∗ ( α ) , respectively. It is well known that par( α ) normalizes con( α ) and par − ( α ) normalizescon − ( α ) (see [4, Lemma 13.1 (a)]; cf. [1, Proposition 3.4] for automorphisms). Theorem 1.4 (a)
Let Ω ⊆ G be the big cell for α . The product map π : con ∗ ( α ) × lev ∗ ( α ) × con −∗ ( α ) → Ω , ( x, y, z ) xyz is an ´etale K -analytic map and surjective. (b) The action lev ∗ ( α ) × con ∗ ( α ) → con ∗ ( α ) , ( x, y ) xyx − is K -analytic,whence the product manifold structure turns con ∗ ( α ) ⋊ lev ∗ ( α ) into a K -analytic Lie group. The product map con ∗ ( α ) ⋊ lev ∗ ( α ) → par ∗ ( α ) , ( x, y ) xy is a surjective group homomorphism and an ´etale K -analytic map. (c) The action lev −∗ ( α ) × con −∗ ( α ) → con −∗ ( α ) , ( x, y ) xyx − is K -analytic, whence the product manifold structure turns con −∗ ( α ) ⋊ lev ∗ ( α ) into a K -analytic Lie group. The product map con −∗ ( α ) ⋊ lev ∗ ( α ) → par −∗ ( α ) , ( x, y ) xy is a surjective group homomorphism and an ´etale K -analytic map. If H is a group and β : H → H an endomorphism, we writeik( β ) := [ n ∈ N ker( β n )for the iterated kernel . If G is a totally disconnected, locally compact groupand β a continuous endomorphism, then ik( β ) ⊆ con( β ). Let us call acontinuous endomorphism of a topological group G contractive if α n ( g ) → e as n → ∞ , for each g ∈ G . If G is a K -analytic endomorphism of a K -analytic Lie group G and α is contractive, then G = con( α ) = con ∗ ( α ) (cf.[11, Proposition 7.10 (a)]). 4 heorem 1.5 (a) If α is ´etale, then ik( α ) is discrete in con ∗ ( α ) . If K hascharacteristic , then both properties are equivalent. (b) If α is ´etale, then con ∗ ( α ) / ik( α ) is an open β -invariant subgroup forsome K -analytic Lie group H and contractive K -analytic automorphism β : H → H of H which extends the K -analytic endomorphism α s in-duced by α s on con ∗ ( α ) / ik( α s ) . In particular, con( α ) / ik( α ) is nilpo-tent. Moreover, con ∗ ( α ) has an open, α -invariant subgroup U which isnilpotent. (c) If char( K ) = 0 , then ik( α s ) is a Lie subgroup of con ∗ ( α ) . Thus Q :=con ∗ ( α ) / ik( α ) has a unique K -analytic manifold structure making thecanonical map con ∗ ( α ) → Q a submersion, and the latter turns Q into a K -analytic Lie group. There exists a K -analytic Lie group H containing Q as an open submanifold and subgroup, and a contractive K -analyticautomorphism β : H → H which extends the K -analytic endomorphism α s induced by α s on Q . Notably, con( α ) / ik( α ) is nilpotent. (d) The K -analytic surjective endomorphism α u of con −∗ ( α ) is ´etale. More-over, con −∗ ( α ) has an open subgroup S with S ⊆ α u ( S ) such that α u | S : S → α u ( S ) is a K -analytic diffeomorphism and α u ( S ) can beregarded as an open submanifold and subgroup of a K -analytic Liegroup H which admits a K -analytic contractive automorphism β : H → H such that α u | S = β − | S . In particular, S and α u ( S ) are nilpotent. Remark 1.6 (a) If α : G → G is a K -analytic automorphism, it was knownpreviously that con( α ) can be given an immersed Lie subgroup structuremodelled on con( L ( α )) such that α | con( α ) becomes a K -analytic contractiveautomorphism; likewise for con − ( α ) = con( α − ) (see [9, Proposition 6.3 (b)]).Hence con( α ) and con − ( α ) are nilpotent in this case (see [7]).(b) If α : G → G is a K -analytic endomorphism and con( α ) is closed in G ,then α | con − ( α ) is a K -analytic automorphism of the Lie subgroup con − ( α )of G such that ( α | con − ( α ) ) − is contractive (cf. [11, Theorem 8.15]), whencecon − ( α ) is nilpotent. Moreover, α | lev( α ) is a K -analytic automorphism of theLie subgroup lev( α ) of G and α | lev( α ) is distal (see [11, Theorem 8.15 andProposition 7.10 (c)]).We also observe: 5 roposition 1.7 In the setting of Theorem , the following holds: (a) ik( α ) ∩ lev ∗ ( α ) is discrete. (b) ker( α ) ∩ par −∗ ( α ) is discrete. (c) ker( α ) ∩ con −∗ ( α ) is discrete. We mention that refined topologies on contraction subgroups of well-behavedautomorphisms (e.g., expansive automorphisms) were also studied in [19]and [12].The following examples illustrate the results.
Example 1.8
Let H be any group and : H → H , x e be the constantendomorphism. If we endow H with the discrete topology, then H becomes a K -analytic Lie group modeled on K , and a K -analytic endomorphism suchthat con( ) = H . As a consequence, contraction groups of endomorphismsdo not have any special group-theoretic properties: any group can occur. Example 1.9
For a prime number p , consider the group ( Z p , +) of p -adicintegers, which is an open subgroup of the local field Q p of p -adic numbersand thus a 1-dimensional analytic Lie group over Q p . The map β : Z p → Z p , z pz is a contractive Q p -analytic endomorphism. Since β is injective, ik( β ) = { } is trivial and hence discrete. We can extend β to the analytic contractiveautomorphism γ : Q p → Q p , z pz . Example 1.10 If H in Example 1.8 is non-trivial, then α := × β : H × Z p → H × Z p is a contractive Q p -analytic endomorphism which cannot extend to a con-tractive automorphism of a Lie group G ⊇ H × Z p as α is not injective.However, { e } × Z p ∼ = Z p is an open α -invariant subgroup and embeds in( Q p , γ ). Likewise, G/ ik( α ) = G/ ( H × { } ) ∼ = Z p embeds in ( Q p , γ ). Example 1.11
Let F be a finite field, K := F (( X )) be the local field of formalLaurent series over F and F [[ X ]] be its compact open subring of formal power6eries. Then G := F Z , with the compact product topology, can be made a2-dimensional K -analytic Lie group in such a way that the bijection φ : G → X F [[ X ]] × F [[ X ]] , ( a k ) k ∈ Z ∞ X k =1 a − k X k , ∞ X k =0 a k X k ! becomes a K -analytic diffeomorphism. The right-shift α : G → G , ( a k ) k ∈ Z ( a k − ) k ∈ Z is an endomorphism and K -analytic, as φ ◦ α ◦ φ − is the restrictionto an open set of the K -linear map K × K → K × K , ( z, w ) ( X − z, Xw ) . Here con( α ) is the dense proper subgroup of all ( a k ) k ∈ N of support boundedto the left, i.e., there is k ∈ Z such that a k = 0 for all k ≤ k . Moreover,con ∗ ( α ) = F (( X )) . Likewise, con − ( α ) is the set of all sequences with support bounded to theright and con −∗ ( α ) ∼ = F (( X )) with the automorphism z X − z . Note thatcon( α ) ∩ con − ( α ) is the subgroup of all finitely-supported sequences, whichis dense in G and dense in both con ∗ ( α ) and con −∗ ( α ). As G is compact,we have lev( α ) = G . Moreover, lev ∗ ( α ) = G , endowed with the discretetopology. Example 1.12
For K as in Example 1.11, G := F [[ X ]] is an open subgroupof ( K , +). The left-shift α : G → G, ∞ X k =0 a k X k ∞ X k =0 a k +1 X k is an endomorphism of G and K -analytic as it coincides with the K -linear map K → K , z X − z on the open subgroup X F [[ X ]]. We have con − ( α ) = F [[ X ]],since ( X n z ) n ∈ N is an α -regressive trajectory for z ∈ G with X n z → n → ∞ . Moreover, con( α ) ⊆ G is the dense subgroup of finitely supportedsequences, ker( α ) = F X , ik( α ) = con( α ), and lev( α ) = G . Also note thatcon − ( α ) = con −∗ ( α ), while con ∗ ( α ) and lev ∗ ( α ) are discrete. Example 1.13 If K is a totally disconnected local field and α a K -linearendomorphism of a finite-dimensional K -vector space E , then E admits theFitting decomposition E = ik( α ) ⊕ F with F := T k ∈ N α k ( E ). 7or a non-discrete K -analytic Lie group G and a K -analytic endomorphism α : G → G it can happen that both T k ∈ N α k ( G ) = { e } and ik( α ) = { e } , evenin the case K = Q p (see Example 1.9). In the following example, ik( α ) = { e } holds, T k ∈ N α k ( G ) = { e } and L ( α ) = 0, whence ik( L ( α )) = L ( G ) and T k ∈ N L ( α ) k ( L ( G )) = { } . Example 1.14 If F is a finite field of positive characteristic p , then theFrobenius homomorphism F (( X )) → F (( X )), z z p is a K -analytic en-domorphism of ( F (( X )) , +) and restricts to an injective contractive endo-morphism α of the compact open subgroup G := X F [[ X ]] of F (( X )). Since ddz (cid:12)(cid:12) z =0 ( z p ) = pz p − | z =0 = 0, we have L ( α ) = 0.As shown in [7], a K -analytic Lie group G is nilpotent if it admits a contrac-tive K -analytic automorphism α . This becomes false for endomorphisms; noteven open subgroups need to exist in this case which are nilpotent. Example 1.15
Let F be the field with 2 elements. Then S := SL ( F (( X )) isa totally disconnected, locally compact group which is compactly generatedand simple as an abstract group. Hence none of the open subgroups of S is soluble (see [26]), whence none of them is nilpotent. Let G ⊆ S be thecongruence subgroup consisting of all ( a ij ) i,j =1 ∈ S such that a ij − δ ij ∈ X F [[ X ]] for all i, j ∈ { , } . Then G is an open subgroup of S and the map α : G → G, (cid:18) a bc d (cid:19) (cid:18) a b c d (cid:19) , which applies the Frobenius to each matrix element, is a contractive K -analytic endomorphism and injective. No open subgroup of G is nilpotent.The pathology also occurs if char( K ) = 0. Example 1.16
Let G := PSL ( Q p ) and α : G → G be the trivial endomor-phism g e . Then G = con( α ) = con ∗ ( α ). As G is a totally disconnectedgroup which is compactly generated and simple, no open subgroup of G issoluble (nor nilpotent, as a special case). We mention that the corresponding Lie algebra endomorphism L ( α ) : L ( G ) → L ( G ), z pz is an isomorphism; thus ik( L ( α )) = { } and T k ∈ N L ( α ) k ( L ( G )) = L ( G ). Preliminaries and notation
We write N := { , , . . . } and N := N ∪{ } . All topological groups consideredin this work are assumed Hausdorff. If X is a set, Y ⊆ X a subset and f : Y → X a map, we call a subset M ⊆ Y invariant under f if f ( M ) ⊆ M ;we say that M is f -stable if f ( M ) = M . In the following, 1 / ∞ . Recallthat a topological field K is called a local field if it is non-discrete and locallycompact [22]. If K is a totally disconnected local field, we fix an absolutevalue | · | on K which defines its topology and extend it to an absolute value,also denoted | · | , on an algebraic closure K of K (see [16, § E is avector space over K , endowed with an ultrametric norm k · k , we write B Er ( x ) := { y ∈ E : k y − x k < r } for the ball of radius r > x ∈ E . If α : E → F is a continuouslinear map between finite-dimensional K -vector spaces E and F , endowedwith ultrametric norms k · k E and k · k F , respectively, we let k α k op := sup (cid:26) k α ( x ) k F k x k E : x ∈ E \ { } (cid:27) ∈ [0 , ∞ [be its operator norm. For the basic theory of K -analytic mappings betweenopen subsets of finite-dimensional K -vector spaces and the corresponding K -analytic manifolds and Lie groups modeled on K m , we refer to [17] (cf.also [2] and [3]). We shall write T p M for the tangent space of a K -analyticmanifold M at p ∈ M . Given a K -analytic map f : M → N between K -analytic manifolds, we write T p f : T p M → T f ( p ) M for the tangent map of f at p ∈ M . Submanifolds are as in [17, Part II, Chapter III, § N is asubmanifold of a K -analytic manifold M and p ∈ N , then the inclusion map ι : N → M is an immersion and we identify T p N with the vector subspace T p ι ( T p ( N )) of T p M . If F ⊆ T p M is a vector subspace and T p N = F , wesay that N is tangent to F at p . If M is a set, p ∈ M and N , N are K -analytic manifolds such that N , N ⊆ M as a set and p ∈ N ∩ N , wewrite N ∼ p N if there exists a subset U ⊆ N ∩ N which is an open p -neighbourhood in both N and N , and such that N and N induce the same K -analytic manifold structure on U . Then ∼ p is an equivalence relation; theequivalence class of N with respect to ∼ p is called the germ of N at p . If G is a K -analytic Lie group, we let L ( G ) := T e G , endowed with its natural Liealgebra structure; if f : G → H is a K -analytic homomorphism between Lie9roups, we abbreviate L ( f ) := T e ( f ) : L ( G ) → L ( H ). If G is a K -analyticLie group and a subgroup H ⊆ G is endowed with a K -analytic Lie groupstructure turning the inclusion map H → G into an immersion, we call H an immersed Lie subgroup of G . This holds if and only if H has an opensubgroup which is a submanifold of G . If we call a map f : M → N between K -analytic manifolds a K -analytic diffeomorphism, then f − is assumed K -analytic as well. Likewise, K -analytic isomorphisms between K -analytic Liegroups (notably, K -analytic automorphisms) presume a K -analytic inversefunction. If U is an open subset of a finite-dimensional K -vector space, weidentify the tangent bundle U with U × E , as usual. If f : M → U is a K -analytic map, we write df : T M → E for the second component of thetangent map T f : T M → T U = U × E . If X and Y are topological spacesand x ∈ X , we say that a map f : X → Y is open at x if f ( U ) is an f ( y )-neighbourhood in Y for each x -neighbourhood U in X . If G is a groupwith group multiplication ( x, y ) xy , we write G op for G endowed with theopposite group structure, with multiplication ( x, y ) yx .We shall use several basic facts (with proofs recalled in Appendix A). Theterminology in (d) is as in [17, Part II, Chapter III, § Let σ : G × X → X , ( g, x ) g.x be a continuous left action of atopological group G on a topological space X , and x ∈ X . Then we have:(a) If the orbit G.x has an interior point, then
G.x is open in X .(b) If σ x : G → G.x , g g.x is open at e , then σ x is an open map.(c) If G is compact and X is Hausdorff, then σ x : G → G.x is an open map.(d) If G is a K -analytic Lie group, X a K -analytic manifold, σ is K -analytic, G.x ⊆ X is open and σ x is ´etale at e , then σ x is ´etale.We shall also use the following fact concerning automorphic actions. Let G and H be K -analytic Lie groups and σ : G × H → H be a left G -action on H with the following properties:(a) σ g := σ ( g, · ) : H → H is an automorphism of the group H , for each g ∈ G .(b) For each g ∈ G , there exists an open e -neighbourhood Q ⊆ H suchthat σ g | Q is K -analytic. 10c) For each x ∈ H , there exists an open e -neighbourhood P ⊆ G suchthat σ x := σ ( · , x ) : G → H is K -analytic on P .(d) There exists an open e -neighbourhood U ⊆ G and an open e -neighbourhood V ⊆ H such that σ | U × V is K -analytic.Then σ is K -analytic.Again, a proof can be found in Appendix A. Likewise for the following fact. Let G be a topological group, ( g n ) n ∈ N be a sequence in G such that { g n : n ∈ N } is relatively compact and ( x n ) n ∈ N be a sequence in G such that x n → e as n → ∞ . Then g n x n g − n → e . Henceforth, let K be a totally disconnected local field with algebraicclosure K and absolute value | · | . If E is a finite-dimensional K -vector spaceand α : E → E a K -linear endomorphism, call ρ ∈ [0 , ∞ ] a characteristicvalue of α if ρ = | λ | for some eigenvalue λ ∈ K of the endomorphism α K := α ⊗ K id K of E ⊗ K K . If R ( α ) ⊆ [0 , ∞ [ is the set of all characteristic valuesof α , then E = M ρ ∈ R ( α ) E ρ for unique α -invariant vector subspaces E ρ ⊆ E such that E ρ ⊗ K K equals thesum of all generalized eigenspaces of α K for eigenvalues λ ∈ K with | λ | = ρ (compare [15, Chapter II, (1.0)]). If a ∈ ]0 , ∞ [ such that a R ( α ), we saythat α is a -hyperbolic . For ρ ∈ [0 , ∞ [ \ R ( α ), let E ρ := { } . For ρ ∈ [0 , ∞ [, wecall E ρ the characteristic subspace of E for ρ . Then E = ik( α ). Moreover, α ( E ρ ) = E ρ for each ρ > α | E ρ : E ρ → E ρ is an isomorphism. For each a ∈ ]0 , ∞ [, we consider the following α -invariant vector subspaces of E : E a := M ρ>a E ρ and E ≥ a := M ρ ≥ a E ρ . By [9, Proposition 2.4], E admits an ultrametric norm k · k which is adapted to α in the following sense:(a) k x k = max {k x ρ k : ρ ∈ R ( α ) } if we write x ∈ E as x = P ρ ∈ R ( α ) x ρ with x ρ ∈ E ρ ;(b) k α | E k op <
1; 11c) For all ρ ∈ R ( α ) such that ρ >
0, we have k α ( x ) k = ρ k x k for all x ∈ E ρ .If ε ∈ ]0 ,
1] is given, then an adapted norm can be found such that, moreover, k α | E k op < ε . Remark 2.6
By (a) in 2.5, we have B Er (0) = Y ρ ∈ R ( α ) B E ρ r (0)for each r >
0, identifying E with Q ρ ∈ R ( α ) E ρ . For each ρ ∈ R ( α ) \ { } , (c)implies that α ( B E ρ r ) = B E ρ ρr (0) for all ρ ∈ R ( α ) \ { } . Let M be a K -analytic manifold and p ∈ M . Let M ⊆ M be an open p -neighbourhood and f : M → M be a K -analytic map such that f ( p ) = p .Let T p ( M ) ρ for ρ > T p M with respect tothe endomorphism T p f of T p M . Let N ⊆ M be a submanifold such that p ∈ N .(a) If a ∈ ]0 ,
1] and T p f is a -hyperbolic, we say that N is a local a -stablemanifold for f around p if T p N = ( T p M ) ρ for each ρ ∈ R ( T p f ) such that ρ <
1, we call N a localstable manifold for f around p .(b) We say that N is a centre manifold for f around p if T p N = ( T p M ) and f ( N ) = N .(c) If b ≥ T p f is b -hyperbolic, we say that N is a local b -unstablemanifold for f around p if T p N = ( T p M ) >b and there exists an open p -neighbourhood P ⊆ N such that f ( P ) ⊆ N . If, moreover, b < ρ foreach ρ ∈ R ( T p f ) such that ρ >
1, we call N a local unstable manifold for f around p .(d) We call N a centre-stable manifold for f around p if f ( N ) ⊆ N and T p N = ( T p M ) ≤ .(e) We call N a local centre-unstable manifold for f around p if T p N =( T p M ) ≥ and there exists an open p -neighbourhood P ⊆ N such that f ( P ) ⊆ N . 12 .9 We mention that a centre manifold for f around p always exists in thesituation of 2.7, by [8, Theorem 1.10], whose germ at p is uniquely determined(noting that the alternative argument in the proof does not require that T p f be an automorphism). For each a ∈ ]0 ,
1] such that T p f is a -hyperbolic, alocal a -stable manifold around p exists, whose germ around p is uniquelydetermined (by the Local Invariant Manifold Theorem in [9]). For each b ∈ [1 , ∞ [ such that T p f is b -hyperbolic, a local b -unstable manifold around p exists, whose germ around p is uniquely determined (by the Local Invari-ant Manifold Theorem just cited). Moreover, a centre-stable manifold for f around p always exists, whose germ at p is uniquely determined (again bythe Local Invariant Manifold Theorem). The germ at p of a local stable manifold for f around p is uniquelydetermined. In fact, ( T p M ) ρ and b > ρ for all ρ ∈ R ( T p f ) ∩ [0 , N ⊆ M is a local a -stable manifold if andonly if it is a local b -stable manifold for f around p . Thus 2.9 applies.Likewise, ( T p M ) >a = ( T p M ) >b for all a, b ∈ [1 , ∞ [ \ R ( T p f ) such that a < ρ and b < ρ for all ρ ∈ R ( T p f ) ∩ ]1 , ∞ [. Hence, a submanifold N ⊆ M is alocal a -unstable manifold for f around p if and only if it is a local b -unstablemanifold. As a consequence: The germ at p of a local unstable manifold for f around p is uniquelydetermined.Let M be a K -analytic manifold, f : M → M be K -analytic and p ∈ M suchthat f ( p ) = p . Let a ∈ ]0 , \ R ( T p f ), b ∈ [1 , ∞ [ \ R ( T p f ) and k · k be anultrametric norm on E := T p M adapted to T p f such that k T p f | E k op < a .Endow vector subspaces F ⊆ E with the norm induced by k·k and abbreviate B Ft := B Ft (0) for t >
0. We shall use the following fact, which is [11,Proposition 7.3]:
There exists
R > a -stable manifold W sa for f around p and a K -analytic diffeomorphism φ s : W sa → B E b R such that φ u ( p ) = 0, W ub ( t ) := φ − u ( B E >b t ) is a local b -unstable manifoldfor f around p for all t ∈ ]0 , R ], and dφ u | T p ( W ub ) = id E >b .See Appendix A for a proof of the following auxiliary result. Lemma 2.13
Let M be a K -analytic manifold, p ∈ M and f : M → M be a K -analytic mapping such that f ( p ) = p . If N ⊆ M is a local centre-unstable manifold for f around p , then for every p -neighbourhood W ⊆ N ,there exists an open p -neighbourhood O ⊆ N such that f ( O ) is open in N and O ⊆ f ( O ) ⊆ W . In particular, f ( O ) is a local centre-unstable manifoldfor f around p . Let U ⊆ G be a compact open subgroup which minimizes the index[ α ( U ) : α ( U ) ∩ U ] . Then U is tidy for α in the sense of [27, Definition 2] (by the main theoremof [27], on p. 405). Thus U = U + U − = U − U + , where U + is the subgroup of all x ∈ U having an α -regressive trajectoryin U and U − := { x ∈ U : ( ∀ n ∈ N ) α n ( x ) ∈ U } . Moreover, U + and U − arecompact subgroups of U (see [27, Definition 4 and Proposition 1]). Nowpar( α ) ∩ U = U − and par − ( α ) ∩ U = U + ,
14y [27, Proposition 11], whence U + and U − are compact open subgroups ofpar − ( α ) and par( α ), respectively. By [4, Lemma 13.1], we havepar( α ) = con( α ) lev( α ) and par − ( α ) = con − ( α ) lev( α ) = lev( α ) con − ( α ) , entailing that Ω := con( α ) lev( α ) con − ( α ) = par( α ) par − ( α ) . The product map p : par( α ) × par − ( α ) → G, ( x, y ) xy is continuous, with image Ω. We get a continuous left action σ : H × G → G of the direct product H := par( α ) × (par − ( α ) op )on G via ( x, y ) .z := xzy . Then Ω = H.e equals the e -orbit. As Ω ⊇ U − U + = U is a neighbourhood of e in G , 2.1 (a) shows that Ω is open in G . Note thatthe orbit map σ e : H → G, ( x, y ) xey = xy equals p . We now restrict σ to a continuous left action τ : K × G → G of thecompact group K := U − × ( U + ) op on G . Then p ( K ) = U = K.e and p | K : K → K.e is the orbit map, which is an open map by 2.1 (c). Since p ( K ) = U is openin G , we deduce that also the map p : H → H.e = Ω is open at e end hencean open map, by 2.1 (b). (cid:3) G, α ) around e We recall the construction of well-behaved e -neighbourhoods from [11, § G be a Lie groupover a totally disconnected local field K and α : G → G be a K -analyticendomorphism. Pick an ultrametric norm k· k on g := L ( G ) which is adaptedto L ( α ). 15 .1 Pick a ∈ ]0 ,
1] such that L ( α ) is a -hyperbolic and a > ρ for each charac-teristic value ρ of L ( α ) such that ρ <
1. Pick b ∈ [1 , ∞ [ such that L ( α ) is b -hyperbolic and b < ρ for each charcteristic value ρ of L ( α ) such that ρ > L ( α ) of g , we then have g < = g = g >b , entailing that g = g b . We find it useful to identify g with the direct product g b ; anelement ( x, y, z ) of the latter is identified with x + y + z ∈ g . Let R > W sa , W c , W ub and the K -analytic diffeomorphisms φ s : W sa → B g b R (0)be as in 2.12, applied with G in place of M , α in place of f and e in placeof p . Abbreviate B Ft := B Ft (0) if t > F ⊆ g is a vector subspace.Using the inverse maps ψ s := φ − s , ψ c := φ − c , and ψ u := φ − u , we define the K -analytic map ψ : B g R = B g b R → G, ( x, y, z ) ψ s ( x ) ψ c ( y ) ψ u ( z ) . Then T ψ = id g if we identify T g = { } × g with g by forgetting the firstcomponent. By the inverse function theorem, after shrinking R if necessary,we may assume that the image W sa W c W ub of ψ is an open identity neighbour-hood in G , and that ψ : B g R → W sa W c W ub is a K -analytic diffeomorphism. In particular, the product map W sa × W c × W ub → W sa W c W ub , ( x, y, z ) xyz (2)is a K -analytic diffeomorphism. We define φ := ψ − , with domain U := W sa W c W ub and image V := B g R . After shrinking R further if necessary, wemay assume that B φt := φ − ( B g t )is a compact open subgroup of G for each t ∈ ]0 , R ] and a normal subgroupof B φR (see [11, 5.1 and Lemma 5.2]). Then B φt = φ − ( B g t ) = ψ ( B g t ) = ψ s ( B g b t ) = W sa ( t ) W c ( t ) W ub ( t )with notation as in 2.12. 16 .2 After shrinking R if necessary, we may assume that W ub ( t ) is a subgroupof G for all t ∈ ]0 , R ] and the set W c ( t ) := φ − c ( B g t ) normalizes W ub ( t ) (see[11, Lemma 8.7]). Since W sa ( t ) is a local a -stable manifold for α , we have α ( W sa ( t )) ⊆ W sa ( t ) for all t ∈ ]0 , R ].After shrinking R , moreover \ n ∈ N α n ( W sa ) = { e } and lim n →∞ α n ( x ) = e for all x ∈ W sa (3)(see [11, 8.8]). In addition, one may assume that α | W c ( t ) : W c ( t ) → W c ( t )is a K -analytic diffeomorphism for each t ∈ ]0 , R ] (see [11, 8.9]). As shown in [11, 8.10], after shrinking R one may assume that, for each t ∈ ]0 , R ] and x ∈ W ub ( t ), there exists an α -regressive trajectory ( x − n ) n ∈ N in W ub ( t ) such that x = x and lim n →∞ x − n = e ;moreover, one may assume that W ub ( t ) ⊆ α ( W ub ( t )) for all t ∈ ]0 , R ]. Since L ( α | W ub ) = L ( α ) | g >b is injective, we may assume that α | W ub is an injectiveimmersion, after possibly shrinking R . After shrinking R if necessary, we may assume that W sa ( t ) is a subgroupof G for each t ∈ ]0 , R ] and that W c ( t ) normalizes W sa ( t ). Using a dynamicaldescription of the local a -stable manifolds as in [8, Theorem 6.6(c)(i)], thiscan be proved like [11, Lemma 8.7]. By [11, 8.1], there exists r ∈ ]0 , R ] such that α ( W ub ( r )) ⊆ W ub and, foreach x ∈ W ub ( r ) \ { e } , there exists n ∈ N such that α n ( x ) W ub ( r ). For each t ∈ ]0 , R ],( B φt ) − := { x ∈ B φt : ( ∀ n ∈ N ) : α n ( x ) ∈ B φt } is a compact subgroup of B φt . Let ( B φt ) + be the set of all x ∈ B φt for whichthere exists an α -regressive trajectory ( x − n ) n ∈ N such that x − n ∈ B φt for all17 ∈ N and x = x . As recalled in Section 3, also B φ + is a compact subgroupof B φt . For each t ∈ ]0 , R ], we have( B φt ) + = W c ( t ) W ub ( t );moreover, ( B φt ) − = W sa ( t ) W c ( t )for all t ∈ ]0 , r ], by Equations (68) and (73), respectively, in [11, proof ofTheorem 8.13]. Moreover, W c ( t ) = ( B φt ) − ∩ ( B φt ) + is a compact subgroup of B φt for each t ∈ ]0 , r ], see [11, Remark 8.14].We shall use the following result concerning local unstable manifolds. Lemma 4.7
There exists a local centre-unstable manifold N for α around e .Its germ at e is uniquely determined. Proof.
The submanifold N := W c W ub of U = W sa W c W ub ∼ = W s × W c × W ub is a local centre-unstable manifold for α around p , as T e N = T ( W c ) ⊕ T ( W ub ) = ( T p M ) ≥ and P := W c W ur is an open e -neighbourhood in N suchthat α ( P ) ⊆ N . If also N ′ is a local centre-unstable manifold for α around p ,then there exists an open p -neighbourhood Q ⊆ N ′ such that α ( Q ) ⊆ N ′ .By Lemma 2.13, there exists an open p -neighbourhood O ⊆ N ′ such that α ( O ) is open in N ′ and O ⊆ α ( O ) ⊆ U ∩ N ′ . Hence, for each x ∈ O we find an α -regressive trajectory ( x − n ) n ∈ N in O suchthat x = x . Since x − n ∈ O ⊆ U for each n ∈ N , we have x ∈ U + = N .Thus O ⊆ N . As the inclusion map ι : O → N is K -analytic and T p ι isthe identity map of ( T p M ) ≥ , the inverse function theorem shows that O contains an open p -neighbourhood W such that W = j ( W ) is open in N and id = j | W : W → W is a K -analytic diffeomorphism. Thus N ′ and N induce the same K -analytic manifold structure on their joint open subset W ,whence the germs of N and N ′ at p coincide. ✷ emma 4.8 We have con( α ) ∩ W sa ( r ) W c ( r ) = W as ( r ) . Moreover, W ub ( r ) isthe set of all x ∈ W c ( r ) W ub ( r ) admitting an α -regressive trajectory ( x − n ) n ∈ N in W c ( r ) W ub ( r ) such that x = x and x − n → e for n → ∞ . Proof. If x ∈ W sa ( r ), then α n ( x ) → e as n → ∞ and thus x ∈ con( α ),by (3). Now assume that x ∈ con( α ) ∩ W sa ( r ) W c ( r ). Then x = yz forunique y ∈ W sa ( r ) and z ∈ W c ( r ). If we had z = e , then we could find t ∈ ]0 , r [ such that z W c ( t ). Then α n ( y ) ∈ W sa ( r ) for each n ∈ N . Since α | W c ( r ) : W c ( r ) → W c ( r ) is a bijection which takes W c ( t ) onto itself, wededuce that α n ( z ) ∈ W c ( r ) \ W c ( t ) for all n ∈ N , entailing that the groupelement α n ( x ) = α n ( y ) α n ( z ) is in B φr \ B φt for each n ∈ N . Hence α n ( x ) e ,contradiction. Thus z = 0 and thus x ∈ W sa ( r ).By 4.3, each x ∈ W ub ( r ) has an α -regressive trajectory of the asserted form.Now let x ∈ W c ( r ) W ub ( r ) and assume there exists an α -regressive trajectory( x − n ) n ∈ N in W r ( r ) W ub ( r ) such that x = x and x − n → e for n → ∞ . Write x − n = y − n z − n with y − n ∈ W c ( r ) and z − n ∈ W ub ( r ). Then y − n → e and z − n → e as n → ∞ . For each n ∈ N , we have y − n +1 z − n +1 = x − n +1 = α ( x − n ) = α ( y − n ) α ( z − n )with α ( y − n ) ∈ W c ( r ) and α ( z − n ) ∈ α ( W ub ( r )) ⊆ W ub . As the product map W c × W ub → W c W ub is a bijection, we deduce that y − n +1 = α ( y − n ) and z − n +1 = α ( z − n ). If we had y = e , we could find t ∈ ]0 , r [ such that y W c ( t ).There would be some N ∈ N such that x − n ∈ W c ( t ) for all n ≥ N . Since α ( W c ( t )) = W c ( t ), this would imply y = α N ( y − N ) ∈ W c ( t ), a contradiction.Thus y = 0 and thus x = z ∈ W ub ( r ). ✷ Remark 4.9
Since W ub ( r ) ⊆ α ( W ub ( r )) by 4.3, ( α n ( W ub ( r )) n ∈ N is an ascend-ing sequence of compact subgroups of con − ( α ). Moreover,con − ( α ) = [ n ∈ N α n ( W ub ( r )) . In fact, for x ∈ con − ( α ) there exists an α -regressive trajectory ( x − n ) n ∈ N such that x = x and x − n → e as n → ∞ . There exists N ∈ N suchthat x − n ∈ W sa ( r ) W c ( r ) W ub ( r ) = B φr for all n ≥ N . For each n ≥ N , theelement x − n of B φr has the α -regressive trajectory ( x − n − m ) m ∈ N in B φr , whence x − n ∈ ( B φr ) + = W c ( r ) W ub ( r ). As x − n − m ∈ W c ( r ) W ub ( r ) and x − n − m → e as m → ∞ , Lemma 4.8 shows that x − n ∈ W ub ( r ) for each n ≥ N . In particular, x − N ∈ W ub ( r ) and x = x = α N ( x − N ) ∈ α N ( W ub ( r )).19 Proof of Theorems 1.2 and 1.3
We retain the notation of Section 4. As is well known, a homomorphism f : H → K between K -analytic Lie groups is K -analytic whenever it is K -analytic on an open e -neighbourhood in H . Applying this to f = id H , wesee that two Lie group structures on H coincide if their germ at e coincides.The uniqueness of the Lie group structures in parts (a), (b), (c), and (d)of Theorem 1.2 therefore follows from the uniqueness statements concerningmanifold germs in 2.9, 2.10, and 2.11; the uniqueness statement in (e) followsfrom Lemma 4.7. It remains to prove the existence of the asserted Lie groupstructures, and that they have the properties described in Theorem 1.3. Contraction groups.
Since W sa is a subgroup of G and a submanifold, it isa Lie subgroup. By Lemma 4.8, we have W sa ⊆ con( α ). If g ∈ con( α ), then { α n ( g ) : n ∈ N } is relatively compact, whence there exists t ∈ ]0 , r ] suchthat α n ( g ) B φt α n ( g ) − ⊆ B φr for all n ∈ N . For each x ∈ W sa ( t ) and each n ∈ N , we then have α n ( gxg − ) = α n ( g ) α n ( x ) α n ( g ) − ∈ B φr , as α n ( x ) ∈ W sa ( t ). As a consequence, gxg − ∈ ( B φr ) − = W sa ( r ) W c ( r ). More-over, gxg − ∈ con( α ) as g ∈ con( α ) and x ∈ W sa ( t ) ⊆ con( α ). Hence gxg − ∈ W sa ( r ), by Lemma 4.8. Being a restriction of the K -analytic conju-gation map G → G , h ghg − , the map W sa ( t ) → W sa , x gxg − is K -analytic. By the Local Description of Lie Group Structures (see Propo-sition 18 in [3, Chapter III, §
1, no. 9]), we get a unique K -analytic manifoldstructure on con( α ) making it a Lie group con ∗ ( α ), such that W sa is an opensubmanifold. As W sa is a submanifold of G , con ∗ ( α ) is an immersed Lie sub-group of G . Now α ( W sa ) ⊆ W sa and α | W sa : W sa → W sa is K -analytic since α is K -analytic and W sa is a submanifold of G . The restriction α s of α to an en-domorphism of the subgroup con ∗ ( α ) coincides with α | W sa on the open subset W sa of con ∗ ( α ), whence α s is K -analytic. If g ∈ con( α ), there exists N ∈ N such that α n ( g ) ∈ B φr for all n ≥ N , whence α n ( g ) ∈ ( B φr ) − = W sa ( r ) W c ( r ).Since also α n ( g ) ∈ con( α ), Lemma 4.8 shows that α n ( g ) ∈ W sa ( r ). As con ∗ ( α )hast W sa as an open submanifold and α n ( g ) → e in W sa as N ≤ n → ∞ , we20ee that ( α s ) n ( g ) = α n ( g ) → e also in con ∗ ( α ). Thus con( α s ) = con ∗ ( α ). Anti-contraction groups.
Being a subgroup of G and a submanifold, W bu isa Lie subgroup. By Lemma 4.8, we have W ub ⊆ con − ( α ). If g ∈ con − ( α ),then there exists an α -regressive trajectory ( g − n ) n ∈ N such that g = g and g − n → e as n → ∞ . Notably, { g − n : n ∈ N } is relatively compact, whencethere exists t ∈ ]0 , r ] such that g − n B φt ( g − n ) − ⊆ B φr for all n ∈ N . For each x ∈ W ub ( t ), there exists an α -regressive trajectory( x − n ) n ∈ N in W ub ( t ) such that x = x and x − n → e as n → ∞ (see 4.3).Then ( g − n x − n ( g − n ) − ) n ∈ N is an α -regressive trajectory for gxg − such that g − n x − n ( g − n ) − → e as n → ∞ and g − n x − n ( g − n ) − ∈ g − n B φt ( g − n ) − ⊆ B φr for all n ∈ N , whence g − n x − n ( g − n ) − ∈ ( B φr ) + = W c ( r ) W ub ( r ) for all n ∈ N and thus gxg − ∈ W ub ( r ), by Lemma 4.8. Being a restriction of the K -analyticconjugation map G → G , h ghg − , the map W ub ( t ) → W ub , x gxg − is K -analytic. Using the Local Description of Lie Group Structures, we geta unique K -analytic manifold structure on con − ( α ) making it a Lie groupcon −∗ ( α ), such that W ub is an open submanifold. As W ub is a submanifoldof G , con −∗ ( α ) is an immersed Lie subgroup of G . Now α ( W ub ( r )) ⊆ W ub and α | W ub ( r ) : W ub ( r ) → W ub is K -analytic since α is K -analytic and W ub is asubmanifold of G . The restriction α u of α to an endomorphism of con −∗ ( α )coincides with α | W ub ( r ) on the open subset W ub ( r ) of con −∗ ( α ), whence α u is K -analytic. If g ∈ con − ( α ), then there exists an α -regressive trajectory( g − n ) n ∈ N such that g = g and g − n → e in G as n → ∞ . Then g − n ∈ con − ( α )for all n ∈ N and we have seen in Remark 4.9 that there exists an N ∈ N suchthat g − n ∈ W ub ( r ) for all n ≥ N . As con −∗ ( α ) and G induce the same topologyon W ub ( r ), we deduce that g − n → e also in con −∗ ( α ). Thus g ∈ con − ( α u ). Parabolic subgroups. ( B φr ) − = W sa ( r ) W c ( r ) is a Lie subgroup of G and asubset of par( α ). If g ∈ par( α ), then { α n ( g ) : n ∈ N } is relatively compact,whence there exists t ∈ ]0 , r ] such that α n ( g ) B φt α n ( g ) − ⊆ B φr n ∈ N . For each x ∈ W sa ( t ) W c ( t ), we have α n ( x ) ∈ W sa ( t ) W c ( t ) forall n ∈ N and thus α n ( gxg − ) = α n ( g ) α n ( x ) α n ( g ) − ∈ α n ( g ) B φt α n ( g ) − ⊆ B φr , whence gxg − ∈ ( B φr ) − = W sa ( r ) W c ( r ). Being a restriction of the K -analyticconjugation map G → G , h ghg − , the map W sa ( t ) W c ( t ) → W sa ( r ) W c ( r ) , x gxg − is K -analytic. By the Local Description of Lie Group Structures, we geta unique K -analytic manifold structure on par( α ) making it a Lie grouppar ∗ ( α ), such that W sa ( r ) W c ( r ) is an open submanifold. As W sa ( r ) W c ( r )is a submanifold of G , par ∗ ( α ) is an immersed Lie subgroup of G . Now α ( W sa ( r ) W c ( r )) ⊆ W sa ( r ) W c ( r ) and α | W sa ( r ) W c ( r ) : W sa ( r ) W c ( r ) → W sa ( r ) W c ( r )is K -analytic. As a consequence, the restriction α cs of α to an endomor-phism of the subgroup par ∗ ( α ) is K -analytic. If g ∈ W sa ( r ) W c ( r ), then { α n ( g ) : n ∈ N } is contained in the compact open subgroup W sa ( r ) W c ( r ) ofpar ∗ ( α ), whence g ∈ par( α cs ). Thus W sa ( r ) W c ( r ) ⊆ par( α cs ), entailing thatthe subgroup par( α cs ) is an e -neighbourhood and hence open in par ∗ ( α ). Antiparabolic subgroups. ( B φr ) + = W c ( r ) W bu ( r ) is a Lie subgroup of G and asubgroup of par − ( α ). If g ∈ par − ( α ), then there exists an α -regressive tra-jectory ( g − n ) n ∈ N such that g = g and { g − n : n ∈ N } is relatively compact.Thus, there exists t ∈ ]0 , r ] such that g − n B φt ( g − n ) − ⊆ B φr for all n ∈ N . For each x ∈ W c ( t ) W ub ( t ) = ( B φt ) + , there exists an α -regressive trajectory ( x − n ) n ∈ N in W c ( t ) W ub ( t ) such that x = x . Then( g − n x − n ( g − n ) − ) n ∈ N is an α -regressive trajectory for gxg − such that g − n x − n ( g − n ) − ∈ g − n B φt ( g − n ) − ⊆ B φr for all n ∈ N , whence gxg − ∈ ( B rφ ) + = W c ( r ) W ub ( r ). Being a restriction ofthe K -analytic conjugation map G → G , h ghg − , the map W c ( t ) W ub ( t ) → W c ( r ) W ub ( r ) , x gxg − is K -analytic. Using the Local Description of Lie Group Structures, we geta unique K -analytic manifold structure on par − ( α ) making it a Lie group22ar −∗ ( α ), such that W c ( r ) W ub ( r ) is an open submanifold. As W c ( r ) W ub ( r )is a submanifold of G , par −∗ ( α ) is an immersed Lie subgroup of G . Since α | W ub ( r ) : W ub ( r ) → W ub is continuous and W ub ( r ) is open in W ub , there exists t ]0 , r ] such that α ( W ub ( t )) ⊆ W ub ( r ). Now α ( W c ( t ) W ub ( t )) ⊆ W c ( r ) W ub ( r ),entailing that the restriction α cu of α to an endomorphism of par −∗ ( α ) is K -analytic. If g ∈ W c ( r ) W ub ( r ) = ( B rφ ) + , then there exists an α -regressivetrajectory ( g − n ) n ∈ N in B φr such that g = g . For each n ∈ N , the se-quence ( g − n − m ) m ∈ N is an α -regressive trajectory in B rφ for g − n , whence g − n ∈ ( B rφ ) + = W c ( r ) W ub ( r ). As par −∗ ( α ) and G induce the same topology on W c ( r ) W bu ( r ), we deduce that g ∈ par − ( α cu ). Thus W c ( r ) W ub ( r ) ⊆ par − ( α cu ),showing that the latter is an open subgroup of par −∗ ( α ). Levi subgroups. W c ( r ) is a Lie subgroup of G and a subgroup of lev( α ). If g ∈ lev( α ) = par( α ) ∩ par − ( α ), our discussion of par ∗ ( α ) and par −∗ ( α ) yield a t ∈ ]0 , r ] such that gW sa ( t ) W c ( t ) g − ⊆ W sa ( r ) W c ( r ) and gW c ( t ) W ub ( t ) g − ⊆ W c ( r ) W ub ( r ), whence gW c ( t ) g − ⊆ W sa ( r ) W c ( r ) ∩ W c ( r ) W ub ( r ) = W c ( r ) . Being a restriction of the K -analytic conjugation map G → G , h ghg − ,the map W c ( t ) → W c ( r ) , x gxg − is K -analytic. Using the Local Description of Lie Group Structures, we get aunique K -analytic manifold structure on lev( α ) making it a Lie group lev ∗ ( α ),such that W c ( r ) is an open submanifold. As W c ( r ) is a submanifold of G ,lev ∗ ( α ) is an immersed Lie subgroup of G . Now α ( W c ( r )) = W c ( r ), entailingthat the restriction α c of α to an endomorphism of lev ∗ ( α ) is K -analytic. If g ∈ W c ( r ), then there exists an α -regressive trajectory ( g − n ) n ∈ N in W c ( r )such that g = g . Moreover, α n ( g ) ∈ W c ( r ) for all n ∈ N . As lev ∗ ( α )and G induce the same topology on W c ( r ), we deduce that g ∈ par − ( α c ) and g ∈ par( α c ). Thus g ∈ lev( α c ), and thus W c ( r ) ⊆ lev( α c ), showing that thelatter is an open subgroup of lev ∗ ( α ). (cid:3) We start with a lemma. 23 emma 6.1 If G is a K -analytic Lie group over a totally disconnected localfield K and α : G → G a K -analytic endomorphism, then the inclusion maps lev ∗ ( α ) → par ∗ ( α ) and lev ∗ ( α ) → par −∗ ( α ) are K -analytic group homomorphisms and immersions. The actions par ∗ ( α ) × con ∗ ( α ) → con ∗ ( α ) and par −∗ ( α ) × con −∗ ( α ) → con −∗ ( α ) given by ( g, x ) gxg − are K -analytic. Proof.
The first assertion follows from the fact that W c ( r ), W sa ( r ) W c ( r ),and W c ( r ) W ub ( r ) are open e -neighbourhoods in lev ∗ ( α ), par ∗ ( α ) and par −∗ ( α ),respectively, and the product maps W sa ( r ) × W c ( r ) → W sa ( r ) W c ( r ) and W c ( r ) × W ub ( r ) → W c ( r ) W ub ( r ) are K -analytic diffeomorphisms.To see that the action of par ∗ ( α ) on con ∗ ( α ) is K -analytic, we verify the hy-potheses of Lemma 2.2. For each g ∈ par ∗ ( α ), the set { α n ( g ) : n ∈ N } is rel-atively compact in G , whence we find t ∈ ]0 , r ] such that α n ( g ) B φt α n ( g ) − ⊆ B φr . For x ∈ W sa ( t ), we have α n ( x ) ∈ W sa ( t ) ⊆ B φt for each n ∈ N , whence α n ( gxg − ) ∈ B φr and thus gxg − ∈ ( B φr ) − = W sa ( r ) W c ( r ). As α n ( gxg − ) = α n ( g ) α n ( x ) α n ( g ) − → e (4)by 2.3, Lemma 4.8 shows that gxg − ∈ W sa ( r ). As the map W sa ( t ) → W sa ( r ), x gxg − is K -analytic, so ist W sa ( t ) → con ∗ ( α ), x gxg − . Next, we notethat if g ∈ W sa ( r ) W c ( r ) and x ∈ W c ( r ), then gxg − ∈ B φr and α n ( gxg − ) = α n ( g ) α n ( x ) α n ( g ) − ∈ B φr for all n ∈ N as W sa ( r ) W c ( r ) and W sa ( r ) are α -invariant. Hence gxg − ∈ ( B φr ) − = W as ( r ) W c ( r ). Moreover, (4) holds by 2.3, whence gxg − ∈ W sa ( r )by Lemma 4.8. Thus W sa ( r ) W c ( r ) × W sa ( r ) → W sa ( r ) ⊆ con ∗ ( α ) , ( g, x ) gxg − is K -analytic. Finally, let x ∈ con ∗ ( α ) be arbitrary. There exists N ∈ N suchthat α n ( x ) ∈ B φr for all n ≥ N . There exists t ∈ ]0 , r ] such that α n ( gxg − x − ) ∈ B φr for all n ∈ { , . . . , N − } and g ∈ W sa ( t ) W c ( t ).24et g ∈ W sa ( t ) W c ( t ). For all n ≥ N we have α n ( g ) ∈ W sa ( t ) W c ( t ) ⊆ B φt ⊆ B φr and α n ( x ) ∈ B φr , whence α n ( gxg − x − ) = α n ( g ) α n ( x ) α n ( g ) − α n ( x ) − ∈ B φr . Hence gxgx − ∈ ( B φr ) − . Since α n ( x ) → e holds and (4), we deduce that α n ( gxg − x − ) = α n ( gxg − ) α n ( x ) − → e as n → ∞ . Thus gxg − x − ∈ W sa ( r ), by Lemma 4.8, entailing that the map W sa ( t ) W c ( t ) → W sa ( r ) ⊆ con ∗ ( α ) , g gxg − x − is K -analytic and hence also the map W sa ( t ) W c ( t ) → con ∗ ( α ), g gxg − ;all hypotheses of Lemma 2.2 are verified.To see that the conjugation action of par −∗ ( α ) on con −∗ ( α ) is K -analytic,again we verify the hypotheses of Lemma 2.2. For each g ∈ par −∗ ( α ),there exists an α -regressive trajectory ( g − n ) n ∈ N with g = g such that { g − n : n ∈ N } is relatively compact in G . There exists t ∈ ]0 , r ] such that g − n B φt ( g − n ) − ⊆ B φr . For each x ∈ W ub ( t ), there exists an α -regressive tra-jectory ( x − n ) n ∈ N in W ub ( t ) such that x = x . Then g − n x − n ( g − n ) − ∈ B φr foreach n ∈ N . As ( g − n − m x − n − m ( g − n − m ) − ) m ∈ N is an α -regressive trajectoryfor g − n x − n ( g − n ) − , we conclude that g − n x − n ( g − n ) − ∈ ( B φr ) + = W c ( r ) W ub ( r )for each n ∈ N . By 2.3, we have g − n x − n ( g − n ) − → e as n → ∞ . (5)Thus Lemma 4.8 shows that gxg − ∈ W ub ( r ). As a consequence, the map W ub ( t ) → W ub ( r ) ⊆ con −∗ ( α ), x gxg − is K -analytic. Next, for g ∈ W c ( r ) W ub ( r ) = ( B φr ) + and x ∈ W c ( r ), there exists an α -regressive trajectory( g − n ) n ∈ N in B rφ with g = g . Moreover, there is an α -regressive trajectory( x − n ) n ∈ N in W ub ( r ) W c ( r ) such that x = x and x − n → e as n → ∞ (seeLemma 4.8). Then g − n x − n ( g − n ) − ∈ B φr for all n ∈ N . Hence gxg − ∈ ( B φr ) + = W c ( r ) W ub ( r ). Moreover, (5) holdsby 2.3, whence gxg − ∈ W ub ( r ) by Lemma 4.8. Thus W c ( r ) W ub ( r ) × W ub ( r ) → W ub ( r ) ⊆ con −∗ ( α ) , ( g, x ) gxg − K -analytic. Finally, let x ∈ con −∗ ( α ) be arbitrary and ( x − n ) n ∈ N be an α -regressive trajectory such that x = x and x − n → e as n → ∞ . Thereexists N ∈ N such that x − n ∈ B φr for all n ≥ N . There exists t ∈ ]0 , r ] suchthat gx − n g − ( x − n ) − ∈ B φr for all n ∈ { , . . . , N − } and g ∈ B φt .Let g ∈ W c ( t ) W ub ( t ) = ( B φt ) + and ( g − n ) n ∈ N be an α -regressive trajectory in B φt such that g = g . For all n ≥ N we have g − n , x − n ∈ B φr and thus g − n x − n ( g − n ) − ( x − n ) − ∈ B φr . Define y − n := g − n x − n ( g − n ) − ( x − n ) − for n ∈ N . For each n ∈ N , thesequence ( y − n − m ) m ∈ N is an α -regressive trajectory for y n in B φr and thus y n ∈ ( B φr ) + . Notably, ( y − n ) n ∈ N is an α -regressive trajectory in ( B φr ) + = W c ( r ) W ub ( r ). Since x − n → e , using (5), we deduce that y n = ( g − n x − n ( g − n ) − )( x − n ) − = α n ( gxg − ) → e as n → ∞ . Hence gxg − x − = y ∈ W ub ( r ), by Lemma 4.8. As a consequence,the map W c ( t ) W ub ( t ) → W ub ( r ) ⊆ con −∗ ( α ) , g gxg − x − is K -analytic and hence also the map W c ( t ) W ub ( t ) → con −∗ ( α ), g gxg − ;all hypotheses of Lemma 2.2 are verified. ✷ Proof of Theorem 1.4. (a) In the Lie group H := con ∗ ( α ) × lev ∗ ( α ) × (con −∗ ( α ) op ) , the subset W sa ( r ) × W c ( r ) × W ub ( r ) is an open identity neighbourhood andthe restriction of π to this set is a K -analytic diffeomorphism onto the openidentity neighbourhood B rφ = W sa ( r ) W c ( r ) W ub ( r ) of G , see (2). Note that H × G → G, (( x, y, z ) , g ) ( x, y, z ) .g := xygz is a K -analytic left action of H on G . Moreover, H.e = Ω is open in G and π : H → Ω is the orbit map σ e . Since π is ´etale at e by the preceding, π is´etale by Lemma 2.1 (d). 26b) By [4, Lemma 13.1 (d)], we have par( α ) = con( α ) lev( α ). Lemma 6.1entails that the conjugation action of lev ∗ ( α ) on con ∗ ( α ) is K -analytic. As theconjugation action is used to define the semi-direct product con ∗ ( α ) ⋊ lev ∗ ( α ),the latter is a K -analytic Lie group and the product map p : con ∗ ( α ) ⋊ lev( α ) → con ∗ ( α ) lev ∗ ( α ) = par ∗ ( α ) , ( x, y ) xy is a group homomorphism. Being the pointwise product of the projectionsonto x and y , the map p is K -analytic. The restriction of p to a map W sa ( r ) × W c ( r ) → W sa ( r ) W c ( r )is a K -analytic diffeomorphism onto the open subset W sa ( r ) W c ( r ) or par ∗ ( α )(cf. (2)). As a consequence, the group homomorphism p is ´etale.(c) By [4, Lemma 13.1 (e)], we have par − ( α ) = con − ( α ) lev( α ). Lemma 6.1entails that the conjugation action of lev ∗ ( α ) on con −∗ ( α ) is K -analytic. Asthe conjugation action is used to define the semi-direct product con −∗ ( α ) ⋊ lev ∗ ( α ), the latter is a K -analytic Lie group and the product map p : con −∗ ( α ) ⋊ lev( α ) → con −∗ ( α ) lev ∗ ( α ) = par −∗ ( α ) , ( x, y ) xy is a group homomorphism. Being the pointwise product of the projectionsonto x and y , the map p is K -analytic. We know that the map q : W c ( r ) × W ub ( r ) → W c ( r ) W ub ( r ) = ( B φr ) + , ( a, b ) ab is a K -analytic diffeomorphism (cf. (2)). Since W c ( r ), W ub ( r ), and ( B φr ) + aresubgroups, we have( B φr ) + = (( B φr ) + ) − = W ub ( r ) − W c ( r ) − = W u b ( r ) W c ( r ) . The restriction of p to the open set W ub ( r ) × W c ( r ) has open image W ub ( r ) W c ( r ) = ( B φr ) + and is given by p ( x, y ) = q ( y − , x − ) − , whence it is a K -analytic diffeomor-phism onto its open image. As a consequence, the group homomorphism p is ´etale. (cid:3) Proof of Theorem 1.5 (a) If α is ´etale, then L ( α s ) = L ( α ) | g b ,the Ultrametric Inverse Function Theorem (see [6, Lemma 6.1 (b)]) provides θ ∈ ]0 , r ] with bθ ≤ R such that α u ( W ub ( t )) ⊇ W ub ( bt ) for all t ∈ ]0 , θ ],exploiting Remark 2.6. Thus α u ( W ub ( t/b )) ⊇ W ub ( t ) for all t ≤ bθ . (6)Let S := ( α u | W ub ( θ ) ) − ( W ub ( bθ )). Then α u ( S ) = W ub ( bθ ), which is an opensubgroup of con −∗ ( α ), and α u | S : S → α u ( S ) is a K -analytic isomorphism.29oreover, ( α u | S ) − : α u ( S ) → S ⊆ α u ( S )maps W ub ( t ) into W ub ( t/b ) for each t ∈ ]0 , bθ ], by (6), whence(( α u | S ) − ) n ( W ub ( bθ )) ⊆ W ub ( θ/b n − )for each n ∈ N . As a consequence, ( α u | S ) − is a contractive endomorphismof α u ( S ). We now define H n := α u ( S ) for each n ∈ N . Using the bondingmaps φ n,m := (( α u | S ) − ) n − m : H m → H n , we can form the direct limit group H := lim −→ H n and give it a K -analytic manifold structure making it a Lie groupand turning each limit map φ n : H n → H into an injective, ´etale, K -analyticgroup homomorphism. As in the proofs of (b) and (c), we obtain a K -analyticautomorphism β of H which extends ( α u | S ) − : α u ( S ) → S ⊆ α u ( S ) and iscontractive. Notably, H is nilpotent. (cid:3) (a) Since α ( W c ( r ) = W c ( r ) and α | W c ( r ) is injective (see 4.2), we see that( α n ) | W c ( r ) = ( α | W c ( r ) ) n is injective for each n ∈ N , entailing that ik( α ) ∩ W c ( r ) = { e } . The asssertion follows as W c ( r ) is an open e -neighbourhoodin lev ∗ ( α ).(b) As the product map W c × W ub → W c W ub , ( x, y ) xy is a bijection, α ( xy ) = α ( x ) α ( y ) holds for ( x, y ) ∈ W c ( r ) × W ub ( r ) and the restrictions of α to mappings W c ( r ) → W c ( r ) ⊆ W c and W ub ( r ) → W ub are injective (see4.3), we deduce that ker( α ) ∩ W c ( r ) W ub ( r ) = { e } . It remains to recall that W c ( r ) W ub ( r ) is an open e -neighbourhood in par −∗ ( α ).(c) We know from Theorem 1.5 (d) that α u is ´etale. Hence ker( α u ) = ker( α ) ∩ con −∗ ( α ) is discrete in con −∗ ( α ). (cid:3) . A Proofs for basic facts in Section 2
Proof for 2.1. (a) Let y ∈ ( G.x ) o . For z ∈ G.x , there exists g ∈ G suchthat gy = z . Then g. ( G.x ) o is an open neighbourhood of z in X , whence z ∈ ( G.x ) o .(b) By hypothesis, V.x is an x -neighbourhood in G.x for each e -neighbourhood30 ⊆ G . If g ∈ G , then each g -neighbourhood contains gV for some V asbefore and σ x ( gV ) = g. ( V.x ) is a neighbourhood of g.x in G.x .(c) If G x ⊆ G is the stabilizer of x , then the quotient topology turns thecanonical map q : G → G/G x into an open map (see, e.g., [20, Lemma 6.2 (a)]).Since G/G x is compact, the induced continuous bijection φ : G/G x → G.x isa homeomorphism. Hence σ x = φ ◦ q is open.(d) For each g ∈ G , the left translation λ g − : G → G , h g − h and the map σ g := σ ( g, · ) : X → X are K -analytic diffeomorphisms. Since σ is a left ac-tion, we have σ x = σ g ◦ σ x ◦ λ g − . As the tangent map T e ( σ x ) : T e ( G ) → T x ( X )is an isomorphism by hypothesis, also T g ( σ x ) = T x ( σ g ) ◦ T e ( σ x ) ◦ T g ( λ g − ) isan isomorphism. Thus σ x is ´etale at g . (cid:3) Proof for 2.2.
For each g ∈ G , (a) entails that the automorphism σ g of H is K -analytic. Since also ( σ g ) − = σ g − is K -analytic, σ g is a K -analytic auto-morphism of H . Let us show that σ is K -analytic on an open neighbourhoodof a given point ( g, x ) ∈ G × H . As the left translation λ g : G → G , z gz and the right translation ρ x : H → H , y yx are K -analytic diffeomor-phisms, it suffices to show that σ ◦ ( λ g × ρ x ) is K -analytic on some open( e, e )-neighbourhood in G × H . Let U and V be as in (d) and P be as in (c).For all ( z, y ) ∈ ( U ∩ P ) × V , we then have σ ( gz, yx ) = σ g ( σ ( z, y ) σ ( z, x )) = σ g ( σ | U × V ( z, y ) σ x | P ( z )) , which is a K -analatic function of ( z, y ) ∈ ( U ∩ P ) × V . (cid:3) Proof of 2.3. If U ⊆ G is an e -neighbourhood, there exists an e -neighbourhood V ⊆ G such that g n V g − n ⊆ U for all n ∈ N . Now x n ∈ V eventually andthus g n x n g − n ∈ U . (cid:3) Proof of Lemma 2.13.
Let P ⊆ N be an open p -neighbourhood suchthat f ( P ) ⊆ N . Then g := f | P : P → N is a K -analytic map such that T p g = T p f | T p N = T p f | ( T p M ) ≥ , which is a linear automorphism of T p N with1 k ( T p g ) − k op ≥ T p M with a norm adapted to T p f and F := T p N ⊆ T p M withthe induced norm. Let φ : U → V be a K -analytic diffeomorphism from anopen p -neighbourhood U ⊆ N onto an open 0-neighbourhood V ⊆ T p N , suchthat φ ( p ) = 0 and dφ | T p N = id T p N . After shrinking U and V , we may assume31hat U ⊆ P and f ( U ) ⊆ W . There exists r > B Fr (0) ⊆ V and f ( φ − ( B Fr (0))) ⊆ U . Then g : B r (0) → V, x φ ( f ( φ − ( x )))is a K -analytic map such that g (0) = 0 and g ′ (0) = T p f | F . By the UltrametricInverse Function Theorem (cf. [6, Lemma 6.1 (b)]), after shrinking r we mayassume that g ( B Fr (0)) is open and g ( B Fr (0)) = g ′ (0)( B Fr (0)) ⊇ B Fr (0) . As a consequence, O := φ − ( B Fr (0)) is an open p -neighbourhood in N suchthat f ( O ) is open in N and O ⊆ f ( O ) ⊆ U ⊆ W . (cid:3) References [1] Baumgartner, U. and G. A. Willis,
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Helge Gl¨ockner , Universit¨at Paderborn, Institut f¨ur Mathematik,Warburger Str. 100, 33098 Paderborn, Germany; [email protected]@math.upb.de