Let G be a topological group acting on a cell complex X

Proof. LetE⊂X_pbe a chamber. By Proposition10.7there is a b∈ B such that b(E)⊂st_Σ_p(op). Recall that B= T U. Thusb=tufor some appropriate choice oft∈ T_pandu∈ U. SinceT stabilizesΣpit follows that

u(E) =t⁻¹tu(E) =tb(E)⊂_Σ_p which proves the claim.

Our goal is to extend this result to the case of theS-arithmetic part of the Borel group. In order to do so we will approximate the action of

p∏∈S

U onX_S by the diagonal action ofU(O_S). The next result makes more clear what we mean if we speak of approximating the action of a group.

10.3 t h e s t r u c t u r e o f t h e c h a r a c t e r s p h e r e 85

acts cocompactly on Σpand thatT(A_p)fixes Σp pointwise. Thus the multiplicativity of the functionsh_i implies that the diagonal action of T(O_S)onΣSis cocompact.

Theorem10.12. The diagonal action ofΓon X_Sis cocompact.

Proof. From Lemma9.46we know thatΓ=T(O_S)U(O_S). Hence the claim follows from Lemma10.10 and Corollary10.11.

10.3 t h e s t r u c t u r e o f t h e c h a r a c t e r s p h e r e We continue our study of the groupΓ=B(O_S).

Definition 10.13. Letδ: Γ→ T(O_S)denote the canonical projection arising from the splitting Γ=T(O_S)U(O_S).

Note that the kernel ofδis preciselyU(O_S). In order to understand the structure of the character sphere S(_Γ) of Γ it will be important to understand the relationship between U(O_S)and the commutator subgroup ofΓ.

Lemma10.14. The commutator subgroup[_Γ,_Γ]lies inU(O_S). The quotient Q:=U(O_S)/[_Γ,Γ]is a torsion group.

Proof. By Lemma9.46Γis the semidirect productT(O_S)U(O_S)where T(O_S)is an abelian group. Thus we get the inclusion[_Γ,_Γ]⊂ U(O_S). For the second claim we recall relation(e)in Proposition9.8 which tells us that

hα(t)x_β(s)hα(t)⁻¹= x_β(t^h^β,αⁱs)

fors ∈_Qandt ∈_Q^∗. In the case β=α∈_Φ⁺ we thus get hα(t)xα(s)hα(t)⁻¹ =xα(t²s).

For a prime p∈Swe thus obtain

[hα(p),xα(s)] =hα(p)xα(s)hα(p)⁻¹xα(s)⁻¹ = xα(p²s)xα(s)⁻¹

=xα(p²s−s) = xα(s)⁽^p²⁻¹⁾.

Thus xα(s)⁽^p²⁻¹⁾ ∈ [_Γ,_Γ] for every α ∈ _Φ⁺ and every s ∈ O_S. Let q := p²−1. Now let u ∈ U(O_S) be an arbitrary element. By Lemma9.46 we can write

∏

n i=1

x_β_i(t_i)

for some β_i ∈_Φ⁺andt_i ∈ O_S. Letube the image ofuinQ. SinceQ is abelian, relation (a)in Proposition9.8gives us

u^q=u^q =

∏

n i=1

x_β_i(t_i)^q=

∏

n i=1

x_β_i(t^q_i).

Now the claim follows since we have just seen thatx_β_i(t^q_i)∈ [_Γ,_Γ].

Proposition10.15. The inclusion i: T(O_S)→_Γinduces an isomorphism i^∗: Hom(_Γ,_R)→Hom(T(O_S),R),χ7→ χ◦i.

The inverse isomorphism is given by

δ^∗: Hom(T(O_S),R)→Hom(_Γ,_R),χ7→χ◦τ.

Proof. Letγ∈ _Γbe an arbitrary element. By Lemma9.46we can write γ = tu witht ∈ T(O_S) and u ∈ U(O_S). By Lemma10.14we know that a power of ulies in[_Γ,_Γ]. Thusχ(u) =0 sinceRis torsion-free.

Hence the restriction ofχtoT(O_S)can be taken to be the mapχin the proposition.

In view of Proposition 10.15it suffices to consider the characters T(O_S)→_Rin order to understand the characters of Γ. We want to split a characterT(O_S)→_Rinto a product of basis elements.

Definition10.16. For everyα∈_∆and every p∈ SletT_α,p denote the subgroup of T(O_S) consisting of elements of the form h_α(pⁿ) with n∈Z. Let furtherT_Sdenote the group generated by the groupsT_α,p.

The following lemma describes the structure ofT_S. Lemma 10.17. The canonical map

f: ^M

p∈S

α∈_∆

Z→ T_S,(n_α,p)₍_p,α_)∈_S_×_∆ 7→

∏

(p,α)∈S×_∆

h_α(pⁿ^α,p) is an isomorphism. In particular T_Scanonically decomposes as ^L

p∈S

α∈_∆

T_α,p. Proof. It suffices to show that ker(f)is trivial. Thus suppose that

γ:=

∏

(p,α)∈S×_∆

hα(pⁿ^α,p)∈ker(f).

Note that in particular γacts trivially on Σp for every p ∈ S. Recall from9.48 and Lemma9.47that the action ofγonΣp ∼=V is given by

γ(x) = _∏

α∈_∆

π(hα(pⁿ^α,p))(x)

= _∏

α∈_∆

t_α,n_α,p(x)

= x− _∑

α∈_∆

n_α,pα^V

and hence n_α,p =0 for every α∈ _∆since ∆is a basis ofV. Now the claim follows since p ∈Swas chosen arbitrarily.

Lemma 10.18. Every element inT(O_S)can be uniquely written as t_εt_S where tε has finite order and t_S∈ T_S.

10.4 e x t e n d i n g c h a r a c t e r s t o h e i g h t f u n c t i o n s 87

Proof. From Lemma9.46we know that every elementγ∈ T(O_S)can be written as γ = _∏

α∈_∆

hα(tα) with tα ∈ O_S^×. Thus tα = εα ∏

p∈S

p^k^α for some appropriate k_α ∈_Zandε_α ∈ {±1}. This gives us

γ=

∏

α∈_∆

h_α(t_α) =

∏

α∈_∆

(h_α(ε_α)

∏

p∈S

h_α(p^k^p,α)). The predicted decomposition now follows by setting

tε :=

∏

α∈_∆

hα(εα)andt_S:=

∏

α∈_∆

∏

p∈S

hα(p^k^p,α).

For the uniqueness we observe that if γ = t_εt_S = t⁰_εt⁰_S has two such decompositions then t⁰⁻_S ¹t_S=t⁰_εt⁻_ε ¹ has finite order. On the other hand Lemma 10.17 tells us that T_S is torsion-free and therefore we have t_S =t⁰_Sand hencet_ε = t⁰_ε.

Corollary 10.19. The inclusionι: T_S →_Γinduces an isomorphism ι^∗: Hom(_Γ,_R)→Hom(T_S,R).

Proof. From Lemma10.18it follows that the inclusionT_S → T(O_S) induces an isomorphism

Hom(T(O_S),R)→Hom(T_S,R). Now the claim follows from Proposition10.15.

In view of Corollary10.19we can now define the following charac-ters ofΓ.

Definition 10.20. For everyα∈ _∆ and p ∈ Sletχ_α,p: Γ→ _Rdenote the unique extension of the character T_S → _R that is induced by h_β(t)7→ hβ,αiv_p(t)for every α∈ ∆. Let further

BG,B(S) ={χα,p: α∈_∆,p∈ S} denote the union of these characters.

The following proposition summarizes the observations above.

Remark10.21. The setB:=BG,B(S)is a basis of Hom(_Γ,_R).

Proof. This follows from the fact thatκ^∗ is non-degenerate and that∆ is a basis of V.

10.4 e x t e n d i n g c h a r a c t e r s t o h e i g h t f u n c t i o n s

In this section we will construct equivariant height functions for the action ofΓonX_S. I.e. for a given characterχ:Γ →_Rwe will define a continuous function h: X_S →_Rsuch that the diagram

X_S R

γ t_χ₍_γ₎

commutes for every γ ∈ Γ. Here we denote by t_χ₍_γ₎: R → _R the translation given by x7→ x+χ(γ). In order to formulate the second restriction that we are going to impose on the height functionshwe have to consider the chamber σ ⊂ ∂_∞X_S. Recall from Remark 9.37 that σ ⊂ ∂_∞V denotes the chamber at infinity which is given as the boundary at infinity of the sector

K={v∈ V: κ^∗(v,α)>0 for everyα∈_∆}

inV. For each p∈ SletKp denote the corresponding sector inΣpand let σ_p := ∂_∞K_p ⊂ ∂_∞Σp be its boundary chamber at infinity. Finally let σ_S⊂∂_∞ΣS denote the boundary chamber at infinity of the sector K_S:= _∏

p∈S

Kp⊂ _Σ_S. In the following we want hto be invariant under the retraction ρ=ρ_Σ_S_,σ.

Definition 10.22. For each p ∈ S let pr_p: ΣS → _Σ_p be the canonical projection. For eachα ∈ _∆let furtherκα,p: Σp → _R,v 7→ κ^∗(α,ιp(v)) be the linear form associated to α via κ^∗. By composing these two functions with the retractionρ_Σ_S_,σ_S: X_S →_Σ_Sand inverting the sign we obtain the height functions

htα,p:=−κα,p◦pr_p◦ρ_Σ_S_,σ_S: X_S→_R.

It follows directly from the definition that the functions htα,p are ρ_Σ_S_,σ_S-invariant, i.e. ht_α,p◦ρ_Σ_S_,σ_S =ht_α,p, and that the restriction of each htα,p toΣS is a linear function. Thus we obtain htα,p ∈ X_S^∗ where X^∗_S denotes the real vector space ofρ_Σ_S_,σ_S-invariant extensions of the linear forms in Σ^∗_S =Hom(_Σ_S,R)(see Definition 4.3). Since∆ is a basis of V it further follows that B^ht_G_,_B(S)_:= {_ht_α,p_:α∈_∆, p∈ S}is a basis of X^∗_S.

Definition 10.23. Let ht : Hom(_Γ,_R) → X^∗_S, χ 7→ htχ denote the isomorphism that extends the bijection

B_G_,_B(S)→ B^ht_G_,_B(S), χ_α,p 7→ht_α,p.

In order to describe the functions ht_α,p more precisely we will identify the apartments Σp, where p∈ S, withV via the equivariant homeomorphism ι_p: Σp→ V from Proposition9.36.

Remark10.24. The diagonal action of an element ∏

α∈_∆

hα(tα)onΣS is given by

x7→ x−

∑

α∈_∆

∑

p∈S

v_p(t_α)ι⁻_p¹(α^V).

10.4 e x t e n d i n g c h a r a c t e r s t o h e i g h t f u n c t i o n s 89

Proof. This follows directly from Lemma9.47and Lemma9.48. We are now ready to prove the equivariance in a specific situation.

Lemma 10.25. Let χ ∈ Hom(_Γ,_R) be a character. For every element γ∈ T(O_S)and every point x∈_Σ_Swe have

htχ(γ(x)) =htχ(x) +χ(γ).

Proof. By the linearity of ht it suffices to proof the statement for the basis characters. Thus let χ = χ_α,p for someα ∈ _∆ and some p ∈ S and let γ = _∏

β∈_∆

h_β(t_β) ∈ T(O_S)be an arbitrary element. For every elementx ∈_Σ_Swe have

ht_χ(γ.x)

=ht_α,p((_∏

β∈_∆

h_β(t_β)).x)

=htα,p(x− _∑

β∈_∆ ∑

q∈S

vq(t_β)ι⁻_q¹(β^V))

=htα,p(x)− _∑

β∈_∆ ∑

q∈S

vq(t_β)htα,p(ι⁻_q¹(β^V))

=htα,p(x) + _∑

β∈∆ ∑

q∈S

vq(tβ)κ_α,p◦pr_p◦ρ_Σ_S_,σ_S(ι⁻_q¹(β^V))

=ht_α,p(x) + _∑

β∈_∆ ∑

q∈S

v_q(t_β)κ_α,p◦pr_p(ι⁻_q¹(β^V))

=ht_α,p(x) + _∑

β∈_∆

v_p(t_β)κ_α,p(ι⁻_p¹(β^V))

=ht_α,p(x) + _∑

β∈_∆

v_p(t_β)κ^∗(α,β^V)

=ht_α,p(x) + _∑

β∈_∆

v_p(t_β)hβ,αi

=htα,p(x) + _∑

β∈_∆

χα,p(h_β(t_β))

=htα,p(x) +χα,p(_∏

β∈_∆

h_β(t_β))

=_ht_χ(x) +χ(γ)

The following lemma summarizes how characters and their height functions behave under the mapsρandδ.

Lemma 10.26. Letχ∈Hom(_Γ,_R)be a character. Let furtherγ∈_Γbe an arbitrary element and let γ=t_γu_γ be the decomposition ofγinto its torus part t_γ ∈ T(O_S)and its unipotent part u_γ ∈ U(O_S). For every x∈ X_Swe have

(a) ht_χ(ρ(x)) =ht_χ(x), (b) χ(δ(γ)) =χ(γ), (c) ρ(uγ(x)) =ρ(x),

(d) ρ(tγ(x)) =tγ(ρ(x)), and (e) ρ(γ(x)) =δ(γ)(ρ(x)).

Proof. The properties(a)and(b)follow directly from the construction of the basis characters and their corresponding height functions. To prove (c) and (d) let x ∈ X be an arbitrary point and let Σ⁰ be an apartment such thatx ∈_Σ⁰ andσ ⊂∂_∞Σ⁰. LetK₁ be a common sector of ΣS and Σ⁰ such that ∂_∞K₁ = σ. From the construction of U(O_S) we know that there is a subsectorK₂ ofK₁that is fixed pointwise by u_γ. Let further Σ⁰⁰ =u_γ(_Σ⁰). Note that the restrictions ofρtoΣ⁰ and Σ⁰⁰ fix K₂. Thus the isomorphismsρ◦uγ: Σ⁰ → _Σ_S andρ: Σ⁰ → _Σ_S fix K₂ and hence coincide. In particular we obtainρ(uγ(x)) = ρ(x) and hence(c). By the same argument we see that the isomorphisms ρ◦tγ: Σ⁰ →_Σ_Sandtγ◦ρ: Σ⁰ →_Σ_S coincide since they coincide on a subsectorK₂ofK₁ which proves(d). By applying the above rules we can derive (e).

ρ(γ(x)) =ρ(t_γu_γ(x)) =ρ(t_γ(u_γ(x)))

=t_γ(ρ(u_γ(x))) =t_γ(ρ(x)) =δ(γ)(ρ(x))

We are now ready to prove the desired equivariance of the height functionshχ.

Corollary 10.27. Letχ: Γ→_Rbe a character. With the notation above we have

hχ(γ(x)) =χ(γ) +hχ(x)for every x ∈X_Sand everyγ∈_Γ.

Proof. In view of Lemma10.25and the properties in Lemma10.26we have

h_χ(γ(x)) = h_χ(ρ(γ(x))) =h_χ(δ(γ)(ρ(x)))

= h_χ(ρ(x)) +χ(δ(γ)) =h_χ(x) +χ(γ). 10.5 s i g m a i n va r i a n t s o f S-a r i t h m e t i c b o r e l g r o u p s In this section we will prove the main results of this paper. In order to state these results we start by recalling and introducing some terminology. Let G = G(_Φ,ρ,Q)be a Chevalley group, letB ⊂ G be a Borel subgroup, and let Γ = B(O_S) for some finite set of prime numbers S ⊂ N. Let further ∆ ⊂ _Φ be the set of simple roots that

10.5 s i g m a i n va r i a n t s o f S-a r i t h m e t i c b o r e l g r o u p s 91

corresponds toB. We consider the action ofG on its corresponding Bruhat-Tits building X_S= _∏

p∈S

Xp that was described in Section10.1. Recall from Definition10.20and Remark 10.21that

BG,B(S) ={χ_α,p: α∈_∆,p∈ S} is a basis of Hom(_Γ,_R)and that

B^ht_G_,_B(S) ={htα,p: α∈_∆, p∈S}

is a basis of X^∗_S. Thus we see that the subset ∆G,B(S) ⊂ S(_Γ) _that is represented by the positive cone of BG,B(S)has the structure of a closed simplex whose set of vertices is represented by BG,B(S).

We recall the following well-known fact about the thickness ofX_p. It follows for example from an application of the orbit-stabilizer theorem to the BN-characterization of panels inX_p.

Remark10.28. The thickness ofX_p is given byp+1.

We are now ready to prove our main result.

Theorem10.29. Let G = G(_Φ,ρ,Q)be a Chevalley group, letB ⊂ Gbe a Borel subgroup, and letΓ= B(O_S)for some finite set of prime numbers S⊂N. Suppose that

1. Φis of type A_n+1, C_n+1, or D_n+1and that

2. every prime factor p∈S satisfies p≥2ⁿin the A_n+1-case, respectively p ≥₂²ⁿ⁺¹in the other two cases.

Then theΣ-invariants ofΓare given by

Σ^k(_Γ) =S(_Γ)\_∆_G_,_B(S)⁽^k⁾for every k ∈_N.

Proof. Let∆⊂_Φbe the set of simple roots that corresponds toB. Let χ: Γ → _R be a character and let ht_χ: X_S → _Rbe its corresponding height function as in Definition10.23. From Corollary10.27we know that htχ is an equivariant extension ofχ. Since the action ofΓonX_Sis cocompact by Theorem10.12and the stabilizers of cells are of type F_∞ by Proposition 10.2, we can apply Theorem2.47. This theorem tells us for every k∈ _N₀ thatχ∈ _Σ^k⁺¹(_Γ)if and only if((X_S)_ht_χ_≥_r)_r_∈_R is essentiallyk-connected. In order to apply Theorem8.18, our geometric main result, we have to check that X_S and ∂_∞X_S satisfy the SOL-property and that Aut(X_S)acts strongly transitively onX_S. The second claim follows from the BN-characterization of X_S(see Theorem9.33).

To get the first claim we note that Remark 10.28 implies that the thickness of th(X_S)is given by th(X_S) =min

p∈S p+1. Thus every linkL in XSsatisfies th(_L)≥ 2ⁿ+_{1 in the} _A_n₊₁ case, respectively th(_L)≥ 2²ⁿ⁺¹+1 in the other two cases. This is exactly what we need to apply Theorem2.39which tells us that Lsatisfies the SOL-property in the

case where Lis irreducible. The SOL-property of the reducible links follows from the sphericity formula of joins (Lemma2.21) applied to the restriction of the join decomposition of spherical buildings ([29, Proposition 1.15]) to the opposite complex of a chamber in L. Note that we also may apply Theorem2.39to∂_∞X_Ssince the boundary at infinity of a thick building always satisfies th(∂_∞X_S) =∞. Before we can apply Theorem 8.18we have to recall from Lemma10.26, that ht_χ isρ_Σ_S_,σ_S-invariant, whereΣS andσ_S ⊂∂_∞ΣSare as in Section10.4. Let B(σ_S) ⊂ S((X_S)^∗)denote the set of classes α_P of functions that are negative onK₀(σ_S)and constant onK₀(P)for some panel Pof σ_Sand the origin 0∈ _Σ_S. The convex hull ofB(σ_S)inS(X^∗_S)will be denoted by∆(σ_S). An application of Theorem8.18to the present setting now gives us the following.

1. [ht_χ] ∈/ _∆(σ_S) if and only if ((X_S)_ht

χ≥r)_r_∈_R is essentially con-tractible.

2. If [ht_χ] ∈ _∆(σ_S)and 0≤ k < dim(_∆(σ_S))then[h]is contained in ∆(σ_S)⁽^k⁺¹⁾\_∆(σ_S)⁽^k⁾ if and only if (X_ht_χ≥r)_r_∈_R is essentially k-connected but not essentially(k+1)-acyclic.

Note that the construction of the functions htα,p (Definition 10.22) implies that their unionB^ht_G_,_B(S)is a system of representatives ofB(σ_S). Since htχis the image of the isomorphism ht : Hom(_Γ,_R)→X_S^∗which extends the bijection

BG,B(S)→B_G^ht_,_B(S), χ_α,p7→ htα,p

we see that [ht_χ] ∈ _∆(σ_S)⁽^k⁾ if and only if [χ] ∈ _∆_G_,_B(S)⁽^k⁾ which proves the claim.

In view of Theorem 2.48 we can translate Theorem 10.29 to the following result about finiteness properties of subgroups ofB(O_S). Corollary 10.30. LetG =G(_Φ,ρ,Q)be a Chevalley group, letB ⊂ Gbe a Borel subgroup, and letΓ= B(O_S)for some finite set of prime numbers S⊂N. Suppose that

1. Φis of type A_n+1, C_n+1, or D_n+1 and that

2. every prime factor p∈S satisfies p≥2ⁿin the A_n+₁-case, respectively p ≥2²ⁿ⁺¹in the other two cases.

Then for every subgroup[_Γ,_Γ]≤H≤ _Γand every k∈_Nwe have H is of type F_k if and only if H*ker(χ)_{for every}χ∈ _∆_G_,_B(_S)⁽^k⁾_.

A L P H A B E T I C A L I N D E X

Σ-invariant,19 abstract cone,38 apartments,15 BN-pair,72 boundary,11

boundary at infinity,10 branching number,38 building,15

Euclidean,15 spherical,15 Busemann function,10 CAT(0)-space,10 CAT(1)-space,10 cell,11

chamber,15 character, 18

character sphere,5,18 Chevalley basis,68 Chevalley group,70 closed interval, 71 coface,11

comparison point,9 comparison triangle,9 convex,45

coroot,74 directions,12

discrete valuation, 73 essential,35

essentiallyn-acyclic,18 essentiallyn-connected,18 Euclidean polysimplices,11 extendibility constant, 46 face,11

facets,11 flow,54 gallery,15

τ-minimal,22

moving towardsτ,22 geodesic,9

geodesic ray,9 geodesic segment,9 geodesic triangle,9 height function,19 horizontal,50 join,12 Levi,50

extended Levi building,49 Levi building,49

link,12

locally bounded above,46 locally uniformly extendible,

lower complex,28 lower face,23,33 nilpotent,69

non-separating boundary,24 open interval,71

opposite,16,32 panel,15 parabolic,50

parabolic building,49 parabolic subgroups,75 projection,15,21

relative link,12 relative star,11 RGD-system,72 sector,17

simple,11 SOL-property,33 special,14 spherical,13

strongly transitively,42 subcomplex

σ-convex,24 σ-length,25 supported,14 superlevelsets,19 thickness,16 Tits system,72

torus subgroup,70 typeFn,18

unipotent subgroup,70 upper complex,28 upper face,23,33 VRGD-system,73

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Im Dokument The Sigma-Invariants of S-arithmetic subgroups of Borel groups (Seite 96-110)