aa r X i v : . [ m a t h . G R ] D ec On lengths on semisimple groups
Yves de CornulierMay 21, 2009
Abstract
We prove that every length on a simple group over a locally compactfield, is either bounded or proper.
Let G be a locally compact group. We call here a semigroup length on G afunction L : G → R + = [0 , ∞ [ such that • (Local boundedness) L is bounded on compact subsets of G . • (Subadditivity) L ( xy ) ≤ L ( x ) + L ( y ) for all x, y .We call it a length if moreover it satisfies • (Symmetricalness) L ( x ) = L ( x − ) for all x ∈ G .We do not require L (1) = 1. Note also that local boundedness weakens themore usual assumption of continuity, but also include important examples likethe word length with respect to a compact generating subset. See Section 2 forfurther discussion. Besides, a length is called proper if L − ([0 , n ]) has compactclosure for all n < ∞ . Definition 1.1.
A locally compact group G has Property PL (respectively strongProperty PL ) if every length (resp. semigroup length) on G is either bounded orproper.We say that an action of a locally compact group G on a metric space is locallybounded if Kx is bounded for every compact subset K of G and x ∈ X . Thisrelaxes the assumption of being continuous. The action is bounded if the orbitsare bounded. If G is locally compact, the action is called metrically proper if forevery bounded subset B of X , the set { g ∈ G | B ∩ gB = ∅} has compact closure. Proposition 1.2.
Let G be a locally compact group. Equivalences: i) G has Property PL;(ii) Any action of G on a metric space, by isometries, is either bounded ormetrically proper;(ii’) Any action of G on a metric space, by uniformly Lipschitz transformations,is either bounded or metrically proper;(iii) Any action of G on a Banach space, by affine isometries, is either boundedor metrically proper. When G is compactly generated, Property PL can also be characterized interms of its Cayley graphs. Proposition 1.3.
Let G be a locally compact group. If G has strong Property PL(resp. Property PL), then for any subset S (resp. symmetric subset) generating G as a semigroup, either S is bounded or we have G = S n for some n . If moreover G is compactly generated, then the converse also holds. I do not know if the converse holds for general locally compact σ -compactgroups. Also, I do not know any example of a locally compact group with Prop-erty PL but without the strong Property PL.If a locally compact group is not σ -compact, then it has no proper lengthand therefore both Property PL and strong Property PL mean that every lengthis bounded. Such groups are called strongly bounded (or are said to satisfy the Bergman Property ); discrete examples are the full permutation group of anyinfinite set, as observed by Bergman [Be] (see also [C]). However the study ofProperty PL is mainly interesting for σ -compact groups, as it is then easy to geta proper length (it is more involved to obtain a continuous proper length; this isdone in [St], based on the Birkhoff-Kakutani metrization Theorem).The main result of the paper is. Theorem 1.4.
Let K be a local field (that is, a non-discrete locally compact field)and G a simple linear algebraic group over K . Then G K satisfies strong PropertyPL. This result was obtained by Y. Shalom [Sh] in the case of continuous Hilbertlengths, i.e. lengths L of the form L ( g ) = k gv − v k for some continuous affineisometric action of G on a Hilbert space, with an action of a group of K -rankone. Some specific actions on L p -spaces were also considered in [CTV].My original motivation was to extend Shalom’s result to actions on L p -spaces,but actually the result turned out to be much more general. However, even forisometric actions on general Banach spaces, we have to prove the result not onlyin K -rank one, but also in higher rank, in which case the reduction to SL requiressome careful arguments. 2he first step is the case of SL ( K ); it is elementary but it seems that it hasnot been observed so far (even for K = R ).Then with some further work, and making use of the Cartan decomposition,we get the general case. In the case of rank one, this second step is straightfor-ward; this was enough in the case of Hilbert lengths considered in [Sh] in viewof Kazhdan’s Property T for simple groups of rank ≥ Remark . It is necessary to consider lengths bounded on compact subsets.Indeed, write R as the union of a properly increasing sequence of subfields K n .(For instance, let I be a transcendence basis of R over Q , write I as the unionof a properly increasing sequence of subsets I n , and define K n as the set of realsalgebraic over Q ( I n ).) If G = G ( R ) is a connected semisimple group, then ℓ ( g ) = min { n | g ∈ G ( K n ) } is an unbounded symmetric (and ultrametric) non-locally bounded length on G . However ℓ is not bounded on compact subsets and { ℓ ≤ n } is dense provided G is defined over K n , and this holds for n large enough.Also, if G = G ( C ) is complex and non-compact, if α is the automorphismof G induced by some non-continuous field automorphism of C , and if ℓ is theword length with respect to some compact generating set, then ℓ ◦ α is anotherexample of a non-locally bounded length neither bounded nor proper.Finally, it is convenient to have a result for general semisimple groups. Proposition 1.6.
Let K be a local field and G a semisimple linear algebraic groupover K . Let L be a semigroup length on G ( K ) . Then L is proper if (and only if )the restriction of L to every non-compact K -simple factor G i ( K ) is unbounded. This proposition relies on Theorem 1.4, from which we get that L is properon each factor G i ( K ), and an easy induction based on the following lemma, ofindependent interest. Lemma 1.7.
Let H × A be a locally compact group. Suppose that H = G ( K ) forsome K -simple linear algebraic group over K . Let L be a semigroup length on G ,and suppose that L is proper on H and A . Then L is proper. Here are some more examples of PL-groups, beyond semisimple groups.
Proposition 1.8.
Let K be a compact group, with a given continuous orthogonalrepresentation on R n for n ≥ , so that the action on the 1-sphere is transitive(e.g. K = SO( n ) or K = SU( m ) with m = n ≥ ). Then the semidirect product G = R n ⋊ K has strong Property PL. Proposition 1.9.
Let K be a non-Archimedean local field with local ring A .Then the group K ⋊ A ∗ has strong Property PL. Note that the locally compact group K ⋊ A ∗ is not compactly generated.3 Discussion on lengths
We observe here that our results actually hold for more general functions thanlengths. Namely, call a weak length a function G → R + which is locally boundedand satisfies (Control Axiom) There exists a non-decreasing function φ : R + → R + such thatfor all x, y , we have L ( xy ) ≤ φ (max( L ( x ) , L ( y )) for all x, y .Note that every semigroup length satisfies the control axiom with φ ( t ) = 2 t .Besides, if L, L ′ are two weak lengths on G , say that L is coarsely bounded by L ′ and write L (cid:22) L ′ if L ≤ u ◦ L ′ for some proper function u : R + → R + , and that L and L ′ are coarsely equivalent, denoted L ≃ L ′ , if L (cid:22) L ′ (cid:22) L . Here is a seriesof remarks concerning various definitions of lengths.1. (Vanishing at 1) Let L be a length (resp. weak length). Set L ′ ( x ) = L ( x )if x = 1 and L ′ (1) = 0. We thus get another length (resp. weak length),and obviously L and L ′ are coarsely equivalent.2. (Continuity) A construction due to Kakutani allows to replace any length bya coarsely equivalent length which is moreover continuous (see [Hj, Theorem7.2]).3. For every weak length L , there exists a semigroup length L coarsely equiv-alent to L . The argument is as follows: we can suppose that φ , thecontrolling function involved in the definition of weak length, is a bijec-tion from R + to [ φ (0) , + ∞ [ satisfying φ ( t ) ≥ t + 1 for all t ; then itmakes sense to define inductively α ( t ) = t/φ (0) for all t ∈ [0 , φ (0)] and α ( t ) = α ( φ − ( t )) + 1 for t ≥ φ (0). By construction, α ( φ ( t )) ≤ α ( t ) + 1for all t . Thus L ′ ( x ) = α ( L ( x )) satisfies the quasi-ultrametric axiom L ′ ( xy ) ≤ max( L ′ ( x ) , L ′ ( y )) + 1, and consequently L = 1 + L ′ is a semi-group length. Moreover, α increases to infinity, so that L is coarsely equiv-alent to L . Note that if L is symmetric, then so is L .4. If L is a length, then L ′ ( x ) = L ( x ) + L ( x − ) is a symmetric length. Wehave ( L bounded) ⇔ ( L ′ bounded) and ( L proper) ⇒ ( L ′ proper), but L ′ can be proper although L is not. In particular, they are not necessarilycoarsely equivalent; when it is the case, L is called coarsely symmetric . Forinstance, the semigroup word length in Z with respect to the generatingsubset { n ≥ − } is not coarsely symmetric.It is well-known that a locally compact group is σ -compact (i.e. a countableunion of compact subsets) if and only if it possesses a proper length. Trivially, thisis a sufficient condition. Let us recall why it is necessary: let ( K n ) be a sequence4f compact subsets covering G ; we can suppose that K has non-empty interior.Define by induction M = K and M n as the set of products of at most 2 elementsin M n − ∪ K n . Then L ( g ) = inf { n | g ∈ M n } satisfies the quasi-ultrametric axiom L ( xy ) ≤ max( L ( x ) , L ( y )) + 1 and is symmetric and proper. Lemma 3.1.
Let G be a locally compact group and K a compact normal subgroup.Then G has Property PL if and only if G/K has Property PL.Proof.
The forward implication is trivial. Conversely if
G/K has Property PLand L is a length on G , then L ′ ( g ) = sup k ∈ K L ( gk ) is a length as well, so is eitherbounded or proper, and L ≤ L ′ ≤ L + sup K L , so L is also either bounded orproper. Lemma 3.2.
Suppose that G has three closed subsets K, K ′ , D with K, K ′ com-pact, and G = KDK ′ . Then a length on G is bounded (resp. proper) if and onlyits restriction to D is so.Proof. Suppose that a length L on G is proper on D . Let ( g n ) in G be boundedfor L . Write g n = k n d n ℓ n with ( k n , d n , ℓ n ) ∈ K × D × K ′ . Then L ( d n ) is bounded.As L is proper on D and bounded on K and K ′ , it follows that ( d n ) = ( k − n g n ℓ − n )is bounded; therefore ( g n ) is bounded as well. So L is proper on all of G . Thecase of boundedness is even easier.As a consequence we get Lemma 3.3.
Let G be a locally compact group and H a cocompact subgroup. If H has (strong) Property PL, then G also has (strong) Property PL. The converse is not true, even when H is normal in G , in view of Proposition1.8. Proof of Propositions 1.8 and 1.9.
Let L be a semigroup length on G . If L is notproper, then there exists an unbounded sequence ( a i ) in R n with L ( a i ) ≤ M forsome M < + ∞ independent on i . Using transitivity of K , if M ′ = M + 2 sup K L ,then for every i , the length L is bounded by M ′ on the sphere S ( a i ) of radius a i centered at 0. As every element of the ball D ( a i ) of radius a i centered at 0 is thesum of two elements of S ( a i ), it follows that L is bounded by 2 M ′ on B ( a i ). As a i → ∞ , L is bounded on R n , and hence L is bounded on all of G .Proposition 1.9 is proved in an analogous way, using the trivial fact that inany non-Archimedean local field K , for any m ≥ n , any element of valuation m is sum of two elements of valuation n . 5 roof of Proposition 1.3. Define L ( g ) as the least n such that g ∈ S n and observethat L is bounded on compact subsets because by the Baire category theorem, S k has non-empty interior for some k . If S is symmetric, then L is a length. Sothe assumption implies that either L is proper (and hence S is bounded) or L isbounded (and hence S n = G for some n ).Conversely, suppose that G is compactly generated and the condition holds.Let L be a non-proper semigroup length (resp. length) on G . Set S n = L − ([0 , n ]),which is symmetric if L is a length. By non-properness, there exists n such that S n is unbounded. As G is compactly generated, S n generates G for some n ≥ n .Then, by assumption, every element of G is product of a bounded number ofelements from S = S n . By subadditivity, this implies that L is bounded on G . Proof of Proposition 1.2. (ii’) ⇒ (ii) is trivial. (i) ⇒ (ii’) Let G act on the non-empty metric space by C -Lipschitz maps, and define L ( g ) = d ( x , gx ) for some x in X . Then L satisfies the inequality L ( gh ) ≤ L ( g ) + CL ( h ) for all g , h . Bythe remarks at the beginning of this Section 2, L is a weak length, so is coarselyequivalent to a length. So L is either proper or bounded. (ii) ⇒ (i) This followsfrom the fact that any length vanishing at 1, is of the form d ( x , gx ) for someisometric action of G on a metric space, and 1. in Section 2.Of course (ii) implies (iii). The converse follows from the construction in [NP,Section 5]: every metric space X embeds isometrically into an affine Banachspace B ( X ), equivariantly, i.e. so that any isometric group action on X extendsuniquely to an action by affine isometries on B ( X ). Let us now proceed to the proof of Theorem 1.4.
Let K be a local field, and D a cocompact subgroup of K ∗ . Proposition 4.1.
Let L be a symmetric length on K ⋊ D . If L is non-proper on D , then L is bounded on K .Proof. Fix W a compact neighborhood of 1, so that L is bounded by a constant M on W . Suppose that the length L is not proper on D : there exists an unboundedsequence ( a n ) in D such that L ( a n ) is bounded by a constant M ′ . Let u be anyelement of the subgroup K . Replacing some of the a n by a − n if necessary, wecan suppose that a n ua − n → L is symmetric). Then for n largeenough, w n = a n ua − n ∈ W , on which L is bounded by M . Writing u = a − n w n a n ,we obtain that L ( u ) ≤ M + 2 M ′ . 6 emark . Proposition 4.1 is false for semigroup lengths. Indeed, the subset { ( x, λ ) ∈ K ⋊ D : | x | ≤ , < | λ | ≤ M } generates K ⋊ D provided M is large enough; the corresponding semigroup wordlength is obviously non-proper, and is easily checked to be unbounded on K . Remark . Proposition 4.1 still holds if the normal subgroup K is replaced bya finite-dimensional K -vector space, D acting by scalar multiplication. SL Denote G = SL ( K ), and D , U , and K the set of diagonal, unipotent, andorthogonal matrices in G . Let L be any semigroup length on G .We have a Cartan decomposition G = KDK , which implies by Lemma 3.2that boundedness and properness of the length L on G can be checked on D .The matrix M = (cid:18) −
11 0 (cid:19) conjugates any matrix in D to its inverse. Itfollows that L ( g ) + L ( g − ) is equivalent to L on D , and hence on all of G . Inother words, we can suppose that the length L is symmetric.So if L is non-proper, then L is bounded on U by Proposition 4.1. Similarly, L is bounded on U t , the lower unipotent subgroup of G (this also follows fromthe fact that U t is conjugate to U by M ). As every element of G is product offour elements in U ∪ U t , we conclude that L is bounded on G . G simply connected Let H → G be the (algebraic) universal covering of G . Then the map H K → G K has finite kernel and cocompact image. Therefore, by Lemmas 3.3 and 3.1 strongProperty PL for G K follows from strong Property PL for H K .So we can assume G algebraically simply connected, and it will be convenientand harmless to identify G with G K . Let d ≥ K -rank of the simplyconnected K -simple group G and D be a maximal split torus in G . The Cartandecomposition tells us that there exists a compact subgroup K of G such that G = KDK (in the case of Lie groups, see [He, Chap. IX 1.]; in the non-Archimedeancase, see Bruhat-Tits [BrT, Section 4.4]).So the proof consists in proving that if a semigroup length L on G is notproper on D , then it is bounded. This case is not necessary for the general case but we wish to point out that thenthe conclusion is straightforward. Indeed, if G is such a group, then its subgroup D is contained in a subgroup isomorphic to SL ( K ) or PSL ( K ) and thereforeevery length on G is either proper or bounded on D .7 .5 General case Remains the case of higher rank groups. Let W be the relative Weyl group of G with respect to D , that is normalizer of D (modulo its centralizer). Let D ∨ ≃ Z d be the group of multiplicative characters of D , that is K -defined homomorphismsfrom D ≃ K ∗ d to the multiplicative group K ∗ .Then by [BoT, Corollary 5.11], the relative root system is irreducible, so thatby [Bk, Chap. V.3, Proposition 5(v)], the action of W on D ∨ ⊗ Z R is irreducible.If u is a function D → R + , we say that a sequence ( a n ) in D is u -bounded if( u ( a n )) is bounded.Let Γ ⊂ D ∨ be the set of α ∈ D ∨ such that every L -bounded sequence ( a n ) in D is also v ◦ α -bounded, where v ( λ ) = log | λ | by definition. Then Γ is a subgroupof D ∨ . It is easy to check that D ∨ / Γ is torsion-free and that Γ is W -invariant.On the other hand, by irreducibility, either Γ = { } or Γ has finite index in D ∨ .As D ∨ / Γ is torsion-free, this means that either Γ = { } or Γ = D ∨ .Suppose that L is not proper. Then there exists an sequence ( a n ) in D whichis L -bounded but not bounded. So there exists α ∈ D ∨ such that v ◦ α ( a n ) isunbounded. It follows that Γ = D ∨ . So Γ = { } . In particular, for every relativeroot α , there exists a sequence ( a n ) which is L -bounded but not α -bounded. Theargument of SL implies that L is bounded on U α , and therefore for any root α , L is bounded on D α = [ U α , U − α ]. As any element of D is a product of d elementsin S D α , we obtain that L is bounded on D . Proof of Lemma 1.7.
By the argument of Lemma 4.3, we can suppose that G issimply connected.Let L be a length on H × A , and suppose that L is proper on both H and A . Suppose that L is not proper. Then there exists a sequence ( h n , a n ) tendingto infinity in H × A so that L ( h n , a n ) is bounded. As L is bounded on compactsubsets and is proper in restriction to the factor A , the sequence ( h n ) tends toinfinity. By Lemma 5.1 below, there exist bounded sequences ( k n ), ( k ′ n ) and u in G ( K ) such that, writing d n = k n h n k ′ n , the sequence of commutators ([ d n , u ]) isunbounded. Note that L ( d n , a n ) is bounded as well.Suppose that L ( d − n , a − n ) is bounded (this holds if L is assumed coarsely sym-metric). Now L ([( d n , a n ) , ( u, L ([ d n , u ] ,
1) is bounded. But this contradictsproperness of the restriction of L to H .If L ( d − n , a − n ) is not assumed bounded, we can go on as follows. First notethat the proof of Lemma 5.1 provides ( d n ) as a sequence in the maximal splittorus D , and we assume this. If W denotes the Weyl group of D in H , then forevery d ∈ D the element Q w ∈ W wdw − of D is fixed by W , so is trivial.8ow the sequence L Y w ∈ W ( w, d n , a n )( w − , ! = L (1 , a | W | n )is bounded. Therefore, by properness on { } × A , the sequence κ n = a | W | n isbounded. Thus, the sequence L (1 , κ − n ) is bounded. Now the sequence L Y w ∈ W −{ } ( w, d n , a n )( w − , = L (( d − n , a − n )(1 , κ n ))is bounded in turn, so L ( d − n , a − n ) is bounded, and this case is settled. Lemma 5.1.