aa r X i v : . [ m a t h . G R ] S e p BOUNDARIES, WEYL GROUPS, AND SUPERRIGIDITY
URI BADER AND ALEX FURMAN
Abstract.
This note describes a unified approach to several superrigidity re-sults, old and new, concerning representations of lattices into simple algebraicgroups over local fields. For an arbitrary group Γ and a boundary actionΓ y B we associate certain generalized Weyl group W Γ ,B and show that anyrepresentation with a Zariski dense unbounded image in a simple algebraicgroup, ρ : Γ → H , defines a special homomorphism W Γ ,B → Weyl H . Thisgeneral fact allows to deduce the aforementioned superrigidity results. Introduction.
This note describes some aspects of a unified approach to a family of ”higher ranksuperrigidity” results, based on a notion of a generalized Weyl group. While thisapproach applies equally well to representations of lattices (as in the original workof Margulis [10]), and to measurable cocycles (as in the later work of Zimmer [18]),in this note we shall focus on representations only. Yet, it should be emphasizedthat our techniques do not involve any cocompactness, or integrability assumptionson lattices, and their generalizations to general measurable cocycles are ratherstraightforward. Hereafter we consider representations into simple algebraic groups;some other possible target groups are discussed in [1], [2].Let k be a local field, and H denote the locally compact group of k -points of someconnected adjoint k -simple k -algebraic group. Consider representations ρ : Γ → H with Zariski dense and unbounded image, where Γ is some discrete countable group.We shall outline a unified argument showing that for the following groups G alllattices Γ in G have the property that such a representation ρ : Γ → H can occuronly as a restriction of a continuous homomorphism ¯ ρ : G → H ; this includes:(a) G = G is the group of ℓ -points of a connected ℓ -simple ℓ -algebraic group ofrk ℓ ( G ) ≥ ℓ is a local field (Margulis [10], [12, § VII]),(b) G = G × G for G , G general locally compact groups, where Γ is assumedto be irreducible (cf. [13], [8], [6]),(c) G = Aut( X ) where X is an e A -building and G has finitely many orbits forits action on the space of chambers of X , Ch( X ).New implications of these results include non-linearity of the exotic e A -groups (de-duced from (c)), and arithmeticity vs. non-linearity dichotomy for irreducible lat-tices in products of topologically simple groups, as in [15], but with integrabilityassumptions removed. Cocycle versions of the above results cover more new ground. U.B. and A.F. were supported in part by the BSF grant 2008267.U.B was supported in part by the ISF grant 704/08.A.F. was supported in part by the NSF grants DMS 0905977.
I. Boundaries and Weyl groups.
We start from some constructions related to the source group G . In rigidity theory,a boundary of a lcsc group G is an auxiliary measure space ( B, ν ) equipped with ameasurable, measure-class preserving action of G , satisfying additional conditionsthat imply existence of measurable Γ-equivariant maps from B to some compacthomogeneous H -spaces. We shall work with the following: Definition 1.
A measurable G -space ( B, ν ) is a G - boundary if:(B1) the action G y B is amenable in the sense of Zimmer [17],(B2) the projection pr : ( B × B, ν × ν ) → ( B, ν ) is ergodic with Polish coefficients (for short, EPC), as defined below.We say that the action G y ( X, µ ) is ergodic with Polish coefficients ( EPC ) iffor every isometric G -action on any Polish metric space ( U, d ), every measurable G -equivariant map F : X → U is µ -a.e. constant. We say that a measure classpreserving G - map π : ( X, µ ) → ( Y, ν ) is
EPC (or that X is EPC relatively to Y ,when the map π is understood) if for every G -action by fiber-wise isometries ona measurable field of Polish metric spaces { ( U v , d v ) } v ∈ V over a standard Borel G -space V , every measurable G -map F : X → U = F v ∈ V U v descends to Y , i.e. thereexists a measurable f : Y → U so that µ -a.e. F coincides with f ◦ π :( X, µ ) F / / π (cid:15) (cid:15) U (cid:15) (cid:15) ( Y, ν ) / / f = = V The relative EPC property (B2) implies the (absolute) EPC property, which in turnimplies ergodicity with unitary coefficients for G y B × B and G y B . Hence G -boundaries are also strong boundaries in the sense of Burger-Monod [4]. Moreover,it can be shown that:(1) If η is a symmetric spread out generating probability measure on a lcscgroup G , then the Poisson-Furstenberg boundary ( B, ν ) of (
G, η ) is a G -boundary (further strengthening [9]).(2) If G is a simple algebraic ℓ -group and P < G is a minimal parabolic, then B = G / P with the Haar measure class is a G -boundary.(3) If ( B i , ν i ) are G i -boundaries, for i = 1 ,
2, then (
B, ν ) = ( B , ν ) × ( B , ν )is a G -boundary for the product G = G × G .(4) If Γ < G is a lattice, then every G -boundary is also a Γ-boundary.Given a G -boundary ( B, ν ) consider the group Aut G ( B × B ) of all measure classpreserving automorphisms of ( B × B, ν × ν ) which are equivariant with respectto the diagonal G -action. The flip involution w flip : ( b, b ′ ) ( b ′ , b ) is an obviousexample of such a map. A generalized Weyl group W G,B associated to a choice ofa G -boundary is Aut G ( B × B ), or a subgroup of Aut G ( B × B ) containing the flip w flip . If Γ is a lattice in G , any G -boundary B is also a Γ-boundary, and we cantake W Γ ,B = Aut G ( B × B ) . We view B and W G,B as auxiliary objects, associated (not in a unique way) to G , and encoding its implicit symmetries. Non-amenable groups have non-trivialboundaries, so their generalized Weyl groups always contain { id , w flip } ∼ = Z / Z . OUNDARIES, WEYL GROUPS, AND SUPERRIGIDITY 3
Presence of additional elements can be viewed as an indication of ”higher rankphenomena”, as in the following examples:
Examples 2. (a) Let G be a non-compact simple algebraic group and B = G / P its flag variety. Then B × B ∼ = G / Z G ( A ) as measurable G -spaces,and the generalized Weyl group coincides with the classical one:W G ,B = Aut G ( G / Z G ( A )) = N G ( A ) / Z G ( A ) = Weyl G . In particular, Weyl
PGL n ( ℓ ) ∼ = S n . Note that Weyl G = Z / Z iff rk ℓ ( G ) ≥ G = G × G be a product of non-amenable factors, take B = B × B and W G,B = { id , w , w , w flip } where the elements w i are the flips of the B i -coordinates. Note that W G,B ∼ = ( Z / Z ) ≤ W G ,B × W G ,B .(c) Let G = Aut( X ) be an e A -group, the Poisson-Furstenberg boundary for thesimple random walk can be realized on the space of chambers of the asso-ciated spherical building, B = Ch( ∂X ), with W G,B ∼ = S . This generalizesthe classical case Weyl PGL ( Q p ) ∼ = S . II. The homomorphism between Weyl groups.
Next we turn to the target group H . Denote by A < P < H the ( k -points of) amaximal k -split torus and minimal parabolic subgroup containing it. The diagonal H -action on H / P × H / P has finitely many orbits, indexed by Weyl H (Bruhatdecomposition) with a unique full-dimensional orbit, corresponding to the longelement w long ∈ Weyl H :(1) H / Z H ( A ) ⊂ H / P × H / P , h Z H ( A ) ( h P , h w long P ) . In the case of H = PGL n ( k ) the groups Z H ( A ) < P correspond to the diagonal andthe upper triangular subgroups, H / P is the space of flags ( E ⊂ E ⊂ · · · ⊂ E n )where E j is a j -dimensional subspace of k n , H / Z H ( A ) is the space of n -tuples ofone-dimensional subspaces ( ℓ , . . . , ℓ n ) with ℓ ⊕ · · · ⊕ ℓ n = k n , and (1) is( ℓ , . . . , ℓ n ) (( ℓ , ℓ ⊕ ℓ , . . . ) , ( ℓ n , ℓ n ⊕ ℓ n − , . . . )) . Here Weyl
PGL n ( k ) = S n acts by permutations on ( ℓ , . . . , ℓ n ) with the long elementw long : ( ℓ , . . . , ℓ n ) ( ℓ n , . . . , ℓ ).We have the following general result: Theorem 3.
Let Γ be a countable group, ρ : Γ → H a homomorphism with un-bounded and Zariski dense image, and ( B, ν ) be a Γ -boundary. Then (i) There exists a unique measurable Γ -equivariant map φ : B → H / P , (ii) The map φ × φ : B × B → H / P × H / P factors through the embedding H / Z H ( A ) → H / P × H / P , (iii) There exists a homomorphism π : W Γ ,B → Weyl H with π ( w flip ) = w long , ( φ × φ ) ◦ w = π ( w ) ◦ ( φ × φ ) ( w ∈ W Γ ,B ) as measurable maps B × B → H / Z H ( A ) ⊂ H / P × H / P . Let us sketch the main ingredients in the proof of this result. Consider k -algebraicactions of H on ( k -points of) k -algebraic varieties, or H - varieties for short. Orbitsof such actions are locally closed, and the actions are smooth , in the sense thatthe space of orbits is standard Borel. Since orbits of ergodic actions cannot beseparated it follows ([19, 2.1.10]) that given an ergodic measure class preservingaction Γ y ( S, σ ) any measurable Γ-equivariant map φ : S → V into a H -variety, URI BADER AND ALEX FURMAN takes values in a single H -orbit H v ⊂ V . This can be used to show the followinggeneral Lemma 4.
Let Γ y ( S, σ ) be an ergodic measure-class preserving action, and ρ : Γ → H a homomorphism. There exists φ : S → V ∼ = H / H , where H < H isan algebraic subgroup, so that for any measurable Γ -map φ : S → V to a H -variety,there is a unique H -map f : H / H → V so that φ = f ◦ φ a.e. on S . In other words, φ : S → H / H is an initial object in the category consistingof measurable Γ-maps φ : S → V to H -varieties, where morphisms between φ i : S → V i ( i = 1 ,
2) are algebraic H -maps f : V → V with φ = f ◦ φ . Theinitial object described in Lemma 4 is necessarily unique up to H -automorphismsof H / H , i.e. up to the action of N H ( H ) / H from the right. This universalitygives a homomorphism from the group of measure-class preserving automorphismsof the Γ-action to the latter groupAut Γ ( S, [ σ ]) → Aut H ( H / H ) = N H ( H ) / H . We apply this construction to the diagonal Γ-action on S = B × B , and use it todeduce Theorem 3 from the following Theorem 5.
Let ρ : Γ → H be as above, and ( B, ν ) be a Γ -boundary. Then: (i) The initial H -object for Γ y ( B, ν ) is φ : B → H / P . (ii) The initial H -object for the diagonal action Γ y ( B × B, ν × ν ) is φ × φ : B × B → H / Z H ( A ) ⊂ H / P × H / P . Note that condition (B1) of amenability of Γ y B yields a measurable Γ-mapΦ : B → Prob( H / P ). Claim (i) asserts that Φ( b ) = δ φ ( b ) are Dirac measures, andthat B admits no Γ-maps to H / H where H is a proper algebraic subgroup of P .The proof of (ii) relies on the relative EPC property of B × B → B in showing that B × B has no Γ-maps to H / H with H a proper subgroup of Z H ( A ). III. Galois correspondence.
Let B be a set and W be a group acting on B × B . This very general datum alonedefines an interesting structure that we shall now briefly describe (see [1] for moredetails).Consider possible quotients p : B → p ( B ) of B , or rather their equivalence classesdetermined by the pull-back of the Boolean algebra from p ( B ) to B . Denote by Q ( B ) the collection of all such (classes of) quotients, ordered by p ≤ p if p = j ◦ p for some j : p ( B ) → p ( B ) or, equivalently, by inclusion of the correspondingBoolean algebras. Let SG ( W ) denote all subgroups of W , ordered by inclusion,and consider maps p W p and V p V between Q ( B ) and SG ( W ) defined asfollows. Given p ∈ Q ( B ) consider the map p : B × B pr −→ B p −→ p ( B ) and define W p = { w ∈ W | p ◦ w = p } . Given a subgroup V ≤ W define the quotient p V : B → p V ( B ) to be the finest onewith V ≤ W p . Then the maps p W p , V p V , between Q ( B ) and SG ( W ), viewedas partially ordered sets, are order-reversing and satisfy V ≤ W p iff p ≤ p V . A pairof order-reversing maps between posets with above property, forms an abstract Galois correspondence ; one of the formal consequences of such a setting is that onecan define the following operations of taking a closure in Q ( B ), SG ( W ): V := W ( p V ) , p := p ( W p ) satisfying V ≤ V = V , p ≤ p = p. OUNDARIES, WEYL GROUPS, AND SUPERRIGIDITY 5
It follows that the collections of closed objects in Q ( B ) and SG ( W ) Q W ( B ) = { p : B → p ( B ) } , SG B ( W ) = { V | V ≤ W } form sub-lattices of Q ( B ), SG ( W ), on which the above Galois correspondence isan order-reversing isomorphism.These constructions can be carried over to the measurable setting, where ( B, B , ν )is a measure space, Q ( B ) consists of measurable quotients (equivalently, completesub- σ -algebras of B ), and W is assumed to preserve the measure class [ ν × ν ]. Examples 6.
With W = W G,B acting on a double of a G -boundary ( B, ν ), as inExamples 2 we get:(a) Let G be a simple algebraic group, B = G / P and W = Weyl G . Then Q Weyl G ( G / P ) = { G / P → G / Q | P ≤ Q ≤ G parabolic } corresponding to Weyl groups Weyl Levi( Q ) embedded in Weyl G . The latticeis a rk ℓ ( G )-dimensional cube.(b) Let G = G × G , B = B × B and W = { id , w , w , w flip } . Then thenon-trivial closed objects are the two factors B i , corresponding to { id , w i } .(c) For an e A -group G = Aut( X ), the non-trivial closed quotients of B =Ch( ∂X ) are face maps, corresponding to { id , (1 , } , { id , (2 , } in S .Let Γ < G be lattice where G is one of the above examples, view the G -boundary( B, ν ) as a Γ-boundary and take W Γ ,B = W . The Galois correspondence abovewas determined by W y B × B alone (without any reference to a G -action); so theconcepts of closed subgroups and closed quotients remain unchanged.Given an unbounded Zariski dense representation ρ : Γ → H , we consider theassociated map φ : B → H / P and homomorphism π : W Γ ,B → Weyl H as inTheorem 3. Then Ker( π ) is a normal subgroup in W Γ ,B which is also closed in theabove sense, and φ factors through a closed quotient corresponding to Ker( π ). Proposition 7.
Let ρ : Γ → H , φ : B → H / P , π : W Γ ,B → Weyl H be as above: (a) If Γ < G is a lattice in a simple algebraic group, then π : W Γ ,B = Weyl G → Weyl H is injective and π (w ( G )long ) = w ( H )long . (b) If Γ < G = G × G , B = B × B , and π : W Γ ,B = ( Z / Z ) → Weyl H isnon-injective, then φ factors through some φ : B pr i −→ B i φ i −→ H / P . where φ i : B i → H / P is a measurable Γ -map for some i = 1 , . (c) If Γ < G = Aut( X ) is an e A -group, then π : S → Weyl H is injective with π ((1 , ( H )long . Case (b) follows from the classification of closed subgroups and correspondingquotients for products; while (a) and (c) are consequences of a general fact ([1])that irreducible Coxeter groups, such as Weyl G , have no non-trivial normal specialsubgroups (same as closed subgroups in our context). The term lattice here refers to a partially ordered set, where any two elements x, y have a join x ∨ y and meet x ∧ y . URI BADER AND ALEX FURMAN
IV. The final step.
Proposition 7 already suffices to deduce some superrigidity results of the type “cer-tain Γ admits no unbounded Zariski dense homomorphisms to certain H ” . Forexample, this is the case if H is a simple k -algebraic group with rk k ( H ) = 1, whileΓ is an exotic e A -group or a lattice in a simple ℓ -algebraic group G with rk ℓ ( G ) ≥ G in Weyl H preserving the long element, can be ruled out inmany other cases, such as G = PGL ( ℓ ) and H = PGL ( k ).However, in the context of Proposition 7 one has more precise information:(a) If Γ is a lattice in a simple algebraic group G , then π : W Γ ,B = Weyl G → Weyl H is an isomorphism of Coxeter groups.(b) If Γ < G = G × G , B = B × B , then π : W Γ ,B = ( Z / Z ) → Weyl H has Im( π ) ∼ = Z / Z .(c) If Γ < G = Aut( X ) is an e A -group, then π : S → Weyl H is an isomorphismof Coxeter groups.The proof of these claims relies on classification of measurable Γ-factors of Γ-boundaries proved by Margulis [11], Bader-Shalom [3], and Shalom-Steger [16],respectively. Let us turn to consequences of such statements. Theorem 8.
Let Γ < G = G × G be a lattice with pr i (Γ) dense in G i ( i = 1 , ),and ρ : Γ → H a Zariski dense unbounded representation. Then ρ extends to G and factors through a continuous homomorphism ρ i : G i → H of a factor. This follows from the fact that π : W Γ ,B → Weyl H has two element image,Proposition 7.(b), and the following general Lemma 9.
Let G y ( B , ν ) be a measure class preserving action of some locallycompact group G , Γ any group, p : Γ → G a homomorphism with dense image;and let φ : B → H / P a measurable Γ -equivariant map. Then there is a continuoushomomorphism ¯ ρ : G → H so that ρ = ¯ ρ ◦ p . The proof of this lemma utilizes the enveloping semigroup of H y H / P , whichcan be identified as the quasi-projective transformations of H / P , introduced in [7].The final treatment of case (a) (the main case of Margulis’s superrigidity) andthe non-linearity result for exotic e A -groups (as in (c)) are deduced by a reductionto some results of Tits on buildings. If G = G is a simple ℓ -algebraic group let∆ = ∆ G denote the spherical building of G , if G = Aut( X ) is an e A -group, let∆ = ∂X denote the spherical building associated to the Affine building X . Let∆ ′ = ∆ H denote the spherical building of H . For a building ∆ denote by Ch(∆) (2) the subspace of Ch(∆) × Ch(∆) consisting of pairs of opposite chambers.
Theorem 10.
Let ∆ and ∆ ′ be two spherical buildings of rank ≥ with the sameWeyl group. Let ν be a probability measure on Ch(∆) with supp( ν ) = Ch(∆) and ( ν × ν )(Ch(∆) (2) ) = 1 , and let φ : Ch(∆) → Ch(∆ ′ ) be a measurable map so that ( φ ∗ ν × φ ∗ ν )(Ch(∆ ′ ) (2) ) = 1 and ( φ × φ ) ◦ w = w ◦ ( φ × φ ) ( w ∈ W ∆ ) . Then there exists an imbedding of buildings ∆ → ∆ ′ which induces a continuousmap Ch(∆) → Ch(∆ ′ ) which agrees ν -a.e. with φ . The proof is now completed by invoking results of Tits, on reconstructing G from ∆ G , and X from ∂X . OUNDARIES, WEYL GROUPS, AND SUPERRIGIDITY 7
References [1] U. Bader and A. Furman,
Superrigidity via Weyl groups: hyperbolic-like targets . preprint.[2] U. Bader, A. Furman, and A. Shaker,
Superrigidity via Weyl groups: actions on the circle .preprint.[3] U. Bader and Y. Shalom,
Factor and normal subgroup theorems for lattices in products ofgroups , Invent. Math. (2006), no. 2, 415–454.[4] M. Burger and N. Monod,
Continuous bounded cohomology and applications to rigidity the-ory , Geom. Funct. Anal. (2002), no. 2, 219–280.[5] M. Burger and S. Mozes, CAT (- )-spaces, divergence groups and their commensurators , J.Amer. Math. Soc. (1996), no. 1, 57–93.[6] M. Burger, S. Mozes, and R. J. Zimmer, Linear representations and arithmeticity of latticesin products of trees , Essays in geometric group theory, Ramanujan Math. Soc. Lect. NotesSer., vol. 9, Ramanujan Math. Soc., Mysore, 2009, pp. 1–25.[7] H. Furstenberg,
A note on Borel’s density theorem , Proc. Amer. Math. Soc. (1976), no. 1,209–212.[8] T. Gelander, A. Karlsson, and G. A. Margulis, Superrigidity, generalized harmonic maps anduniformly convex spaces , Geom. Funct. Anal. (2008), no. 5, 1524–1550.[9] V. A. Kaimanovich, Double ergodicity of the Poisson boundary and applications to boundedcohomology , Geom. Funct. Anal. (2003), no. 4, 852–861.[10] G. A. Margulis, Discrete groups of motions of manifolds of nonpositive curvature , Proceed-ings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 2, Canad.Math. Congress, Montreal, Que., 1975, pp. 21–34 (Russian).[11] ,
Finiteness of quotient groups of discrete groups , Funkts. Anal. Prilozh. (1979),28–39.[12] , Discrete subgroups of semisimple Lie groups , Ergebnisse der Mathematik und ihrerGrenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag,Berlin, 1991.[13] N. Monod,
Superrigidity for irreducible lattices and geometric splitting , J. Amer. Math. Soc. (2006), no. 4, 781–814.[14] , Continuous bounded cohomology of locally compact groups , Lecture Notes in Math-ematics, vol. 1758, Springer-Verlag, Berlin, 2001.[15] ,
Arithmeticity vs. nonlinearity for irreducible lattices , Geom. Dedicata (2005),225–237.[16] Y. Shalom and T. Steger. unpublished.[17] R. J. Zimmer,
Amenable ergodic group actions and an application to Poisson boundaries ofrandom walks , J. Functional Analysis (1978), no. 3, 350–372.[18] R. J. Zimmer, Strong rigidity for ergodic actions of semisimple Lie groups , Ann. of Math.(2) (1980), no. 3, 511–529.[19] ,
Ergodic theory and semisimple groups , Monographs in Mathematics, vol. 81,Birkh¨auser Verlag, Basel, 1984.
Technion, Haifa
E-mail address : [email protected] University of Illinois at Chicago, Chicago
E-mail address ::