Boundaries, Weyl groups, and Superrigidity

This note describes a unified approach to several superrigidity results, old and new, concerning representations of lattices into simple algebraic groups over local fields. For an arbitrary group $\Gamma$ and a $\Gamma$-boundary $B$ we associate certain generalized Weyl group $W_{\Gamma,B}$ and show that any representation with a Zariski dense unbounded image in a simple algebraic group, $\rho:\Gamma\to \mathbf{H}$, defines a special homomorphism $W_{\Gamma,B}\to {\rm Weyl}(\mathbf{H})$. This general fact allows to deduce the aforementioned superrigidity results.


Introduction.
This note describes some aspects of a unified approach to a family of "higher rank superrigidity" results, based on a notion of a generalized Weyl group. While this approach applies equally well to representations of lattices (as in the original work of 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 emphasized that our techniques do not involve any cocompactness, or integrability assumptions on lattices, and their generalizations to general measurable cocycles are rather straightforward. 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 some connected 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 all lattices Γ in G have the property that such a representation ρ : Γ → H can occur only as a restriction of a continuous homomorphismρ : G → H; this includes: (a) G = G is the group of ℓ-points of a connected ℓ-simple ℓ-algebraic group of rk ℓ (G) ≥ 2 where ℓ is a local field (Margulis [10], [12,§VII]), (b) G = G 1 ×G 2 for G 1 , G 2 general locally compact groups, where Γ is assumed to be irreducible (cf. [13], [8], [6]), (c) G = Aut(X) where X is an A 2 -building and G has finitely many orbits for its action on the space of chambers of X, Ch(X).
New implications of these results include non-linearity of the exotic A 2 -groups (deduced from (c)), and arithmeticity vs. non-linearity dichotomy for irreducible lattices in products of topologically simple groups, as in [15], but with integrability assumptions removed. Cocycle versions of the above results cover more new ground.

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 a measurable, measure-class preserving action of G, satisfying additional conditions that imply existence of measurable Γ-equivariant maps from B to some compact homogeneous H-spaces. We shall work with the following: is amenable in the sense of Zimmer [17], (B2) the projection pr 1 : (B×B, ν×ν) → (B, ν) is ergodic with Polish coefficients (for short, EPC), as defined below.
We say that the action G (X, µ) is ergodic with Polish coefficients (EPC ) if for 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 class preserving 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 on a measurable field of Polish metric spaces there exists a measurable f : Y → U so that µ-a.e. F coincides with f • π: The relative EPC property (B2) implies the (absolute) EPC property, which in turn implies ergodicity with unitary coefficients for G B × B and G B. Hence Gboundaries 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 lcsc group G, then the Poisson-Furstenberg boundary (B, ν) of (G, η) is a Gboundary (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.
is a G-boundary for the product G = G 1 × G 2 . (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 class preserving automorphisms of (B × B, ν × ν) which are equivariant with respect to the diagonal G-action. The flip involution w flip : is an obvious example of such a map. A generalized Weyl group W G,B associated to a choice of a 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 can take 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-trivial boundaries, so their generalized Weyl groups always contain {id, w flip } ∼ = Z/2Z.
Presence of additional elements can be viewed as an indication of "higher rank phenomena", as in the following examples: (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: II. The homomorphism between Weyl groups.
Next we turn to the target group H. Denote by A < P < H the (k-points of) a maximal 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 (Bruhat decomposition) with a unique full-dimensional orbit, corresponding to the long element w long ∈ Weyl H : In the case of H = PGL n (k) the groups Z H (A) < P correspond to the diagonal and the upper triangular subgroups, H/P is the space of flags ( is the space of n-tuples of one-dimensional subspaces (ℓ 1 , . . . , ℓ n ) with ℓ 1 ⊕ · · · ⊕ ℓ n = k n , and (1) is Here Weyl PGLn(k) = S n acts by permutations on (ℓ 1 , . . . , ℓ n ) with the long element w long : (ℓ 1 , . . . , ℓ n ) → (ℓ n , . . . , ℓ 1 ). We have the following general result:  We apply this construction to the diagonal Γ-action on S = B × B, and use it to deduce Theorem 3 from the following Theorem 5. Let ρ : Γ → H be as above, and (B, ν) be a Γ-boundary. Then:

III. Galois correspondence.
Let B be a set and W be a group acting on B × B. This very general datum alone defines an interesting structure that we shall now briefly describe (see [1] for more details). Consider Given a subgroup V ≤ W define the quotient p V : B → p V (B) to be the finest one with V ≤ W p . Then the maps p → W p , V → p V , between Q(B) and SG(W ), viewed as partially ordered sets, are order-reversing and satisfy V ≤ W p iff p ≤ p V . A pair of order-reversing maps between posets with above property, forms an abstract Galois correspondence; one of the formal consequences of such a setting is that one can define the following operations of taking a closure in Q(B), SG(W ): It follows that the collections of closed objects in Q(B) and SG(W ) form sub-lattices 1 of Q(B), SG(W ), on which the above Galois correspondence is an 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, complete sub-σ-algebras of B), and W is assumed to preserve the measure class [ν × ν]. 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 above was determined by W B × B alone (without any reference to a G-action); so the concepts of closed subgroups and closed quotients remain unchanged.
Given an unbounded Zariski dense representation ρ : Γ → H, we consider the associated map φ : B → H/P and homomorphism π : W Γ,B → Weyl H as in Theorem 3. Then Ker(π) is a normal subgroup in W Γ,B which is also closed in the above 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 Case (b) follows from the classification of closed subgroups and corresponding quotients 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 special subgroups (same as closed subgroups in our context).
IV. The final step. Proposition 7 already suffices to deduce some superrigidity results of the type "certain Γ admits no unbounded Zariski dense homomorphisms to certain H". For example, this is the case if H is a simple k-algebraic group with rk k (H) = 1, while Γ is an exotic A 2 -group or a lattice in a simple ℓ-algebraic group G with rk ℓ (G) ≥ 2. Embedding of Weyl G in Weyl H preserving the long element, can be ruled out in many other cases, such as G = PGL 4 (ℓ) and H = PGL 3 (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.
Theorem 8. Let Γ < G = G 1 × G 2 be a lattice with pr i (Γ) dense in G i (i = 1, 2), 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 1 (B 1 , ν 1 ) be a measure class preserving action of some locally compact group G 1 , Γ any group, p : Γ → G 1 a homomorphism with dense image; and let φ 1 : B 1 → H/P a measurable Γ-equivariant map. Then there is a continuous homomorphismρ 1 : G 1 → H so that ρ =ρ 1 • p.
The proof of this lemma utilizes the enveloping semigroup of H H/P, which can 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) and the non-linearity result for exotic A 2 -groups (as in (c)) are deduced by a reduction to 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 A 2 -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.
The proof is now completed by invoking results of Tits, on reconstructing G from ∆ G , and X from ∂X.