Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds
aa r X i v : . [ m a t h . D S ] J un ARITHMETICITY, SUPERRIGIDITY AND TOTALLYGEODESIC SUBMANIFOLDS OF COMPLEX HYPERBOLICMANIFOLDS
URI BADER, DAVID FISHER, NICHOLAS MILLER, AND MATTHEW STOVER
To Domingo Toledo with admiration on the occasion of his 75th birthday
Abstract.
For n ≥
2, we prove that a finite volume complex hyper-bolic n -manifold containing infinitely many maximal properly immersedtotally geodesic submanifolds of dimension at least two is arithmetic,paralleling our previous work for real hyperbolic manifolds. As in thereal hyperbolic case, our primary result is a superrigidity theorem forcertain representations of complex hyperbolic lattices. The proof re-quires developing new general tools not needed in the real hyperboliccase. Our main results also have a number of other applications. Forexample, we prove nonexistence of certain maps between complex hy-perbolic manifolds, which is related to a question of Siu, that certainhyperbolic 3-manifolds cannot be totally geodesic submanifolds of com-plex hyperbolic manifolds, and that arithmeticity of complex hyperbolicmanifolds is detected purely by the topology of the underlying com-plex variety, which is related to a question of Margulis. Our resultsalso provide some evidence for a conjecture of Klingler that is a broadgeneralization of the Zilber–Pink conjecture. Introduction
Throughout this paper, a geodesic submanifold will always mean a prop-erly immersed totally geodesic subspace, and a geodesic submanifold is called maximal if it is not contained in a proper geodesic submanifold of larger di-mension. In this paper, we will prove the following.
Theorem 1.1 (Arithmeticity) . Suppose that n ≥ and M is a finite volumecomplex hyperbolic n -manifold containing infinitely many maximal totallygeodesic submanifolds of dimension at least . Then M is arithmetic. In previous work, we proved arithmeticity of real hyperbolic manifoldscontaining infinitely many maximal geodesic subspaces of dimension at leasttwo [2, Thm. 1.1], which answered a question independently due to AlanReid and Curtis McMullen [16]. As in that case, Theorem 1.1 can be restatedpurely in terms of homogeneous dynamics, where one obtains the exact samestatement as [2, Thm. 1.5]. We also note that, after learning of our result,Baldi and Ullmo recently gave a very different proof of Theorem 1.1 in thespecial case of totally geodesic complex subvarieties [4].
To answer Reid and McMullen’s question we introduced the notion of compatibility of an algebraic group H over a local field k with a semisimpleLie group G and used this with G = SO ( n,
1) to prove a superrigiditytheorem for certain representations of real hyperbolic lattices [2, Thm. 1.6].However, for G = SU( n,
1) superrigidity with compatible targets is notenough to prove arithmeticity. Thus, in this paper we introduce new toolsfor proving superrigidity theorems in rank one with target groups that arenot compatible with G . While these new tools can be used for SO ( n, G = SU( n,
1) to prove Theorem 1.1 and the other applications describedlater in this introduction.It is well-known that if G is a connected adjoint semisimple Lie groupwith no compact factors and Γ < G is an irreducible nonarithmetic lattice,then G is necessarily isomorphic to either PO ( n,
1) or PU( n,
1) for some n ≥
2. Therefore, combining [2, Thm. 1.1] and Theorem 1.1 with the factthat geodesic subspaces of arithmetic manifolds are arithmetic yields:
Corollary 1.2 (Finiteness) . Let M be a nonarithmetic finite volume irre-ducible locally symmetric space of noncompact type. Then (1) M contains only finitely many maximal geodesic subspaces of dimen-sion at least , and (2) M contains only finitely many nonarithmetic geodesic subspaces ofdimension at least . Note that a complex hyperbolic n -manifold can contain geodesic sub-manifolds of dimension k that are real hyperbolic (2 ≤ k ≤ n ) or complexhyperbolic (2 ≤ k ≤ n − § , M as an algebraic variety in Theorem 1.6 below.We now state our general superrigidity result. Part (1) mirrors [2, Thm.1.6]. However, the tools developed in [2] cannot be used to prove part (2). RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 3
The heart of this paper develops new tools for proving superrigidity andapplies them to prove part (2).
Theorem 1.3 (Superrigidity) . Let G be SU( n, for n ≥ , W < G bea noncompact connected almost simple subgroup, and Γ < G be a lattice.Suppose that k is a local field, H is a connected adjoint k -algebraic group,and ρ : Γ → H ( k ) is a homomorphism with unbounded, Zariski dense image.Moreover, suppose that there is a faithful irreducible representation of H ( k ) on a k -vector space V of dimension at least two and a W -invariant, ergodicmeasure ν on ( G × P ( V )) / Γ that projects to Haar measure on G/ Γ . If either (1) the pair consisting of k and H is compatible with G , or (2) k = R and H ( R ) ∼ = PU( n, ,then ρ extends to a continuous homomorphism from G to H ( k ) . We briefly recall the definition of compatibility from [2]. Let P be a min-imal parabolic subgroup of G and U its unipotent radical. A pair consistingof a local field k and a k -algebraic group H is said to be compatible with G iffor every nontrivial k -subgroup J < H and any continuous homomorphism τ : P → N H ( J ) / J ( k ), where N H ( J ) is the normalizer of J in H , we havethat the Zariski closure of τ ( U ′ ) coincides with the Zariski closure of τ ( U )for every nontrivial subgroup U ′ < U . We will extend this definition andelaborate on it in § Remark 1.4.
Margulis asked when superrigidity holds for arithmetic com-plex hyperbolic lattices [34, Prob. 9], and our results provide a partial answerto his question. Indeed, Theorem 1.3 proves superrigidity of certain repre-sentations of arithmetic complex hyperbolic lattices, and we will describeapplications of this later in the introduction. We also note that there area number of previous superrigidity results for particular representations ofcomplex hyperbolic lattices. For example, see the famous work of Toledo [55]and Corlette [14] on what are now known as maximal representations , andsee more recent work of Burger–Iozzi [8], Pozzetti [46], and Koziarz–Maubon[30] for more on results of this kind and historical remarks. Another famousexample is the proof by Klingler of superrigidity of low-dimensional repre-sentations of fundamental groups of fake projective planes, which he thenused to deduce their arithmeticity [28]; see [29] for a more general resultalong these lines.We now describe the general ideas behind the proof of Theorem 1.3(2).The failure of PU( n,
1) to be compatible with SU( n,
1) can be measuredprecisely, and we describe this in §
3. Specifically, we define an incompatibilitydatum for a group G with respect to an algebraic group H over a local field k as a measure of the failure of compatibility. When G is a real semisimplegroup, Proposition 3.2 shows that the only relevant case is when k is R or C and Lemma 3.3 proves that an incompatibility datum for the pair ( k, H )is associated with a particular parabolic subgroup Q of H ( k ). U BADER, D FISHER, N MILLER, AND M STOVER
When G = SU( n, k = R , and H ( k ) = PU( n, chains on the boundary of complex hyperbolic space. This problemwas solved by Pozzetti [46, Thm. 1.6] (see Theorem 6.5 for a statement inour language), and this allows us to complete the proof of Theorem 1.3(2).More generally, the tools developed in § § M is a finite volume complex hyperbolic n -manifold, n ≥ < SU( n,
1) is the associated lattice. Canonically associatedwith Γ is its adjoint trace field ℓ , and the Zariski closure of Γ under theadjoint representation is an adjoint simple ℓ -algebraic group G . The latticeembedding Γ ֒ → PU( n,
1) is determined by a real place v of ℓ so that G ( ℓ v ) ∼ = PU( n, ℓ v denotes the completion of ℓ at a place v . ThenΓ is arithmetic if and only if Γ is precompact in G ( ℓ v ) for all infinite andfinite places v = v of ℓ .In proving the analogue of Theorem 1.1 in the real hyperbolic setting, thepairs ℓ v and G one encounters are always compatible with SO ( n,
1) and sothe analogue of Theorem 1.3(1) is all we needed. Certain pairs of interest tous in the proof of Theorem 1.1, particularly ( R , PU( r, s )) with 2 ≤ s ≤ r < n and r + s = n + 1, are compatible with SU( n, ℓ v = R and G ( ℓ v ) ∼ = PU( n, C , SL n +1 ( C )).To finish the proof of Theorem 1.1, we apply Simpson’s celebrated resultson linear representations of K¨ahler groups [52] and their generalization tothe quasi-projective case. These methods actually lead to a more generaltheorem about the possibilities for the adjoint trace field ℓ of a lattice Γin SU( n,
1) and the associated ℓ -algebraic group. In § Theorem 1.5 (Hodge type and integral) . Let Γ < SU( n, be a lattice, ℓ be its adjoint trace field, and G be the absolutely almost simple simplyconnected ℓ -algebraic group canonically associated with Γ . Then: (1) ℓ is totally real and the quadratic extension ℓ ′ /ℓ associated with G as a group of type A n is totally complex; RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 5 (2) for each real place v of ℓ , G ( ℓ ⊗ v R ) is isomorphic to SU( r v , s v ) forsome r v , s v ≥ with r v + s v = n + 1 ; (3) Γ is integral, i.e., there is an action of G ( ℓ ) on an ℓ -vector space V and an O ℓ -lattice L ⊂ V so that a finite index subgroup of Γ stabilizes L .In particular, if v is the place of ℓ associated with the lattice embedding of Γ in SU( n, , then Γ is arithmetic if and only if G ( ℓ v ) ∼ = SU( n + 1) for allarchimedean places v = v of ℓ . The first two statements and their proofs were described to us by DomingoToledo, and the third follows from work of Esnault–Groechenig [19]. Toprove Theorem 1.1, we only need parts (1) and (2) of Theorem 1.5, and inthe cocompact case these are immediate from Simpson’s result that rigidrepresentations of K¨ahler groups are of Hodge type [52, Lem. 4.5]. Thereis in fact considerable overlap between the cases in Theorem 1.1 coveredby Theorems 1.3(1) and 1.5. Theorem 1.5(3) rules out the cases where ℓ v is nonarchimedean, but these cases are perhaps more easily handled byTheorem 1.3(1), since any simple algebraic group over a nonarchimedeanlocal field is very easily seen to be compatible with G .We now describe some other applications and interpretations of our resultsin the language of algebraic and complex geometry.One application of Theorem 1.1 is to provide evidence for a conjectureof Klingler [27, Conj. 1.12]. Let M = B n / Γ be a finite volume complexhyperbolic manifold of complex dimension n ≥
2. As discussed above, thereexists a totally real number field with fixed embedding v : ℓ → R andan almost simple ℓ -algebraic group G with G ( ℓ ⊗ v R ) ∼ = SU( n,
1) suchthat, up to passing to a subgroup of finite index, Γ ⊆ G ( O ℓ ), where O ℓ denotes the ring of integers of ℓ . Let H denote the Weil restriction of scalarsfrom ℓ to Q of G , which is a semisimple Q -group with Γ ⊆ H ( Z ). Anyfaithful representation ρ : H → GL( V ) defined over Z induces a polarizable Z -variation of Hodge structure V on M .As soon as Γ is nonarithmetic, the group H ( ℓ ⊗ Q R ) admits at leasttwo noncompact factors and one easily checks that any totally geodesicsubvariety of M is atypical for ( M, V ) in the sense of [27, Def. 1.7]. One canin fact check that maximal totally geodesic subvarieties of M are optimalin the sense of [27, Def. 1.8]. Then [27, Conj. 1.12] implies that M containsat most finitely many maximal totally geodesic subvarieties. Therefore, ourresults confirm this consequence of [27, Conj. 1.12]. See [4] for more aboutthese connections.An application in another direction is to Margulis’s question as to whetherarithmeticity can be detected at the topological level. We explain in § U BADER, D FISHER, N MILLER, AND M STOVER manifold. In fact, arithmeticity is recognized by the structure of the inter-section product on cohomology. For simplicity, we state here our result forcomplex dimension 2 and refer to § X is a smooth complex projective variety with canonicaldivisor K X and D ⊂ X is a (possibly empty) smooth divisor such that M = X r D is a complex hyperbolic 2-manifold for which X is a smoothtoroidal compactification of M . We recall that any complex projective curve C on X satisfies(1) 3 C · C + 3 deg( D ∩ C ) ≥ − K X · C + 2 D · C, with respect to the intersection pairing on H ( X ). Note that D is empty ifand only if M is compact, where Equation (1) reduces to3 C · C ≥ − K X · C. We also note that every complex hyperbolic manifold admits a finite cov-ering with a compactification as assumed in Theorem 1.6; see the proof ofTheorem 1.5 for a precise discussion. Therefore there is no loss of generalityin making the assumptions in Theorem 1.6.
Theorem 1.6 (Arithmeticity and the intersection pairing) . If X containsinfinitely many complex projective curves C where equality holds in (1) then M is arithmetic. Equivalently, if M is nonarithmetic, then there are onlyfinitely many curves C on X for which equality holds in (1) . Another application of Theorem 1.3 is to ruling out behavior of mappingsbetween complex hyperbolic manifolds. For example, Siu asked whetheror not there are surjective holomorphic mappings M → M between ballquotients with 2 ≤ dim( M ) < dim( M ) [53]. The only progress thus faris by Koziarz and Mok, who ruled out the case of holomorphic submersions[32]. Using Theorem 1.3 we prove the following in § Theorem 1.7 (Nonexistence of certain maps) . Let M be a finite volumecomplex hyperbolic manifold of complex dimension n ≥ containing anequidistributed family { Z i } of geodesic submanifolds, all of dimension atleast two. Suppose that N is a finite volume complex hyperbolic manifold N with dim C ( N ) = d ≤ n and f : M → N is a continuous map such that f ( Z i ) is contained in a proper geodesic submanifold of N for all i and f ∗ ( π ( M )) ≤ π ( N ) < PU( m, is Zariski dense. Then d = n and f is homotopic to a finite cover. Our last application of our results is the following. In § RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 7
Theorem 1.8 (Restricting geodesic submanifolds) . There are both closedand noncompact finite volume hyperbolic -manifolds that are not isomet-ric to an immersed totally geodesic submanifold of a complex hyperbolic n -manifold for any n . We now discuss the organization of this paper. Section 2 starts by dis-cussing SU( n,
1) and some of its subgroups. In § n,
1) with certain algebraic groupsover local fields and study the failure of PU( n,
1) to be compatible withSU( n, § § § §
7, we prove Theorem 1.5 and give someadditional algebraic setup for the proof of Theorem 1.1, which is containedin §
8. Then § Acknowledgments.
The authors thank Domingo Toledo for describing howSimpson’s work leads to Theorem 1.5 and Olivier Biquard for correspon-dence about the nonuniform case. We thank Bruno Klingler for explain-ing the relationship to his conjecture discussed in the introduction and toVenkataramana, Nicolas Bergeron, and Emmanuel Ullmo for insightful con-versations. We also thank Jean Lecureaux and Beatrice Pozzetti for conver-sations related to the proof of Theorem 1.3(2). Bader was supported by theISF Moked 713510 grant number 2919/19. Fisher was supported by NSFDMS-1906107. Stover was supported by Grant Number 523197 from theSimons Foundation/SFARI and NSF DMS-1906088.2.
Preliminaries on
SU( n, G ∼ = SU( n,
1) with n ≥
2. We will consider G as being the group of realpoints of a real algebraic group.2.1. The group G and its standard subgroups. In this subsection wefix some notation and give convenient coordinates for certain subgroups of G that will be important in what follows.We start by giving a convenient matrix representation for G . The groupU( n,
1) is often described as the automorphism group of the hermitian form h ( x , . . . , x n +1 ) = n X i =1 | x i | − | x n +1 | , U BADER, D FISHER, N MILLER, AND M STOVER on C n +1 . Under the linear change of variables y = √
22 ( x + x n +1 ) , y = x , . . . , y n = x n , y n +1 = √
22 ( x − x n +1 )this form becomes(2) h ( y , . . . , y n +1 ) = y y n +1 + y n +1 y + n X i =2 | y i | . Hereafter, we will view G as the subgroup of SL n +1 ( C ) preserving h . Remark 2.1.
For n = 1, the stabilizer of h in SL ( C ) is the image of thehomomorphism: SL ( R ) → SL ( C ) (cid:18) a bc d (cid:19) (cid:18) a ib − ic d (cid:19) This explicitly realizes the well-known isomorphism SU(1 , ∼ = SL ( R ). Wewill tacitly assume from here forward that n ≥ e i denote the i th standard basis vector in C n +1 given by y i = 1 and y j = 0 for j = i . Note that the restriction of h to the complex linespanned by the vector e − e n +1 is negative definite. We denote by K ≤ G the stabilizer in G of this line. Then K also stabilizes the h -orthogonalcomplement of this line, namely the complex hyperplane spanned by thevectors e , . . . , e n and e + e n +1 , and note that h is positive definite on thishyperplane. One sees easily that K ∼ = S(U( n ) × U(1)) ≃ U( n ) , and that it is a maximal compact subgroup of G . In particular, every com-pact subgroup of G is conjugate to a subgroup of K .For 1 ≤ m ≤ n let W cm ≤ G be the subgroup of G fixing each of the n − m standard basis vectors e m +1 , . . . , e n ∈ C n +1 , and note that W cm ∼ = SU( m, W cm ∩ GL n +1 ( R ) < GL n +1 ( C ) , which is isomorphic to SO( m,
1) and we denote its identity component by W rm , thus W rm ∼ = SO ( m, Definition 2.2.
The subgroups W c , . . . , W cn and W r , . . . , W rn of G definedabove are said to be the standard almost simple subgroups , or for short just the standard subgroups of G .Identifying K \ G with complex hyperbolic n -space B n , the standard sub-groups of G have a special relationship with its totally geodesic subspaces.It is shown in § § n -space with real di-mension at least two is isometric to either real hyperbolic m -space for some2 ≤ m ≤ n or complex hyperbolic m -space for some 1 ≤ m ≤ n . RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 9
Moreover, the group G acts transitively on the collection of totally ge-odesic subspaces of any given type , where the type of a totally geodesicsubspace describes whether it is isometric to real or complex hyperbolicspace of a given fixed dimension. We note that a real hyperbolic 2-planehas a different type from a complex hyperbolic line. Indeed, the former hasa constant sectional curvature − /
4, while the latter has − W ≤ G , W ∩ K is a maximalcompact subgroup of W and that( W ∩ K ) \ W ∼ −→ K \ KW ⊆ K \ G, is a totally geodesic embedding of a real or complex hyperbolic space ofthe corresponding type in K \ G . The following proposition summarizes theabove discussion. Proposition 2.3 ([21, § . The totally geodesic subspaces of (real) di-mension at least two in the symmetric space K \ G are exactly the subsets ofthe form K \ KW g for an element g ∈ G and a standard subgroup W ≤ G corresponding to the type of the given totally geodesic subspace. Proposition 2.3 will play a prominent role in § G . Proposition 2.4.
The standard subgroups of G are noncompact, connected,almost simple, closed subgroups, and every noncompact, connected, almostsimple, closed subgroup of G is conjugate to a unique standard subgroup of G . Definition 2.5.
Given a noncompact, connected, almost simple, closed sub-group of G , we say that its type is the type of the unique standard subgroupof G to which it is conjugate. Remark 2.6.
The analogue of Proposition 2.4 for SO ( n,
1) holds as well:every noncompact, connected, almost simple, closed subgroup of SO ( n, ( m,
1) for some2 ≤ m ≤ n . The proof of this fact is similar to the proof of Proposition 2.4to be presented below, only it is easier and well-known, so we will omit it.The proof of Proposition 2.4 will be derived simultaneously with the proofof Lemma 2.7 given below. For a subgroup S ≤ G , we use the notation S + to denote the closed subgroup of G generated by all the one-dimensionalunipotent subgroups of S . We note that S + is necessarily connected, and itis either a unipotent subgroup or a noncompact, almost simple subgroup of G . This follows from the fact that G has rank 1. Indeed, if S + is containedin a parabolic subgroup P then it is contained in P + , which is the unipotentradical of P , hence S + is unipotent. Otherwise, S + has a trivial unipotentradical, hence it is semisimple, and since it has no compact factor, it followsthat it is almost simple and noncompact. Lemma 2.7.
Fix a standard subgroup W ≤ G and let N ≤ G be its nor-malizer. Then the following results hold. (1) If S ≤ G is a connected, almost simple, closed subgroup that pre-serves K \ KW and acts transitively on it, then S = W . (2) The stabilizer in G of the totally geodesic subspace K \ KW is N .Moreover, N ⊆ KW thus N/W is compact and N + = W . (3) Assume that S ≤ G is a closed intermediate subgroup W ≤ S ≤ N .Then S + = W and K \ KW = K \ KS = K \ KN . (4) Let S ≤ G be a closed subgroup containing W . Then there exists k ∈ K and a standard subgroup W ≤ G such that kS + k − = W and K \ KS = K \ KS + = K \ KW k. Moreover, K \ KS is a totally geodesic subspace of K \ G whose vol-ume measure coincides, up to normalization, with the push-forwardof Haar measure on S to ( S ∩ K ) \ S ≃ K \ KS .Proofs of Proposition 2.4 and Lemma 2.7. We first prove part (1) of thelemma. Let N be the stabilizer in G of K \ KW and observe that S isa subgroup of N and S = S + ≤ N +0 . It follows that N +0 is not unipotent,thus it is connected and almost simple. We will show that in fact S = N +0 .Since N acts by isometries on the symmetric space Z = W ∩ K \ W ≃ K \ KW we have a natural continuous homomorphism from N to the groupof isometries, Isom( Z ), endowed with the compact open topology. It goesback to Elie Cartan that the image of S coincides with the identity com-ponent of Isom( Z ), as S is semisimple connected group of isometries of thesymmetric space Z . The same holds for N +0 . Since both S and N +0 are al-most simple, the restriction of the above homomorphism to each has a finitekernel. It follows that S has finite index in N +0 , hence indeed S = N +0 , since N +0 is connected.In the above discussion S was arbitrary, thus applying it in the specialcase S = W we conclude that W = N +0 . Therefore S = W holds a priori and this proves part (1).Next we prove part (2) of the lemma. By the previous discussion we havethat N ≤ N , since W = N +0 is normal in N . Observe that K \ KW is theunique W -invariant totally geodesic subspace of its type. Indeed, given any W -invariant totally geodesic subspace of the same type Z ⊆ K \ G and z ∈ Z ,the function d ( · , zW ) measuring distance to the W -orbit of z is constant on K \ KW . This implies that zW and K \ KW have identical boundaries inthe visual compactification of K \ G . Since K \ G has negative curvature, thisimplies that zW ⊂ K \ KW , hence Z = K \ KW .It also follows that N ≤ N , so N = N and N is indeed the stabilizerin G of the totally geodesic subspace K \ KW . We furthermore see that N + = N +0 = W . For every n ∈ N , it also follows that K \ Kn = K \ Kw forsome w ∈ W , thus N ⊆ KW . This proves part (2). RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 11
Part (3) of the lemma follows immediately from part (2). Indeed, noticethat W = W + ≤ S + ≤ N + = W implies that S + = W and the sequence ofinclusions K \ KW ⊆ K \ KS ⊆ K \ KN ⊆ K \ K ( KW ) = K \ KW then has equality everywhere.We now turn to the proof of Proposition 2.4. The fact that the standardsubgroups are pairwise nonconjugate, noncompact, connected, almost sim-ple, and closed subgroups of G is obvious, so we only need to show thatany other such group is conjugate to a standard one. Let S ≤ G be a non-compact, connected, almost simple, closed subgroup. By the Karpelevich–Mostow Theorem [26, 41] for some h ∈ G the S -orbit K \ KhS ⊆ K \ G istotally geodesic. Thus, by Proposition 2.3, there exist an element g ∈ G anda standard subgroup W ≤ G such that K \ KhS = K \ KW g . Rewriting wehave K \ Khg − S g = K \ KW and we conclude that S g preserves K \ KW and acts transitively on it. By part (1) we get that S g = W and this provesProposition 2.4.We are now in a position to prove part (4) of the lemma. Note that W ≤ S implies that W = W + ≤ S + . It follows that S + is not unipotent,thus it is connected and almost simple. Using Proposition 2.4 there existan element g ∈ G and a standard subgroup W ≤ G such that ( S + ) g = W .We conclude that W g ≤ W . We claim that the group W g , which is anoncompact, connected, almost simple, closed subgroup of W , is conjugatein W to a standard subgroup of W . Indeed, when W = W cm for some1 ≤ m ≤ n this follows from Proposition 2.4, replacing the role of G by W ∼ = SU( m, W = W rm for some 2 ≤ m ≤ n this follows fromRemark 2.6 applied to W ≃ SO ( m, W must be of the same type as W g , so it must be W itself.Therefore, there is an h ∈ W such that W hg = ( W g ) h = W . It followsthat hg ∈ N , thus by part (2), hg = kw for some k ∈ K and w ∈ W . Thesequence of equations kS + k − = kwS + w − k − = hgS + g − h − = hW h − = W , implies that ( kSk − ) + = kS + k − = W , and therefore W ≤ kSk − ≤ N , where N is the normalizer of W in G . By part (3) we get K \ KS + k − = K \ K ( S + ) k = K \ KW = K \ KS k = K \ KSk − , and upon applying k on the right we conclude that indeed K \ KS = K \ KS + = K \ KW k. This is a totally geodesic subspace of K \ G by Proposition 2.3. The finalstatement follows from the essential uniqueness of an S -invariant measure on( S ∩ K ) \ S ≃ K \ KS . This proves part (4) and thus completes the proof. (cid:3) Parabolic subgroups of G and some of their subgroups. In thissubsection we continue the discussion of special subgroups of G begun in theprevious one. Now we focus mostly on parabolic subgroups of G and someof their subgroups. Recall that e , . . . , e n +1 denotes the standard basis of C n +1 and that G is the subgroup of SL n +1 ( C ) preserving the form h givenin Equation (2).Let P < G be the stabilizer of the isotropic line spanned by e . Thisis a parabolic subgroup of G and all proper parabolic subgroups of G areconjugate to P , since G has real rank 1. Some important subgroups of P are A = λ I n −
00 0 λ − : λ ∈ R ∗ ,M = θ T
00 0 θ : θ ∈ U(1) ,T ∈ U( n − θ = det( T ) − ,U = − v ∗ ib − k v k I n − v : v ∈ C n − , b ∈ R . Here v ∗ denotes the complex conjugate transpose of v . Note that U isisomorphic to the real (2 n − H n − ( R ) andthat A ∼ = R ∗ .One checks that P is generated by M , A , and U . Note that A is a maximal R -split torus of G , U is the unipotent radical of P , and M is a compactreductive group that commutes with A . Thus the Langlands decompositionof P is given by P = M AU .Every element of P can be represented in an obvious way by parameters( λ, θ, T, v, b ) using the coordinates introduced above. Such a representationis unique up to the order two intersection group M ∩ A , that is, ( λ, θ, T, v, b )represents the same element as ( − λ, − θ, T, v, b ). In the remainder of thissubsection we will use this representation often. We will also representelements of M , A , and U by ( θ, T ), λ , and ( v, b ) in the obvious way.Let C < M be the subgroup of scalar matrices in G . These scalars arethe ( n + 1) st roots of unity, thus C is a cyclic group of order n + 1. Notethat C is the center of G , and it is easy to see that it is also the center of P . RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 13
The group U is a two step nilpotent group with center Z = ib I n
00 0 1 : b ∈ R . This is a characteristic subgroup of U , hence it is a normal subgroup of P .Under the identification Z ∼ = R , the conjugation action of P on Z is givenby the homomorphism: P → R ∗ + < R ∗ ∼ = GL ( R )( λ, θ, T, v, b ) λ Since R ∗ + has no nontrivial compact subgroups, we obtain the following,which we record for future reference. Lemma 2.8.
Every compact subgroup of P commutes with Z . The quotient group
U/Z is naturally identified with C n − by the map( v, b ) v . Under this identification, the conjugation action of P on U/Z isgiven by the homomorphism: P → R ∗ · U( n − < GL n − ( C )( λ, θ, T, v, b ) λθ − T Here R ∗ < GL n − ( C ) is considered as the group of real scalar matrices.Note that this homomorphism induces isomorphisms M/C ∼ = U( n −
1) and
M A/C ∼ = R ∗ · U( n − M A/C on(
U/Z ) r { } is transitive and faithful. This is crucial in proving the followinglemma, which describes the normal subgroups of P . Lemma 2.9. If N E P is a normal subgroup, either U ≤ N or N ≤ CZ .Proof. Consider the quotient map θ : P → P/CZ . Then θ ( N ) ∩ θ ( U ) isnormal in P/CZ , hence transitivity of the
M A/C conjugation action on(
U/Z ) r { } implies that this intersection is either trivial or all of θ ( U ).Assume the latter case. We get that U C ≤ N CZ , thus
U C ≤ ( U C ∩ N ) CZ .Taking commutators and using the centrality of CZ in CU , we get Z = [ U, U ] = [
U C, U C ] ≤ [( U C ∩ N ) CZ, ( U C ∩ N ) CZ ] = [ CU ∩ N, CU ∩ N ]and we deduce that Z ≤ N . It follows that U C ≤ CN , but U is connectedand CN/N is totally disconnected, hence U ≤ N and we are done.Therefore, it remains to consider the case where θ ( N ) ∩ θ ( U ) is trivial. Itfollows that θ ( N ) commutes with θ ( U ), as they are both normal subgroupsof P/CZ . Since the action of θ ( M A ) ∼ = M A/C on θ ( U ) ∼ = U/Z is faithful,we deduce that N is in the kernel of the natural map P = M AU → M A/C , that is N ≤ U C . By triviality of θ ( N ) ∩ θ ( U ) = θ ( N ) ∩ θ ( U C ), we see that N is in the kernel of θ , hence N ≤ CZ as desired. (cid:3) Let
D < G be the subgroup stabilizing the plane spanned by e and e n +1 . Thus D also stabilizes the h -orthogonal complement of this plane, thesubspace spanned by e , . . . , e n , and we have: D = θa iθb T − iθc θd : a, b, c, d ∈ R , ad − bc = 1 ,θ ∈ U(1) , T ∈ U( n − ,θ = det( T ) − . We will be particularly interested in the group P ∩ D stabilizing both theline spanned by e and the plane spanned by e and e n +1 . Lemma 2.10.
We have P ∩ D = M AZ .Proof.
This is easily deduced from the matrix representations of P and D . (cid:3) The following lemma is easy, nevertheless we give a detailed proof due toits importance to our considerations.
Lemma 2.11.
Fix d ≥ and consider the group R ∗ · U( d ) ⋉ C d ≤ GL d ( C ) ⋉ C d . Its group of outer automorphisms,
Out( R ∗ · U( d ) ⋉ C d ) , is of order 2 and itsnontrivial element corresponds to complex conjugation of GL d ( C ) ⋉ C d .Proof. For brevity, we set S = R ∗ · U( d ) and V = C d . We consider anautomorphism τ : S ⋉ V → S ⋉ V and will show that, up to an innerautomorphism, τ is either trivial or complex conjugation.Identifying V with R d in the standard way, we identify its group ofcontinuous automorphisms with the the real general linear group GL d ( R ).For an operator t ∈ GL d ( R ) and v ∈ V we denote the corresponding actionby t · v . We identify the conjugation action of S on V as the linear actionof S < GL d ( R ) and for s ∈ S and v ∈ V , define s · v = svs − = v s .We have that τ ( S ) < S ⋉ V is a Levi subgroup. As all Levi subgroupsare conjugate, we assume as we may that τ ( S ) = S . Let α denote theautomorphism of S induced by τ . Since V is a characteristic subgroup of S ⋉ V , we have τ ( V ) = V . Thus τ induces a continuous automorphism of V corresponding to a fixed element t ∈ GL d ( R ). For s ∈ S and v ∈ V wehave ts · v = t · v s = τ ( v s ) = τ ( v ) τ ( s ) = ( t · v ) α ( s ) = α ( s ) t · v. Thus α ( s ) = s t . We conclude that t is contained in N , where N is thenormalizer of S in GL d ( R ), and α is the automorphism of S obtained byconjugating by t . RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 15
We claim that
S < N is of index two, where the nontrivial coset is gener-ated by complex conjugation. We fix g ∈ N and argue that, up to complexconjugation, g is in S . Note that N commutes with the group of real scalars R ∗ < S and the group U(1) < S consisting of modulus 1 complex scalarsis characteristic. Indeed, U( d ) < S is characteristic, as it is the uniquemaximal compact subgroup of S , and U(1) is its center. Thus g normalizes U (1).Since the unique nontrivial automorphism of U(1) is complex conjugation,we can assume that g acts trivially on U(1). We conclude that g commuteswith the group of complex scalars C ∗ , hence g is C -linear, i.e., g ∈ GL d ( C ).If d = 1 then g ∈ GL d ( C ) = S , and we are done. We thus assume d ≥ d ) < U( d ) is also characteristic, we get that g normalizes it as well. However, SU( d ) is a maximal subgroup of SL d ( C ),hence it equals its own normalizer in SL d ( C ). It follows that the normalizerof SU( d ) in GL d ( C ) is C ∗ · SU( d ) = S , thus g ∈ S . This proves the claim.We conclude that indeed, up to an inner automorphism, τ is either trivialor complex conjugation. (cid:3) Proposition 2.12.
Consider P = P/CZ ∼ = R ∗ · U( n − ⋉ C n − and let τ : P → P be a continuous homomorphism with one-dimensional kernel.Then there exists an inner automorphism i : P → P such thateither τ = θ ◦ i or τ = c ◦ θ ◦ i, where θ : P → P is the obvious quotient map and c : P → P corresponds tocomplex conjugation on R ∗ · U( n − ⋉ C n − as discussed in Lemma 2.11.In particular, τ is surjective, ker( τ ) = ker( θ ) = CZ and, up to precom-posing τ by an inner automorphism of P , τ ( P ∩ D ) = θ ( P ∩ D ) .Proof. We first note that Lemma 2.9 implies that Z ≤ ker( τ ) ≤ CZ , sincedim( U ) = 2 n − ≥ Z ∼ = R is the identity component of CZ . Bydimension considerations, τ ( P ) is Zariski dense in P , as P is Zariski con-nected. Thus C ≤ ker( τ ), as P has trivial center, hence ker( τ ) = CZ and τ induces a continuous injection τ : P → P such that τ = τ ◦ θ .A continuous injective endomorphism of a Lie group with finitely manyconnected components is surjective, so τ is necessarily an automorphism. ByLemma 2.11 we get that either ¯ τ = c ◦ inn(¯ g ) or ¯ τ = inn(¯ g ) for some ¯ g ∈ P ,where c denotes complex conjugation. Fixing g ∈ P such that ¯ g = θ ( g ) weget that either τ = c ◦ θ ◦ inn( g ) or τ = θ ◦ inn( g ), proving the first part ofthe proposition.It remains to show that τ ( P ∩ D ) = θ ( P ∩ D ) up to an inner automorphism.This follows from Lemma 2.10, since θ ( P ∩ D ) = θ ( M A ) corresponds to thesubgroup C ∗ · SU( n − ⊂ C ∗ · SU( n − ⋉ C n − , which is invariant undercomplex conjugation. (cid:3) Compatibility and measuring incompatibility
While much of this section applies more broadly, in what follows G willalways denote SU( n, § § G . In this paper, we will also need to measurethe extent to which compatibility fails. This leads us to begin with thefollowing sequence of definitions. Definition 3.1.
Let P be a minimal parabolic subgroup of G and U itsunipotent radical. An incompatibility datum for G is a tuple ( k, H , J , U ′ , τ ),where k is a local field, H is a k -algebraic group, J < H is a nontrivial k -subgroup, U ′ < U is a nontrivial proper subgroup, and τ : P → N H ( J ) / J ( k )is a continuous homomorphism such that the Zariski closure of τ ( U ′ ) is notequal to the Zariski closure of τ ( U ). Here N H ( J ) denotes the normalizer of J in H . We then have: • When k , H , and J are fixed, τ is an incompatible homomorphism for J if there exists U ′ < U as above so the corresponding tuple formsan incompatibility datum for G . If no such U ′ exists, then τ is calleda compatible homomorphism . • When k and H are fixed, J is an incompatible subgroup if there existsan incompatible homomorphism for J . Otherwise J is a compatiblesubgroup . • When k is fixed, H is an incompatible k -group for G if there is anontrivial k -subgroup J < H that is an incompatible subgroup forthe pair ( k, H ). Otherwise we call H a compatible k -group . • A local field k is incompatible with G if there exists an incompatible k -group. Otherwise k is called compatible . Proposition 3.2.
Every nonarchimedean local field is compatible with G .Proof. If k is a nonarchimedean local field, then for any k -group H and k -subgroup J < H , N H ( J ) / J ( k ) is totally disconnected. Therefore everycontinuous homomorphism τ : P → N H ( J ) / J ( k ) must be trivial on theconnected subgroup U < P . Compatibility clearly follows. (cid:3)
In view of Proposition 3.2 we will be concerned in the rest of this subsec-tion with archimedean local fields, i.e., k = R or k = C . We then have thefollowing general lemma. Lemma 3.3.
Let ( k, H , J , U ′ , τ ) be an incompatibility datum for G where k is R or C . Suppose that S is the Zariski closure in H of the preimageof τ ( P ) under the map N H ( J )( k ) → N H ( J ) / J ( k ) . Then S is not reductive.In particular, if H is reductive then S is contained in a proper parabolicsubgroup of H . RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 17
Proof.
We first observe that if H is reductive and S is not reductive then S is contained in a proper parabolic subgroup of H by [7, § S is not reductive.By extending scalars if necessary, we can assume that k = C . We identifyalgebraic groups with their C -points, writing H for H ( C ) with similar nota-tion for the other groups. We assume that S is reductive and will prove thatits Lie algebra has a noncentral nilpotent ideal, which is a contradiction.Incompatibility of τ implies that τ ( U ) is nontrivial, since τ ( U ′ ) must bea proper subgroup of τ ( U ). Note that [ P, U ] = U , thus τ ( U ) is a noncen-tral normal nilpotent subgroup of τ ( P ). We now consider G , H , and theirsubgroups as Lie groups, τ as a morphism of Lie groups, and denote theirLie algebras by the corresponding Gothic letters. Since s is reductive, theideal j E s has a direct complement i E s , i.e., s ∼ = i ⊕ j . The image of thecomposition u → s / j → i → s defines a nontrivial noncentral nilpotent ideal in s , giving the desired con-tradiction and thus proving the lemma. (cid:3) In view of Theorem 1.5(1), we will be concerned exclusively with the case k = R and H ( R ) = PU( r, s ) with r + s = n + 1. For the proof of Theorem1.7, we will care about the case r + s < n + 1. In particular, we will needthe following technical result. Proposition 3.4.
Let the tuple ( k, H , J , U ′ , τ ) be an incompatibility datumfor G with k = R and H = PU( r, s ) where r + s ≤ n + 1 , r ≥ s . Then (1) r = n and s = 1 , i.e., H = PU( n, , and (2) J is the center of the unipotent radical of a proper parabolic subgroupof H and τ has one-dimensional kernel.In particular, H = PU( r, s ) is compatible with G for either r + s < n + 1 or s > .Proof. We work directly with real points of real algebraic groups and aban-don the bold face notation. We again let S denote the Zariski closure of thepreimage of τ ( P ) under the map N H ( J ) → N H ( J ) /J . Lemma 3.3 impliesthat S ≤ N H ( J ) is contained in some maximal parabolic subgroup Q < H .Set N = N Q ( J ) and note that S ≤ N ≤ Q . Restricting the codomain, weconsider τ as a map from P to N/J ≤ N H ( J ) /J .The group U( r, s ) acts on C r + s preserving the hermitian form h r,s ( x , . . . , x n +1 ) = r X i =1 | x i | − s X i =1 | x r + i | , and Q is the projectivization of the stabilizer in U( r, s ) of a totally isotropiccomplex subspace V ⊂ C r + s . Let d = dim C ( V ) and note that d ≤ s , as s is the dimension of a maximal h r,s -isotropic subspace C r + s . A parabolicsubgroup of H is conjugate to Q if it is the stabilizer of a d -dimensionaltotally isotropic subspace. A parabolic subgroup is opposite to Q if it is the stabilizer of a maximal totally isotropic subspace V ′ disjoint from V suchthat the restriction of h r,s to V ⊕ V ′ is nondegenerate.A Levi factor of Q is obtained by intersecting Q with an opposite parabolicsubgroup Q ′ . If V ′ is the totally isotropic subspace associated with Q ′ , then L = Q ∩ Q ′ stabilizes the subspaces V , V ′ and ( V ⊕ V ′ ) ⊥ , and these subspacesform a direct sum decomposition of C r + s . Taking V ⊕ V ′ and ( V ⊕ V ′ ) ⊥ withthe induced forms, we identify GL( V ) with the stabilizer of V in U( V ⊕ V ′ )under the action T ( v, v ′ ) = ( T v, ( T − ) ∗ v ′ ). This leads to the identification L ∼ = P(GL( V ) × U(( V ⊕ V ′ ) ⊥ )) ∼ = P(GL d ( C ) × U( r − d, s − d )) , and we see that K ∼ = P(U( d ) × U( r − d ) × U( s − d )) is a maximal compactsubgroup of L .We use the notation introduced in § M < P is locallyisomorphic to U( n − S → S/J is real algebraic, thus thecompact subgroup τ ( M ) < S/J has a compact lift f M < S such that themap S → S/J restricts to a local isomorphism f M → τ ( M ). Incompatibilityof τ means that U cannot be in the kernel of τ , and it follows from Lemma 2.9that ker( τ ) ≤ CZ . In particular, τ is almost injective on M .We therefore get that f M is locally isomorphic to τ ( M ), which is locallyisomorphic to M , therefore f M is locally isomorphic to U( n − L bea Levi factor of Q that contains f M and K < L be a maximal compactsubgroup containing f M . We conclude that d , s , r and n are parameterssatisfying 1 ≤ d ≤ s ≤ r ≤ n and r + s ≤ n + 1 such that the groupP(U( d ) × U( r − d ) × U( s − d )) contains a subgroup locally isomorphic toU( n − r = n , which will imply s = 1 and will prove (1). Assumefor contradiction r ≤ n −
1. Note that the possibility d = s = r is excluded.Indeed, in this case P(U( d ) × U( r − d ) × U( s − d )) = PU( d ) does not containa subgroup locally isomorphic to U( n −
1) by dimension considerations, as d ≤ r ≤ n −
1. If n = 2 we necessarily have 1 = d = s = r , which gives acontradiction. We thus have n ≥
3. Thus the commutator subgroup in thesubgroup of P(U( d ) × U( r − d ) × U( s − d )) which is locally isomorphic toU( n −
1) is almost simple, hence it projects almost injectively to one of thegroups PU( d ), PU( r − d ), or PU( s − d ). Since this commutator subgroupis locally isomorphic to SU( n − n − d , r − d , or s − d . Since s − d ≤ r − d ≤ n − n − ≤ d and we conclude that d = s = r = n −
1. This gives a contradiction. Hence we must have r = n ,completing the proof of (1).We now have that H = PU( n, G = SU( n, → G/C ∼ = PU( n,
1) = H, RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 19 with finite kernel. Let P ′ be the preimage of Q under this surjection. Then P ′ < G is a proper parabolic subgroup, and hence is conjugate to P . Pre-composing the above map with the corresponding conjugation we get a sur-jection θ : G → H such that P = θ − ( Q ) and C = θ − ( e ).Therefore P is locally isomorphic to Q and dim( Q ) = dim( P ). Note that τ factors through an injection P/ ker( τ ) → N/J = N Q ( J ) /J . Recalling thatker( τ ) ≤ CZ , we have that dim(ker( τ )) ≤
1. We then have the followingchain of inequalitiesdim( P ) − ≤ dim( P ) − dim(ker( τ )) = dim( P/ ker( τ ))(3) ≤ dim( N ) − dim( J ) = dim( N/J ) ≤ dim( Q ) − dim( J )= dim( P ) − dim( J )that all follow easily from the above observations.We claim that dim( N ) = dim( Q ). If not, then dim( N ) < dim( Q ) and wesee from (3) that dim( P ) − < dim( P ) − dim( J ) so dim( J ) = 0. In otherwords, J < Q is finite, and so θ − ( J ) < P is also finite, hence compact.Then θ − ( J ) commutes with Z < P by Lemma 2.8. Setting Z = θ ( Z ), J commutes with Z < Q , and in particular Z ≤ N .Since Z ⊳ P is normal, Z ⊳ Q is normal and hence J Z ⊳ N is normal.Consider the natural map π : N/J → N/J Z . Since Z is one-dimensional, Z is one-dimensional, and so ker( π ) = J Z/J is one-dimensional. Recall thatker( τ ) ≤ CZ , so it is at most one-dimensional. We conclude that ker( π ◦ τ )is at most two-dimensional. However, dim( U ) ≥
3, thus ker( π ◦ τ ) ≤ CZ byLemma 2.9, and in particular ker( π ◦ τ ) is at most one-dimensional.Considering the injective map P/ ker( π ◦ τ ) → N/J Z , we obtain thecontradictory chain of inequalities:dim( P ) − ≤ dim( P ) − dim(ker( π ◦ τ )) = dim( P/ ker( π ◦ τ )) ≤ dim( N ) − dim( J Z ) = dim(
N/J Z )= dim( N ) − < dim( Q ) −
1= dim( P ) − N ) = dim( Q ) as claimed.We now have that N ≤ Q and dim( N ) = dim( Q ). Since Q is Zariskiconnected, we conclude that N = Q and J ⊳ Q is normal. From Equation(3) we get that dim( J ) ≤
1. If J was finite, it would be central in Q , since J ⊳ Q is normal, Q is Zariski connected, and any discrete normal subgroup ofa connected group is central. However Q has trivial center and J is nontrivialby the incompatibility assumption, thus we have that dim( J ) = 1.Then θ − ( J ) is normal in P and one-dimensional. Lemma 2.9 impliesthat θ − ( J ) ≤ CZ and we conclude that J ≤ Z = θ ( CZ ). Since dim( J ) = 1and Z ∼ = Z ∼ = R is connected, we must have J = Z . Noting that Q < H is a proper parabolic subgroup and
Z < Q is the center of its unipotentradical, we have shown that J is indeed the center of the unipotent radicalof a proper parabolic subgroup of H . Finally, using the fact that dim( J ) = 1we get from Equation (3) that dim(ker( τ )) = 1. This completes the proofof (2). (cid:3) Preliminaries for the proof of Theorem 1.3
Existence of equivariant maps.
Assume that k is a local field and H is a connected adjoint k -algebraic group. Let Γ < G be a lattice andconsider a homomorphism ρ : Γ → H ( k ) with unbounded, Zariski denseimage. Assume that W < G is a closed noncompact subgroup for whichthere exists a faithful irreducible representation of H ( k ) on a k -vector space V of dimension at least two and a W -invariant ergodic measure ν on thebundle ( G × P ( V )) / Γ that projects to Haar measure on G/ Γ. The proof ofthe following result is exactly the same as the proof of [2, Prop. 4.1].
Proposition 4.1.
Under the assumptions above, there is a proper non-compact k -algebraic subgroup L < H and a measurable W -invariant and Γ -equivariant map φ : G → H / L ( k ) . We can also view φ as a measurable Γ -map from W \ G to H / L ( k ) . Algebraic representations.
Here we recall the work of Bader andFurman [3] that will be used in the proof of Theorem 1.3. We also refer to[2, § k is a local field and H is a connected adjoint k -algebraic group.We fix a lattice Γ < G and a Zariski dense representation ρ : Γ → H ( k ).For a closed subgroup T < G , a T -algebraic representation of G consists of: • a k -algebraic group I , • a k -( H × I )-algebraic variety V that is a left H -space and a right I -space on which the I -action is faithful, • a Zariski dense homomorphism τ : T → I ( k ), • an algebraic representation of G on V , by which we mean an almost-everywhere defined measurable map φ : G → V ( k ) such that φ ( tgγ − ) = ρ ( γ ) φ ( g ) τ ( t ) − for every γ ∈ Γ, every t ∈ T , and almost every g ∈ G .The data for a T -algebraic representation is denoted by I V , τ V , and φ V .A T -algebraic representation is coset T -algebraic representation when V is the coset space H / J for some k -algebraic subgroup J < H and I is a k -subgroup of N H ( J ) / J , where N H ( J ) is the normalizer of J in H and H / J is endowed with the standard right action of N H ( J ) / J . The collection of T -algebraic representations of G forms a category. The Howe–Moore theoremimplies that if T is noncompact then the T -action on G/ Γ is mixing, henceweakly mixing. In this case, by [3, Thm. 4.3], the above category has aninitial object which is a coset T -algebraic representation. RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 21
Definition 4.2.
Suppose that G , Γ, and T are as above. The initial objectin the category of T -algebraic representations of G is called the gate or,when wishing to emphasize its dependence on T , the T -gate .We recall one last definition from [2, § S, T < G , consider their gates φ S , V S and φ T , V T . We say that thesegates have the same map if V S = V T and φ T , φ S agree almost everywhere.We then have the following. Lemma 4.3 (cf. [2, Lem. 4.4]) . Assume that
T < G is a closed noncompactsubgroup. Assume T = T ⊳ T ⊳ · · · ⊳ T n = T ′ is a sequence of subgroups of G such that T i − is normal in T i for each i = 1 , . . . , n . Then the gates for T and T ′ can be chosen to have the same map.Proof. By induction we can assume that n = 1, that is T < T ′ ≤ N G ( T ).Then [2, Lem. 4.4] implies we can assume that the gates for T and N G ( T )have the same map φ : G → H / J ( k ). Since φ is an N G ( T )-algebraic repre-sentation, it is also a T ′ -algebraic representation. Let φ ′ : G → H / J ′ ( k ) bethe gate in the category of T ′ -algebraic representations. It follows that wecan find an H -equivariant k -map H / J ′ → H / J . Viewing φ ′ as a T -algebraicrepresentation, we can also find an H -equivariant k -map H / J → H / J ′ . By[3, Cor. 4.7] we see that the gates for T and T ′ can indeed be chosen to havethe same map. (cid:3) The proof of Theorem 1.3 (1)In this section we state and prove Proposition 5.1 and then use it toprove Theorem 1.3(1). Proposition 5.1 will also be used in the proof ofTheorem 1.3(2) in the next section. Throughout this section we rely on thenotation, definitions, and results of the previous sections, particularly thenotion of incompatibility given in Definition 3.1 and the notion of a gatefrom Definition 4.2.
Proposition 5.1.
Let G be SU( n, for some n ≥ , Γ < G be a lattice,and W < G be a noncompact connected almost simple subgroup. Supposethat k is a local field, H is a connected adjoint k -algebraic group, and that ρ : Γ → H ( k ) is a homomorphism with unbounded, Zariski dense image.Assume moreover that there is a faithful irreducible representation of H ( k ) on a k -vector space V of dimension at least two and a W -invariant, ergodicmeasure ν on ( G × P ( V )) / Γ that projects to Haar measure on G/ Γ .Suppose that U ′ < W is a nontrivial unipotent subgroup, consider thecategory of U ′ -algebraic representations of G , and let Ψ : G → H / J ( k ) bethe corresponding gate, where J < H is a k -algebraic subgroup and Ψ is ameasurable map that is ( U ′ × Γ) -equivariant with respect to a homomorphism τ : U ′ → N H ( J ) / J ( k ) . Then τ extends to a continuous homomorphism τ : P → N H ( J ) / J ( k ) with τ | U ′ = τ such that Ψ is ( P × Γ) -equivariant withrespect to τ . Furthermore: (1) If J is trivial, then ρ extends to a continuous homomorphism from G to H ( k ) . (2) If J is nontrivial, then it is an incompatible subgroup of H and τ isan incompatible homomorphism. We note that Proposition 5.1 is essentially proved in [2, § J and τ , rather thancompatibility of the group H . Apart from that and minor differences dueto the fact that G = SU( n,
1) here rather than SO( n, Proof of Proposition 5.1.
By [7, §
3] we can find a proper parabolic subgroup
P < G containing U ′ . Let M , A , U , and Z be as in § G has realrank 1, P is minimal parabolic and all its unipotent elements are containedin U , hence U ′ < U .Now consider the group U ′ Z < U , where Z is the center of U . We thenhave U ′ ⊳ U ′ Z ⊳ U ⊳ P , and applying Lemma 4.3 we see that the gates for U ′ and P can be chosen to have the same map. Therefore, τ extends to acontinuous homomorphism τ : P → N H ( J ) / J ( k ) with respect to which Ψ is( P × Γ)-equivariant.Assume J is trivial. As A < P , Ψ is A -equivariant through τ | A . Trivi-ality of J implies that this must be the gate in the category of A -algebraicrepresentations. Then Lemma 4.3 implies that τ | A extends to a homomor-phism N G ( A ) → H ( k ) for which Ψ is N G ( A )-equivariant, where N G ( A ) isthe normalizer of A in G .Since N G ( A ) contains a Weyl element for A we have h P, N G ( A ) i = G .Since Ψ is equivariant for both P and N G ( A ), using [3, Prop. 5.1] andfollowing the end of the proof of [3, Thm. 1.3], we conclude that ρ : Γ → H ( k )extends to a continuous homomorphism b ρ : G → H ( k ). This proves theproposition when J is trivial.We now assume that J is nontrivial but τ is compatible and derive a con-tradiction. Compatibility of τ implies that the Zariski closures of τ ( U ′ ) and τ ( U ) coincide. Proposition 4.1 implies that there exists a proper noncompact k -algebraic subgroup L (cid:12) H and a measurable W -invariant, Γ-equivariantmap φ : G → H / L ( k ).The W -invariant map φ is also U ′ -invariant, as U ′ < W . Thus it factorsthrough Ψ : G → H / J ( k ) and G → H / J ( k ) → ( H / J ) / ¯ τ ( U ′ )( k ) = ( H / J ) / ¯ τ ( U )( k )by U ′ -invariance, where τ ( U ′ ) = τ ( U ) are the corresponding Zariski closures.Since Ψ is U -equivariant, the above composition is U -invariant, and it followsthat φ is also U -invariant. Since φ is also W -invariant and h U, W i = G , weobtain that φ : G → H / L ( k ) is an essentially constant Γ-equivariant map,hence ρ (Γ) has a fixed point on H / L ( k ). This is impossible since ρ (Γ) isZariski dense in H and L is a proper algebraic subgroup of the connectedadjoint group H . This is a contradiction, which completes the proof. (cid:3) RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 23
We are now prepared to prove part (1) of Theorem 1.3.
Proof of Theorem 1.3 (1) . Suppose U ′ < W is a nontrivial unipotent sub-group and consider the category of U ′ -algebraic representations with gateΨ : G → H / J ( k ), where J < H is a k -algebraic subgroup and Ψ is a mea-surable map that is ( U ′ × Γ)-equivariant with respect to a homomorphism τ : U ′ → N H ( J ) / J ( k ).We claim that J is trivial. Indeed, if it was nontrivial then it wouldbe incompatible by Proposition 5.1, contradicting the compatibility of H with SU( n, ρ extends to acontinuous homomorphism from G to H ( k ). This completes the proof. (cid:3) The proof of Theorem 1.3 (2)We begin by stating Proposition 6.1 and then use it and Proposition 5.1to prove Theorem 1.3(2). The rest of this section is then devoted to theproof of Proposition 6.1.
Proposition 6.1.
Suppose that G is SU( n, for n ≥ and Γ < G is alattice. Let H = PU( n, , Z < H be the center of the unipotent radical ofa proper parabolic subgroup
Q < H , and ρ : Γ → H be a homomorphismwith unbounded, Zariski dense image. Assume that there exists a continuoushomomorphism τ : P → Q/Z with one-dimensional kernel and a measurablemap
Φ : G → H/Z that is ( P × Γ) -equivariant with respect to the left Γ -action and the right P -action on H/Z via τ . Then ρ extends to a continuoushomomorphism from G to H . Using this, we prove part (2) of Theorem 1.3.
Proof of Theorem 1.3 (2) given Proposition 6.1.
Here k = R and H is the k -algebraic group corresponding to H . Let U ′ < W be a nontrivial unipotentsubgroup and consider the category of U ′ -algebraic representations of G and the corresponding gate Ψ : G → H / J ( k ), where J < H is a k -algebraicsubgroup and Ψ is a measurable map that is ( U ′ × Γ)-equivariant withrespect to a homomorphism τ : U ′ → N H ( J ) / J ( k ). By Proposition 5.1, τ extends to a continuous homomorphism τ : P → N H ( J ) / J ( k ) such that Ψis ( P × Γ)-equivariant with respect to ¯ τ .If J is trivial then we are done by Proposition 5.1. We therefore assumethat J is nontrivial. It follows, again by Proposition 5.1, that τ is incompat-ible. Proposition 3.4 implies that J is the center of the unipotent radical ofa proper parabolic subgroup of H and ker( τ ) is one-dimensional. By Propo-sition 6.1 we conclude that ρ indeed extends to a continuous homomorphismfrom G to H ( k ) = PU( n, (cid:3) The proof of Proposition 6.1 will be given in § fiber products inthe measured category and the incidence geometry of chains on the idealboundary of complex hyperbolic space. Fiber products.
We begin by recalling some standard definitions. Wewill consider the category of Lebesgue spaces and their morphisms, wherea Lebesgue space is a standard Borel space endowed with a measure class.An almost everywhere defined measurable map between Lebesgue spaces issaid to be measure class preserving if the preimage of a null set is null. Amorphism of Lebesgue spaces is an equivalence class of almost everywheredefined measure class preserving measurable maps, where two such mapsare equivalent if they agree almost everywhere.We now recall the definition of the fiber product in this category. Supposethat
X, Y, Z are Lebesgue spaces endowed with probability measures µ X , µ Y ,and µ Z , respectively. Let φ : X → Z and ψ : Y → Z be maps such that themeasures are compatible in the sense that φ ∗ µ X and ψ ∗ µ Y are in the samemeasure class as µ Z . The corresponding set theoretic fiber product is X × Z Y = { ( x, y ) ∈ X × Y : φ ( x ) = ψ ( y ) } ⊆ X × Y. For µ Z almost every z ∈ Z , disintegration of φ and ψ give measures ν X,z and ν Y,z on the corresponding fibers φ − ( z ) ⊂ X and ψ − ( z ) ⊂ Y such that µ X = Z Z ν X,z dµ Z µ Y = Z Z ν Y,z dµ Z . We then define a measure µ X × Z µ Y = Z Z ( ν X,z × ν Y,z ) dµ Z , on X × Y whose equivalence class is supported on the set theoretic fiberedproduct X × Z Y . Further, note that this measure class is independent of thechoices of representatives for φ and ψ and the choices of representatives for µ X , µ Y , and µ Z . We thus view this measure as a measure class on X × Z Y and call it the fiber product measure . For details, see for instance [49, p.265]. We will need the following lemma. Lemma 6.2.
Consider Lebesgue spaces X , Y , and Z endowed with mor-phisms X → Z and Y → Z . Assume X ′ , Y ′ , and Z ′ are standard Borelspaces endowed with Borel maps X ′ → Z ′ and Y ′ → Z ′ , and moreoverthat there are almost everywhere defined measurable maps f : X → X ′ , g : Y → Y ′ , and h : Z → Z ′ such that the following diagram commutes: X ′ Y ′ X YZ ′ Z f gh RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 25
Consider the product map f × g : X × Y → X ′ × Y ′ and the fiber productmeasure class µ X × Z µ Y on X × Z Y ⊂ X × Y . Then for µ X × Z µ Y almostevery ( x, y ) , one has that ( f × g )( x, y ) ∈ X ′ × Z ′ Y ′ .Proof. Using the choices and notation introduced above, we have( f × g ) ∗ ( µ X × Z µ Y ) = Z Z ′ ( f ∗ ν X,z × g ∗ ν Y,z ) dh ∗ µ Z , and conclude that ( f × g ) ∗ ( µ X × Z µ Y ) = f ∗ µ X × Z ′ g ∗ µ Y . That is, thepush forward of the fiber product measure is the fiber product of the pushforward measures. It follows that this measure is indeed supported on theset theoretical fiber product X ′ × Z ′ Y ′ . We then get that:( µ X × Z µ Y ) (cid:0) ( f × g ) − (( X ′ × Y ′ ) r ( X ′ × Z ′ Y ′ )) (cid:1) = ( f × g ) ∗ ( µ X × Z µ Y )(( X ′ × Y ′ ) r ( X ′ × Z ′ Y ′ ))= ( f ∗ µ X × Z ′ g ∗ µ Y )(( X ′ × Y ′ ) r ( X ′ × Z ′ Y ′ ))= 0This proves the lemma. (cid:3) Real algebraic varieties as Lebesgue spaces.
Assume V is a realalgebraic variety. Then the volume form on V gives rise to a well-definedmeasure class on V , thus V has a natural structure as a Lebesgue space.This is easily seen to be a functor from the category of real algebraic varietiesand real regular dominant maps to the category of Lebesgue spaces and theirmorphisms. For this, note that if V is open and Zariski dense in V then theinclusion V ⊂ V is an isomorphism of Lebesgue spaces, as the complementof V is of lower dimension, hence is null. The fact that the above functorpreserves fiber products is easy but fundamental to this paper, so we recordit here. Lemma 6.3.
Let X , Y , and Z be real algebraic varieties and let φ : X → Z , ψ : Y → Z be real regular dominant maps. Consider the corresponding fiberproduct X × Z Y in the category of real algebraic varieties. Then the mea-sure class of its volume form coincides with the corresponding fiber productmeasure class of the volume forms on X , Y , and Z . The incidence geometry of chains.
Let B n denote complex hyper-bolic n -space. Then G = SU( n,
1) acts transitively on B n with point sta-bilizers conjugates of K = S(U( n ) × U(1)), the maximal compact subgroupof G . We can also identify B n with the unit ball in C n with its Bergmanmetric. Note that B is naturally identified with the Poincar´e disk model ofhyperbolic 2-space.Let ∂ B n denote the ideal boundary of B n , which is homeomorphic to the(2 n − S n − . Identify ∂ B n with the space of isotropic lines in C n +1 with respect to a hermitian form h of signature ( n, P \ G for P < G a minimal parabolic subgroup. The spaceof pairs of points ∂ B n × ∂ B n is then identified with P \ G × P \ G and the Zariski open subset ( ∂ B n ) (2) consisting of pairs of distinct points is identifiedwith M A \ G , i.e., the space of oriented geodesics in B n .A totally geodesic holomorphic embedding f : B ֒ → B n induces an em-bedding f ∞ : ∂ B ≃ S ֒ → ∂ B n , and f ∞ ( ∂ B ) is called a chain on ∂ B n . Denote the space of chains on ∂ B n by C . Chains were originally studied by Cartan [10] and we refer to [21]for basic facts about them. Chains are in one-to-one correspondence withtwo-dimensional subspaces of C n +1 on which h is nondegenerate of signature(1 , G -action on C is transitive and the stabilizers areconjugate in G to D = S(U(1 , × U( n − C ≃ D \ G .Given a natural number k , we denote by C k the space whose elements area chain and k points lying on that chain. That is, C k = { ( c, x , . . . , x k ) : c ∈ C , x , . . . , x k ∈ c } . Let C ( k ) denote the subset of C k where the points are distinct, i.e., C ( k ) = { ( c, x , . . . , x k ) ∈ C k : x i = x j for i = j } . Note that C = C and C = C (1) is the set of pairs consisting of a chain anda point on that chain. Thus C ≃ D \ G and C ≃ ( P ∩ D ) \ G = M AZ \ G .In particular, these spaces have real algebraic variety structures and weendow them with the corresponding volume measure classes, which are alsothe unique G -invariant measure classes. We have the projection C → C forgetting the point, which corresponds to the projection ( P ∩ D ) \ G → D \ G .This map is real algebraic and G -equivariant. The space C k can be seen asthe k th fiber power of the above projection and is thus also endowed with anatural real algebraic variety structure and a corresponding measure class.The space C ( k ) is a dense Zariski open subset of C k and then is also endowedwith a real algebraic variety structure and a corresponding measure class.In particular, the inclusion C ( k ) ⊂ C k is an isomorphism of Lebesgue spaces.In this paper we will primarily be concerned with C (2) and C (3) . A distinctpair of points x, y ∈ ∂ B n determines a unique chain, which is easily seen byconsidering x and y as isotropic lines in C n +1 and taking their span. Thusthere are G -equivariant isomorphisms M A \ G ≃ ( ∂ B n ) (2) ≃ C (2) .We now consider the subset C k = { ( c, x , . . . , x k ) ∈ C k : x i = x j for some i = j } ⊂ C k consisting of k -tuples of points containing at least two distinct points andthe subset C k = { ( c, x , . . . , x k ) ∈ C k : x i = x for i = 1 } ⊂ C k where the first point in each chain is different from the others. We note that C ( k ) ⊂ C k ⊂ C k ⊂ C k , RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 27 and these are all dense Zariski open subsets of C k . In particular, they areall isomorphic in the category of Lebesgue spaces. Remark 6.4.
The space C k can be alternatively described as: (cid:8) ( x , . . . , x k ) : x i ⊂ C n +1 an isotropic line , dim C span { x , . . . , x n } = 2 (cid:9) . It is this space that is considered in Pozzetti’s chain rigidity theorem, whichis Theorem 6.5 below.We now make an observation that will be useful later. For each k wedefine α k : C k → C by ( c, x , . . . , x k ) ( c, x ). In particular we have themap α : C (2) = C → C that forgets the second point of each chain. Weobserve that α k : C k → C can be identified in the category of algebraicvarieties with the ( k − st fibered power of α . It follows by Lemma 6.3that C k is isomorphic to the ( k − st fibered power of α also in the categoryof Lebesgue spaces.Note that C (2) ≃ M A \ G , but G is not transitive on C ( k ) already for k = 3.Indeed, C (3) has two G -orbits distinguished by the cyclic orientations of thegiven triple of points on each chain. The stabilizer in G of each elementof C (3) is a compact subgroup, namely a conjugate of M < G . It followsthat the measure class on C (3) is the sum of the unique G -invariant measureclasses on the two G -orbits. Each of C , C , and C are obtained by addingsome lower-dimensional manifolds, which are null sets, to C (3) .6.4. Pozzetti’s chain rigidity theorem.
In [10], Cartan studied the in-cidence geometry of chains in ∂ B n and showed that a map preserving thisgeometry must come from an isometry of B n . His work was later generalizedby Burger–Iozzi to almost everywhere defined measurable maps under theadded assumption that the map preserves orientation [8]. This assumptionwas later removed by Pozzetti [46], and this is the result we will need.In the formulation of the theorem below we use the notation introducedin § ∂ B n represents the variety of isotropic lines in C n +1 , andunder this identification, C represents triples of isotropic lines that span atwo-dimensional subspace. Briefly, Pozzetti’s chain rigidity theorem is thata measurable map from ∂ B n to ∂ B n that takes triples of points on a chainto triples of points on a chain is necessarily rational. Restated precisely inour language, this is the following. Theorem 6.5 (Thm. 1.6 [46]) . For n ≥ , let φ : ∂ B n → ∂ B n be a mea-surable map whose essential image is Zariski dense. Endow ( ∂ B n ) withthe measure class [ µ ] associated with the volume form on C and considerthe induced map φ : ( ∂ B n ) → ( ∂ B n ) . If the [ µ ] -essential image of φ iscontained in C , then φ agrees almost everywhere with a rational map.On the proof of Theorem 6.5. The statement of [46, Thm. 1.6] assumes thatthe target of φ is the Shilov boundary S m ,m associated with SU( m , m )when 1 < m < m . However, nowhere in the proof of the theorem does Pozzetti use the assumption that 1 < m , and hence her result also holdsfor S ,m , m >
1, and in particular for S ,n . The Shilov boundary is thespace of maximal isotropic subspaces for the relevant hermitian form, hence S ,n ≃ ∂ B n . In other words, Pozzetti’s proof applies without alterationto give Theorem 6.5 as stated. The reason for Pozzetti’s assumption that m > (cid:3) We recall from § C can be identified with the fibered square ofthe forgetful map α : C (2) → C and that C (2) ≃ ( ∂ B n ) (2) . Upon making thisidentification, we consider the map α : ( ∂ B n ) (2) → C and regard C as thefibered square of α . We then obtain the following corollary of Theorem 6.5. Corollary 6.6.
Let G be SU( n, for n ≥ , Γ < G be a lattice, H be PU( n, , and ρ : Γ → H be a homomorphism with unbounded, Zariskidense image. Assume that φ : ∂ B n → ∂ B n and ψ : C → C are measurable, Γ -equivariant maps, where Γ acts on the domain through its inclusion into G and on the target via ρ . Then the following assertions hold. (1) The essential image of φ : ( ∂ B n ) → ( ∂ B n ) is contained in the set ( ∂ B n ) (2) of distinct points. (2) Considering the restricted map φ (2) : ( ∂ B n ) (2) → ( ∂ B n ) (2) , if α ◦ φ (2) and ψ ◦ α agree almost everywhere as maps from ( ∂ B n ) (2) to C , then ρ extends to a continuous homomorphism from G to H . The content of (2) in Corollary 6.6 is that the boundary map φ sends chainsto chains and that the induced map on chains is ψ . Proof.
We first prove (1). Note that ( ∂ B n ) (2) ⊂ ( ∂ B n ) is open and Zariskidense, hence of full measure. Moreover ( ∂ B n ) (2) is isomorphic to M A \ G as a G -space and so it is Γ-ergodic by Howe–Moore. It follows that the essentialimage of φ is either contained in ( ∂ B n ) (2) or in its complement, which isthe diagonal in ( ∂ B n ) . If the latter were true then φ would be essentiallyconstant, and its essential image is invariant under Γ. This would implythat ρ (Γ) is contained in a proper parabolic subgroup of H , contradictingZariski density of ρ (Γ). It therefore follows that the essential image of φ iscontained in the set of pairs of distinct points, ( ∂ B n ) (2) . This proves (1).To prove (2), assume that α ◦ φ (2) and ψ ◦ α agree almost everywhere. Bya well-known lemma of Margulis [62, Lem. 5.1.3], to show that ρ extendsit suffices to show that φ is rational. Therefore it suffices to show thatTheorem 6.5 applies.Endow ( ∂ B n ) with the measure class [ µ ] associated with the volume formon C . We will show that the [ µ ]-essential image of φ is contained in C .In fact, we will show that it is contained in the subset C of C . Since C is conull in C , we view [ µ ] as the measure class associated with the volumeform on C . RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 29
As indicated above, we identify C with the fibered square of the map α from ( ∂ B n ) (2) to C , which is naturally a subset of ( ∂ B n ) (2) × ( ∂ B n ) (2) , eventhough C is a subset of ( ∂ B n ) by its original definition. This is clarifiedby the commutative diagram( ∂ B n ) (2) × C ( ∂ B n ) (2) (cid:8) ( x, y ) , ( x, z ) ∈ ( ∂ B n ) (2) : y, z = x (cid:9) C (cid:8) ( x, y, z ) ∈ ( ∂ B n ) : y, z = x (cid:9) ≃ ≃ in which the top two spaces are subsets of ( ∂ B n ) (2) × ( ∂ B n ) (2) and the bottomtwo spaces are subsets of ( ∂ B n ) containing the support of [ µ ]. Notice thatthis diagram also commutes with applying φ (2) × φ (2) to the top line and φ to the bottom line.Further, the assumption that α ◦ φ (2) and ψ ◦ α agree almost everywhereallows us to apply Lemma 6.2 with: X = Y = X ′ = Y ′ = ( ∂ B n ) (2) f = g = φ (2) Z = Z ′ = C h = ψ This allows us to conclude that the essential image of φ (2) × φ (2) is con-tained in ( ∂ B n ) (2) × C ( ∂ B n ) (2) with respect to the fibered measure class of( ∂ B n ) (2) × C ( ∂ B n ) (2) on ( ∂ B n ) (2) × ( ∂ B n ) (2) . In view of the above discus-sion, it follows that, with respect to the measure class associated with C on ( ∂ B n ) , the essential image of φ is contained in C . This completes theproof. (cid:3) Proof of Proposition 6.1.
Let G be SU( n,
1) for n ≥ < G be a lattice. Suppose that H = PU( n, Z < H is the center of theunipotent radical of a proper parabolic subgroup
Q < H , and ρ : Γ → H isa homomorphism with unbounded, Zariski dense image. Assume that thereexists a continuous homomorphism τ : P → Q/Z with one-dimensionalkernel and a measurable map Φ : G → H/Z that is ( P × Γ)-equivariant withrespect to the left Γ-action via ρ and the right P -action through τ . We mustshow that ρ extends to a continuous homomorphism from G to H .We will freely use the notation introduced in § Q with P/C we consider the natural map θ : P → P/CZ ∼ = Q/Z . For a fixed p ∈ P , the map G → H/Z given by g Φ( gp − ) is ( P × Γ)-equivariantwith respect to the left Γ-action and the right P -action via τ ◦ inn( p ). Wecan then replace τ with τ ◦ inn( p ), hence Proposition 2.12 allows us toassume that τ is surjective, ker( τ ) = CZ , and τ ( P ∩ D ) = θ ( P ∩ D ).The composition of Φ : G → H/Z with
H/Z → H/Q is right P -invariant, and hence gives rise to a Γ-map P \ G → H/Q . Upon identi-fying P \ G with H/Q ≃ ∂ B n , we consider the above composition as a Γ-equivariant measurable map φ : ∂ B n → ∂ B n . By Corollary 6.6(1), the map φ : ( ∂ B n ) → ( ∂ B n ) restricts to a map φ (2) : ( ∂ B n ) (2) → ( ∂ B n ) (2) . Next we consider the map α obtained by composing the natural identifi-cation ( ∂ B n ) (2) ≃ C (2) with the forgetful map α : C (2) → C . In other words, α maps the pair of distinct points ( x, y ) to the pair consisting of the uniquechain through x and y and the point x . Identify M A \ G with ( ∂ B n ) (2) and C with ( H/Z ) /θ ( P ∩ D ), where the latter identification comes from the factthat θ ( P ∩ D ) = τ ( P ∩ D ).Our goal is now to prove the existence of the dashed arrows in the follow-ing diagram of ( P × Γ)-equivariant measurable maps:
GM A \ G ( ∂ B n ) (2) ( ∂ B n ) (2) ( P ∩ D ) \ G C C ( H/Z ) /θ ( P ∩ D ) ≃ φ (2) α α ≃ ψ ′ ψ ≃ For this, the composed map G → ( H/Z ) /θ ( P ∩ D ) is ( P ∩ D )-invariant,since θ ( P ∩ D ) = τ ( P ∩ D ), thus it descends to a map from ( P ∩ D ) \ G to( H/Z ) /θ ( P ∩ D ), which proves the existence of ψ ′ . The existence of ψ is thenobtained by pre- and post-composing with the corresponding isomorphisms.We thus have that α ◦ φ (2) and ψ ◦ α agree almost everywhere, and thereforewe can apply Corollary 6.6(2) and conclude that ρ extends to a continuoushomomorphism from G to H . This proves the proposition. (cid:3) Remark 6.7.
The use of incidence geometry to prove rigidity theorems goesback at least to Mostow’s use of work of Tits in his proof of Mostow Rigidity[42]. Rigidity of chain preserving maps is older, going back to Cartan’s 1932paper [10]. An important application of rigidity of chain preserving maps inthe study of representations of discrete groups was by Burger and Iozzi in [8].The idea of exploiting triples of points on a chain goes back to earlier work ofToledo on rigidity of certain surface group representations into PU( n,
1) [55],and Toledo attributes this general idea to Thurston. As mentioned above,we cannot use Cartan’s result because our boundary map is only measurableand we cannot use Burger–Iozzi’s because they assume orientability of themap on chains, but Pozzetti chain rigidity theorem suffices for our purposes.
Remark 6.8.
Margulis and Mohammadi proved a version of Theorem 1.1for cocompact lattices in SO(3 ,
1) using incidence geometry. It is possible touse the methods of this section combined with results in [2] to produce theinput for their incidence geometry result, i.e., to prove the existence of a cir-cle preserving map between boundaries. However, the proof of Theorem 1.1in the real hyperbolic case is easier to complete using compatibility as wedid in [2]. More generally though, it is possible for one to apply Lemma 3.3
RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 31 above and use the methods of this paper to turn some problems about homo-morphisms from lattices in SO ( n,
1) to incompatible targets into incidencegeometry problems for boundary maps.7.
Algebraic groups associated with lattices in
SU( n, andTheorem 1.5 We begin by establishing some notation we need for the proof of Theo-rem 1.5. Let Γ < SU( n,
1) be a lattice, n ≥
2. Following the discussionof [2, § ℓ = Tr(Ad(Γ)) is both a number field and a minimal fieldof definition for Γ. Let G denote the connected adjoint ℓ -algebraic groupdefined by the Zariski closure of Γ under the adjoint representation.Then G is the adjoint form of a unique simply connected ℓ -group G oftype A n [54]. In other words, there is a quadratic extension ℓ ′ /ℓ and acentral division algebra D over ℓ ′ of degree d | ( n + 1) with involution σ ofthe second kind so that G is the special unitary group SU ( n +1) /d ( D, h ) forsome nondegenerate σ -hermitian form h on D ( n +1) /d . Recall that σ is ofthe second kind if its restriction to the center of D is the nontrivial Galoisinvolution of ℓ ′ /ℓ . The image of Γ under the adjoint representation lies in G ( ℓ ) and the kernel of SU( n, → PU( n,
1) is cyclic of order ( n + 1), hencewe can replace Γ with a subgroup of finite index and assume that Γ < G ( ℓ ).With this setup, we now prove Theorem 1.5. Proof of Theorem 1.5.
Let Γ < SU( n,
1) be a lattice, and retain all previ-ous definitions and notation from this section. As described above, we canassume that Γ < G ( ℓ ). For each place v of ℓ , we obtain a homomorphism ρ v : Γ ֒ → G v = G ( ℓ v ) , where ℓ v is the completion of ℓ associated with v . Let v be the placeassociated with the lattice embedding of Γ into SU( n, ρ v is locally rigid for all v . Recall that localrigidity of ρ v is the vanishing of the cohomology group H ( ρ v , g v ), where g v is the Lie algebra of G v . The assumption that Γ < G ( ℓ ) implies that thereis an ℓ -form g ℓ of su ( n,
1) so that g v ∼ = g ℓ ⊗ ℓ ℓ v for all v . Note that vanishingof H ( ρ v , g v ) is equivalent to vanishing of H (Γ , g ℓ ) since H ( ρ v , g v ) ∼ = H (Γ , g ℓ ) ⊗ ℓ ℓ v . However, H (Γ , g ℓ ) is trivial since H ( ρ v , g v ) is trivial by local rigidity ofthe lattice embedding of Γ into SU( n, ρ v forall v .We now make some technical reductions needed to apply some resultsas they are stated in the literature. First, without loss of generality wecan pass to a finite index torsion-free subgroup of Γ, since ℓ , ℓ ′ , and G are commensurability invariants. When Γ is cocompact, we can therefore Punctured disk base withpoint at infinity at originRadial cusp cross-section Loop aroundcompactifying divisorGeneral abelianvariety fiberCompactifying abelian variety { Figure 1.
Smooth compactification of a cusp.assume that B n / Γ is a compact K¨ahler manifold. When Γ is not cocompact,we can assume that B n / Γ is a smooth quasiprojective variety admitting asmooth toroidal compactification by a smooth divisor D (e.g., see [1, 39]).More precisely in the noncompact case, we can assume by passing toa finite index subgroup that the cusps of B n / Γ are diffeomorphic to bun-dles over the punctured disk with fiber an abelian variety. The associatedperipheral subgroup of Γ is a two-step nilpotent group with infinite cycliccenter naturally realized as a torsion-free lattice in the unipotent radical ofthe Borel subgroup of PU( n,
1) (cf. the structure of U in § B n / Γ to obtain a smooth projectivevariety by adding a certain abelian variety above the puncture in the disk.See Figure 1 and see [24, § v is archimedean, i.e., that ℓ v is R or C . Rigidity of ρ v then implies that the real Zariski closure of ρ v (Γ), namely G v , is a groupof Hodge type . See [52, Lem. 4.5] when Γ is cocompact. In general see [38, § B n / Γ. Since G is of absolute typeA n , considering [52, p. 50-51], we conclude that v is real and G v must beSU( r v , s v ) for an appropriate pair r v , s v , completing the proof of the firsttwo parts of the theorem.It remains to prove that Γ is integral. Indeed, the last statement ofTheorem 1.5 is an immediate consequence of the previous statements alongwith the definition of arithmeticity. Integrality will follow from a theoremof Esnault–Groechenig [19, Thm. 1.1] once we verify that their assumptionshold in our setting. For this, suppose that v is nonarchimedean. RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 33
Since ρ v is locally rigid and has determinant one, it remains to verifythat ρ v has quasi-unipotent monodromy at infinity. See [19, §
2] for theprecise definition. Indeed, loops around the compactifying divisor of B n / Γare associated with central elements of peripheral subgroups of Γ, whichare naturally unipotent subgroups of G ( ℓ ). These elements clearly remainunipotent under ρ v , hence ρ v has quasi-unipotent monodromy at infinity.We then conclude from [19, Thm. 1.1] that Γ is integral, which completesthe proof of the theorem. (cid:3) The proof of Theorem 1.1
Equidistribution on G/ Γ . This subsection describes the equidistri-bution results needed to show that Theorem 1.3 applies to prove Theo-rem 1.1. Each result in this section has a direct analogue in [2], and here weprovide the necessary modifications for the SU( n,
1) setting.Throughout this section we fix a lattice Γ < G . Recall that a measure µ on G/ Γ is called homogeneous if there exists a closed subgroup S ≤ G anda closed S -orbit in G/ Γ such that µ is the push-forward of Haar measureon S along this orbit. If W ≤ S is a closed subgroup with respect to which µ is ergodic, we call the measure W -ergodic . Given a homogeneous, W -ergodic measure µ , we will refer to its support, supp( µ ), as a homogeneous, W -ergodic subspace of G/ Γ.We ask the reader to recall the notation introduced in § K ≤ G and we considerthe symmetric space K \ G . The locally symmetric space K \ G/ Γ will bedenoted by M . In Definition 2.2 we introduced the standard (almost sim-ple) subgroups of G , namely certain copies in G of the group SU( m,
1) for1 ≤ m ≤ n and SO ( m,
1) for 2 ≤ m ≤ n which, by Proposition 2.4, arerepresentatives of conjugacy classes of all noncompact, connected, almostsimple, closed subgroups of G .In what follows “geodesic subspace” will always mean a properly im-mersed totally geodesic subspace of either M or its universal cover K \ G .Our goal in this subsection is to prove the following proposition, which trans-lates the existence of infinitely many maximal geodesic subspaces of M intoa statement about measures on G/ Γ that are invariant under a standardsubgroup of G . Proposition 8.1 (Cf. Prop. 3.1 in [2]) . The following are equivalent: (1)
The complex hyperbolic space M = K \ G/ Γ contains infinitely manymaximal geodesic subspaces of dimension at least . (2) There exists a standard subgroup
W < G and an infinite sequence { µ i } of W -invariant, W -ergodic measures on G/ Γ with proper sup-port for which Haar measure on G/ Γ is a weak- ∗ limit of the µ i . (3) There exists a standard subgroup
W < G and an infinite sequence { µ i } of homogeneous, W -ergodic measures on G/ Γ for which Haarmeasure on G/ Γ is a weak- ∗ limit of the µ i . As noted in [2], work of Ratner [50] shows that (2) and (3) are equivalent,and therefore it remains to prove that (1) and (3) are equivalent. For this,we closely follow the strategy in [2, § π : G/ Γ → K \ G/ Γ = M for the natural projection map. Note that π is a proper map. We begin withthe following lemma. Lemma 8.2.
The following hold: (1)
Let S be a closed subgroup of G such that W ≤ S for some standardsubgroup W < G , and suppose there exists h ∈ G for which Sh Γ / Γ is a closed S -orbit. Then Z = π ( Sh Γ / Γ) is a closed totally geodesicsubspace of M of dimension at least . Up to normalization, the dim( Z ) -volume of Z is the push-forward of Haar measure on Sh Γ / Γ via the projection map π . (2) Under the assumptions of part (1) , Z = M if and only if S = G and Z = π ( W g Γ / Γ) for some g ∈ G if and only if W ≤ S ≤ N , where N is the normalizer of W in G . In the latter case, N h Γ / Γ is alsoclosed with projection π ( N h Γ / Γ) = Z . (3) Conversely, every totally geodesic subspace Z in M of dimension atleast has finite measure. Moreover, there is a standard subgroup W of G with normalizer N , an intermediate subgroup W ≤ S ≤ N ,and an element h ∈ G such that Γ ∩ S h − is a lattice in S h − and Sh Γ / Γ is a homogeneous, W -ergodic subspace of G/ Γ for which Z = π ( W h Γ / Γ) = π ( Sh Γ / Γ) .Proof. The fact that Z is closed follows from the fact that π is a propermap and the assumption that Sh Γ / Γ is closed. By Lemma 2.7(4), K \ KSh is a totally geodesic subspace of K \ G , thus Z , which is the image of K \ KS under the covering map K \ G → M , is a totally geodesic subspace of M .The statement about the measure also follows from Lemma 2.7(4), and thisproves part (1).Next we consider part (2). Clearly S = G implies Z = M and the factthat Z = M implies S = G follows by applying Lemma 2.7(1) in the specialcase W = G . If W ≤ S ≤ N then Z = π ( Sh Γ / Γ) = π ( W h Γ / Γ) , by Lemma 2.7(3). Conversely, assume Z = π ( W g Γ / Γ) for some g ∈ G . Thenboth K \ KSh and K \ KW g cover Z in K \ G , thus they are totally geodesicsubspaces of the same type. Therefore K \ KSh = K \ KW g for some g ∈ G by Proposition 2.3, and Lemma 2.7(4) implies that K \ KSh = K \ KS + h .Hence we obtain the sequence of equalities K \ Khg − ( S + ) g h − = K \ KS + hg − = K \ KShg − = K \ KW, and conclude that ( S + ) g h − preserves K \ KW and acts transitively on it.Thus by Lemma 2.7(1) we see that ( S + ) g h − = W . RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 35
Since W ≤ S , we have ( S + ) g h − = W = W + ≤ S + and deduce that S + = ( S + ) g h − = W . Since S normalizes S + it follows that S ≤ N . Then W is cocompact in N , so S is cocompact in N by Lemma 2.7(2). It followsthat N h Γ / Γ is closed in G/ Γ, since Sh Γ / Γ is closed by hypothesis. That π ( N h Γ / Γ) = Z follows from Lemma 2.7(3), and this completes the proof ofpart (2).We now prove part (3). Fix a totally geodesic subspace Z in M of di-mension at least 2. The fact that Z has a finite volume is a well-knownconsequence of the existence of a thick-thin decomposition. See [20] for de-tailed argument in real hyperbolic space that is easily adapted to any rankone symmetric space.The preimage of Z under the Γ-invariant map K \ G → M is a collectionof mutually disjoint totally geodesic subspaces on which Γ acts. We fix oneof these, which by Proposition 2.3 is of the form K \ KW g for some g ∈ G and a standard subgroup W ≤ G . By Lemma 2.7(2), the subgroup of G that stabilizes this totally geodesic subspace is N g − , thus the subgroup ofΓ stabilizing it is Γ ∩ N g − . By Lemma 2.7(4) we have that K \ KW g = K \ KN g = K \ KgN g − which is isomorphic as an N g − -space to ( K g − ∩ N g − ) \ N g − . It follows that Z = K \ KgN g − Γ / Γ ≃ ( K g − ∩ N g − ) \ N g − / (Γ ∩ N g − ) . Since Z has a finite volume and K g − ∩ N g − is a compact subgroup of N g − ,we conclude that Γ ∩ N g − is a lattice in N g − . By [48, Thm. 1.13] we obtainthat N g Γ / Γ is closed in G/ Γ and conclude that it is a closed homogeneous N -orbit of finite volume for which the associated Haar measure on this orbitis a homogeneous measure.However N g Γ / Γ may not be W -ergodic, even though it is W -invariant.To complete the proof, it remains to show that there exists an intermediatesubgroup W ≤ S ≤ N and an h ∈ G such that Sh Γ / Γ is a homogeneous, W -ergodic subspace of G/ Γ. Let µ be a W -ergodic measure in the ergodicdecomposition of Haar measure on N g Γ / Γ, and let S denote the stabilizer of µ in N . Then W ≤ S ≤ N by W -invariance, and Ratner’s theorem impliesthat µ is S -homogeneous. Write the corresponding closed homogeneousspace as Sh Γ / Γ for some h ∈ G . Since Sh Γ / Γ ≃ S/ (Γ ∩ S h − ) we see thatΓ ∩ S h − is a lattice in S h − . Since h is in the N -homogeneous space N g Γ / Γ,we have that
N h Γ / Γ =
N g Γ / Γ. Using Lemma 2.7(4) twice, we see that π ( W h Γ / Γ) = π ( Sh Γ / Γ) = π ( N h Γ / Γ) = π ( N g Γ / Γ) = π ( W g Γ / Γ) = Z, which completes the proof. (cid:3) We now collect some useful facts about limits of homogeneous, W -ergodicmeasures that we need in the proof of Proposition 8.1. Despite the differencein presentation, the next three results are direct analogues of results in thereal hyperbolic case [2, Thm 3.3] and use an essentially identical argument.All of these results are relatively straightforward consequences of work of Dani–Margulis [15, Thm. 6.1] and Mozes–Shah [43]. We first show thatthere is no escape of mass for the sequence of measures under consideration.
Lemma 8.3.
Let
W < G be a closed, connected, almost simple subgroup of G that is generated by unipotent elements. If { µ i } is a sequence of homo-geneous, W -ergodic measures on G/ Γ that weak- ∗ converges to a measure µ in the space of all finite Radon measures, then µ is not the zero measure.Proof. Since the claim is invariant under conjugation, we can assume byProposition 2.4 that W is a standard subgroup of G .Let C be any fixed compact set in M whose interior contains the compactcore of M . See [5, Thm. 10.5] for the existence of a compact core in thissetting. Then C has the property that C ∩ Z contains a nonempty opensubset of Z for any closed geodesic subspace Z of M . Set F = π − ( C ).Applying [15, Thm. 6.1] for ǫ = 1 /
2, there exists a compact set F ′ ⊆ G/ Γsuch that(4) 12 ≤ T λ (cid:0)(cid:8) t ∈ [0 , T ] | u t x ∈ F ′ (cid:9)(cid:1) = 1 T Z T χ F ′ ( u t x ) dt, for every one-parameter unipotent subgroup { u t } of G , every x ∈ F , andevery T ≥
0, where χ F ′ is the characteristic function of F ′ and λ is Lebesguemeasure on R . We claim that µ ( F ′ ) ≥ /
2, which proves that µ is not thezero measure.Fix a one-parameter unipotent subgroup U = { u t } of W . Then W -ergodicity and the Howe–Moore theorem imply that each µ i is U -ergodic.The measures µ i are homogeneous and W -ergodic, so Lemma 8.2(1) impliesthat for each i there exists a closed totally geodesic subspace Z i of M withdimension at least 2 such that π ∗ µ i is a constant multiple of the volumemeasure on Z i . Since C ∩ Z i contains an open subset of Z i , we see that π ∗ µ i ( C ) = µ i ( π − ( C )) > . In particular, there exists a U -generic point x i ∈ F for each µ i . The Birkhoffergodic theorem applied to χ F ′ combined with Equation (4) then impliesthat µ i ( F ′ ) ≥ /
2. Therefore, µ ( F ′ ) ≥ /
2, which completes the proof. (cid:3)
Corollary 8.4.
Under the assumptions of Lemma 8.3, µ is a homogeneous, W -ergodic probability measure on G/ Γ . Moreover, µ is ergodic with respectto any nontrivial subgroup generated by unipotent elements of its stabilizerin G .Proof. Let S denote the stabilizer of µ in G . Then µ is not the zero measureby Lemma 8.3, thus [43, Cor. 1.3] shows that µ is a homogeneous, S + -ergodicprobability measure on G/ Γ. Since any nontrivial subgroup generated byunipotents in S is a noncompact subgroup of S + , µ is ergodic with respectto any such subgroup by the Howe–Moore theorem. In particular, thisapplies to W , as W ≤ S because the set of W -invariant measures is weak- ∗ closed. (cid:3) RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 37
We now apply [43, Thm. 1.1] to understand the relationship between thesupport of the µ i and the support of µ . Theorem 8.5.
Let
W < G be a closed, connected, almost simple subgroupgenerated by unipotent elements and { µ i } be a sequence of homogeneous, W -ergodic measures on G/ Γ that weak- ∗ converges to µ . Then there exista sequence of elements { g i } in G and a natural number i such that for all i ≥ i , the measures g i µ are homogeneous, W -ergodic probability measureson G/ Γ with supp( µ i ) ⊆ g i supp( µ ) .Proof. We fix unipotent subgroups U , . . . , U s that generate W . By theHowe–Moore theorem, µ i is U j -ergodic for every i ∈ N and every 1 ≤ j ≤ s .Therefore, for each i , the set of points for which µ i is U j -generic for all1 ≤ j ≤ s is of full measure and hence is dense in supp( µ i ).By Corollary 8.4, µ is a nonzero homogeneous W -ergodic probability mea-sure and supp( µ ) is a nonempty, closed homogeneous subspace of G/ Γ. Fixa point x ∞ in supp( µ ). Then by the above we can find a sequence of points { x i } converging to x ∞ such that, for each i , x i ∈ supp( µ i ) and x i is a U j -generic point for every 1 ≤ j ≤ s . We also fix a sequence { g i } of elementsof G converging to the identity such that g i x ∞ = x i for each i .Applying [43, Thm. 1.1] to each unipotent subgroup U j , we find a naturalnumber i j such that for all i ≥ i j the measure µ is U g − i j -invariant andsupp( µ i ) ⊆ g i supp( µ ). Let i = max { i , . . . , i s } . We now have that forevery i ≥ i , supp( µ i ) ⊆ g i supp( µ ) and we are left to show that for everysuch i , the measure g i µ , which is clearly a homogeneous probability measure,is in fact W -ergodic. Equivalently, we are left to show that for every i ≥ i ,the measure µ is W g − i -ergodic.Fix i ≥ i . We have that µ is U g − i j -invariant for every 1 ≤ j ≤ s . Weconclude that µ is W g − i -invariant, as W g − i is the group generated by theunipotent subgroups U g − i j , 1 ≤ j ≤ s . It follows from Corollary 8.4 thatindeed µ is W g − i -ergodic, which completes the proof of the theorem. (cid:3) We finally have all of the necessary ingredients to prove Proposition 8.1,which has an essentially identical proof to that of [2, Prop. 3.1]. As remarkedimmediately following its statement, it suffices to show that (1) and (3) areequivalent.
Proof of Proposition 8.1.
We first prove that (3) implies (1). Recall that { µ i } is a sequence of homogeneous, W -ergodic measures with weak- ∗ limit µ , which is Haar measure on G/ Γ. Then π ∗ µ is the volume measure on M and hence supp( π ∗ µ ) = M . As the measures µ i are homogeneous, π ∗ µ i is a constant multiple of the volume form on some closed totally geodesicsubspace of M by Lemma 8.2(1), therefore supp( π ∗ µ i ) is contained in some closed maximal geodesic subspace Z i of M . Then M = supp( π ∗ µ ) is con-tained in closure of ∪ Z i , and we conclude that there must be infinitely manydistinct Z i in this union. This implies (1).Finally, we show that (1) implies (3). Assume there are infinitely manydistinct closed maximal totally geodesic subspaces { Z i } of M . By Lemma8.2(3), for every i there exists a standard subgroup W i ≤ G , a homogeneous W i -ergodic measure µ i on G/ Γ, and an element h i ∈ G such that Z i = π ( W i h i Γ / Γ) = π ( X i ), where X i = supp( µ i ). Since the collection of standardsubgroups of G is finite, we may and do pass to a subsequence for which thesubgroups W i all coincide. We denote this common standard subgroup by W . Upon passing to a further subsequence, which we still denote by { µ i } ,we assume that the µ i weak- ∗ converge to a probability measure µ that ishomogeneous and W -ergodic by Corollary 8.4. The proof will be completeonce we show that µ is Haar measure on G/ Γ.For a contradiction, assume µ is not Haar measure. Since µ is homoge-neous, X ∞ = supp( µ ) is a closed homogeneous subspace Sh Γ / Γ of G/ Γ,where h ∈ G and S is the stabilizer of µ . By the assumption that µ is notHaar measure on G/ Γ, S < G is a proper subgroup. By Theorem 8.5, thereexists a sequence of elements { g i } in G and a natural number i such thatfor all i ≥ i , the measures g i µ are homogeneous, W -ergodic probabilitymeasures on G/ Γ with supp( µ i ) ⊆ g i supp( µ ). Upon passing again to a sub-sequence, we assume that i = 1, thus for every i , X i ⊂ g i X ∞ and g i µ is ahomogeneous, W -ergodic probability measure on G/ Γ.We will find a contradiction by showing that the spaces Z i all coincidewith π ( X ∞ ), contradicting the assumption that they are all distinct. Fromnow on we fix a natural number i and will argue that Z i = π ( X ∞ ).By Lemma 8.2(1), π ( g i X ∞ ) is a totally geodesic subspace of M , andby Lemma 8.2(2) it is a proper subspace of M , since the stabilizer of g i µ ,namely S g i , is a proper subgroup of G . From X i ⊆ g i X ∞ we have that Z i = π ( X i ) ⊆ π ( g i X ∞ ) and, by the maximality assumption on the totallygeodesic subspace Z i , we conclude that Z i = π ( g i X ∞ ). Since g i µ is W -invariant and its stabilizer is S g i , we have that W ≤ S g i . We also have π ( S g i g i h Γ / Γ) = π ( W h i Γ / Γ), by the sequence of equations π ( S g i g i h Γ / Γ) = π ( g i Sh Γ / Γ) = π ( g i X ∞ ) = Z i = π ( W h i Γ / Γ) . Lemma 8.2(2) implies that W ≤ S g i ≤ N , where N is the normalizer of W .It follows from Lemma 2.7(3) that W = ( S g i ) + . As W ≤ S , we also havethat W g i ≤ ( S g i ) + = W , hence W g i = W and in particular, g i ∈ N . UsingLemma 8.2(2) again, we conclude that N g i h Γ / Γ is a closed homogeneoussubspace of G/ Γ such that π ( S g i g i h Γ / Γ) = π ( N g i h Γ / Γ). Conjugating by g − ∈ N , the inclusion W ≤ S g i ≤ N implies that W ≤ S ≤ N and an-other application of Lemma 8.2(2) gives π ( X ∞ ) = π ( Sh Γ / Γ) = π ( N h Γ / Γ).Therefore, Z i = π ( g i X ∞ ) = π ( S g i g i h Γ / Γ) = π ( N g i h Γ / Γ) = π ( N h Γ / Γ) = π ( X ∞ ) . RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 39
Thus Z i = π ( X ∞ ), and this completes the proof that µ is Haar measure,hence we have shown that (1) implies (3). (cid:3) Remark 8.6.
It is also worthy of mention here that there has been con-siderable previous work on equidistribution in the context of Shimura va-rieties and special subvarieties. For example, see deep work of Clozel–Ullmo on equidistribution for strongly special subvarieties [13]. Also see[57, 60, 58, 40, 31] for subsequent results in this direction.8.2.
Setup for the proof of Theorem 1.1.
The purpose of this sectionis to collect the necessary final setup so that we can apply Theorem 1.3 inthe proof of Theorem 1.1, which is given in § G = SU( n, < G is a lattice such that the com-plex hyperbolic orbifold K \ G/ Γ contains infinitely many maximal geodesicsubspaces. Pass to a subsequence so that they are all either real or complexhyperbolic subspaces of the same type. By Lemma 8.2(3), there is a fixedstandard subgroup
W < G with normalizer N and elements g i ∈ G so that∆ i = N g − i ∩ Γ , is a lattice in N g − i associated with the i th maximal geodesic subspace.Specifically, ∆ i acts on the totally geodesic subspace K \ KN g − i of K \ G with finite covolume. We will need the following general result. Proposition 8.7 (Cf. Prop. 3.3 [2]) . In addition to the above assumptions,suppose that k is a local field and H is a connected adjoint k -algebraic group.If ρ : Γ → H ( k ) is a representation so that ρ (∆ i ) has proper Zariski closurein H ( k ) for infinitely many i , then there is a k -vector space V of dimensionat least two, an irreducible representation of H on V , and a W -invariant,ergodic measure ν on the bundle ( G × P ( V )) / Γ that projects to Haar measureon G/ Γ .Proof. The proof is almost exactly the same as [2, Prop. 3.3], so we onlysketch the proof. The assumption on ρ (∆ i ) implies that we can pass to asubsequence to assume that the Zariski closures of the ρ (∆ i ) are all containedin proper, nontrivial, k -algebraic subgroups J i < H all of which have thesame dimension d >
0. Let h be the Lie algebra of H over k and j i ⊂ h thesubalgebra associated with J i . We then take V to be the d th exterior powerof h , so each j i defines a point in P ( V ).Let S i be the closed subgroup of N g − i determined by Lemma 8.2(3). Then∆ i is a lattice in S i . Consequently, for each i we obtain a measurable section S i / ∆ i → ( G × P ( V )) / Γ. Let ν i be the push-forward of Haar measure on S i / ∆ i and ν be an ergodic component of the weak- ∗ limit of the ν i . Ratner’stheorem, the assumption that the geodesic submanifolds are distinct andmaximal, and Proposition 8.1 implies that ν is a W -invariant, W -ergodicmeasure that projects to Haar measure on G/ Γ. (cid:3) The proof of Theorem 1.1.
Suppose that Γ < SU( n,
1) is a latticeso that B n / Γ contains infinitely many maximal geodesic subspaces. Let ℓ bethe adjoint trace field of Γ and H be the connected adjoint algebraic groupassociated with Γ as in §
7. Given a place v of ℓ , let ℓ v be the completion of ℓ at this place and ρ v : Γ → H ( ℓ v ) = H v be the natural inclusion. We havea place v so that H v ∼ = PU( n,
1) and ρ v (Γ) is the lattice embedding. Toprove that Γ is arithmetic we must show that ρ v (Γ) is precompact for all v = v .Assume ρ v (Γ) is not precompact for some v = v . Let k = ℓ v , and W and { ∆ i } be as in § ρ v (∆ i ) has proper Zariski closurein H v . Then Proposition 8.7 applies to produce a vector space V as in theconclusion of the proposition and a W -invariant measure ν on ( G × P ( V )) / Γthat projects to Haar measure on G/ Γ.Then Theorem 1.5 along with Propositions 3.2 and 3.4 imply that eitherthe pair ( k, H v ) is compatible with G , or k = R and H v ∼ = PU( n, ρ v (Γ) is unbounded, Theorem 1.3 applies and ρ v extends to a continuoushomomorphism b ρ v : G → H v , and this is a contradiction as explained in [2, § ρ v (Γ) must be precompact for all v = v , which proves thatΓ is arithmetic. (cid:3) Examples, other results, and final comments
Submanifolds of arithmetic manifolds.
In this section, we brieflygive three examples exhibiting some of the possibilities for geodesic subman-ifolds of arithmetic quotients of B n .The general construction is as follows. See [54]. Let ℓ be a totally realnumber field and ℓ ′ a totally imaginary quadratic extension of ℓ . Supposethat D is a central simple division algebra over ℓ ′ of degree d admitting aninvolution τ of second kind , i.e., so that the restriction of τ to ℓ ′ is the non-trivial Galois involution of ℓ ′ /ℓ . For r ≥ τ extends to an anti-involutionof the matrix algebra M r ( D ) by τ -conjugate transposition, denoted x x ∗ .Note that under any embedding of ℓ ′ in C , this involution extends to complexconjugate transposition on M r ( D ) ⊗ ℓ ′ C ∼ = M rd ( C ).An element h ∈ M r ( D ) is called τ -hermitian if h ∗ = h , and then we candefine the ℓ -algebraic unitary group G with ℓ -points G ( ℓ ) = { x ∈ SL r ( D ) : x ∗ hx = h } . Choosing a maximal order O of D , we obtain an arithmetic groupΓ O = { x ∈ SL r ( O ) : x ∗ hx = h } < G ( ℓ ) . Let n = rd −
1. Choosing D and h so thatRes ℓ/ Q ( G )( R ) ∼ = SU( n, × SU( n + 1) [ ℓ : Q ] − , one has that the projection of Γ O to SU( n,
1) is an arithmetic lattice. Forany n ≥
2, all arithmetic subgroups of SU( n,
1) are commensurable withsome such Γ O . RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 41
Example 9.1.
When d = 1, we have D = ℓ . Then h is a τ -hermitian formon an ℓ ′ -vector space V of dimension n + 1, O is the ring of integers of ℓ ′ , and Γ O is sometimes called an arithmetic lattice of simplest type . Forconcreteness, we take ℓ = Q , ℓ ′ = Q ( α ) with α = −
1, and h to be thehermitian form fixed in § O = { g ∈ SL n +1 ( Z [ α ]) : g ∗ hg = h } , is the subgroup of SL n +1 ( Z [ α ]) preserving the form h and G is a Q -algebraicgroup with G ( R ) ∼ = SU( n, B n / Γ O contains all possible types of geodesic submani-folds of a complex hyperbolic n -manifold. Indeed, let { e i } be the standardbasis for our vector space V ∼ = C n +1 . Restricting the form to the span of { e , . . . , e m , e n +1 } visibly gives an arithmetic subgroup Λ O < SU( m,
1) con-tained in Γ O , where SU( m,
1) denotes the standard subgroup in the notationof § m -submanifolds for all m . Considering the real span of this subspaceinstead, the restriction of h now defines a quadratic form on R n +1 stabi-lized by the standard SO ( m,
1) subgroup, hence one similarly finds realhyperbolic submanifolds of every dimension between 2 and n . Example 9.2.
At the other extreme, assume that d = n +1 is prime, so G ( ℓ )is a subgroup of the group SL ( D ) of units of D with reduced norm 1. Fixa maximal order O of D and consider the arithmetic group Γ O < SU( n, n = 2 all fake projective planes arise from thisconstruction [11].Were B n / Γ O to contain a proper geodesic subspace that is complex hyper-bolic, then we would obtain an injection M r ( D ′ ) ֒ → D , where D ′ is a centralsimple ℓ ′ -division algebra of degree d ′ for some totally complex subfield ℓ ′ of ℓ ′ whose intersection ℓ with ℓ is totally real. Moreover, rd ′ divides n + 1.Since n + 1 is prime, we claim that r = 1 and d ′ = n + 1. Indeed, thecase r = d and d ′ = 1 is impossible since the algebra M n +1 ( ℓ ′ ) containssubalgebras of degree 1 < e < n + 1. After taking the tensor product with ℓ ′ , this contradicts the fact that D has prime degree and hence containsno such subalgebras. Now we rule out the case that ℓ ′ is a proper subfieldof ℓ ′ . To see this, note that in this case the ℓ -algebraic unitary group G associated with D now contains H ⊗ ℓ ℓ , where H is the ℓ -algebraic unitarygroup associated with D ′ . However, H is noncompact at exactly one realplace of ℓ and G is consequently noncompact at exactly [ ℓ : ℓ ] real placesof ℓ . Since G is noncompact at exactly one real place of ℓ , we have that ℓ = ℓ and D = D ′ .It follows that B n / Γ O contains no complex hyperbolic geodesic subspaces.A similar argument shows that B n / Γ O also contains no real hyperbolic sub-spaces of any dimension m ≥
2. In other words, the only properly immersedgeodesic subspaces of B n / Γ O are the closed geodesics. Example 9.3.
We leave it to the reader to show that the case M r ( D ) for r > D of possibly composite degree d > Geodesic submanifolds and the intersection pairing.
In this sec-tion, we give some background on the algebraic and complex geometry ofcomplex hyperbolic manifolds and explain the proof of Theorem 1.6. Forsimplicity we restrict to the case complex dimension two, then give referencesfor the analogous results in higher dimension at the end of the section.If M is a closed complex hyperbolic 2-manifold, then the associated com-plex hyperbolic manifold M is a smooth projective surface of general typewhose Chern numbers satisfy c ( M ) = 3 c ( M ) [23]. We recall that c ( M ) isthe self-intersection of the canonical divisor K M ∈ H ( M ) and c ( M ) is theEuler number of M . Moreover, Yau’s famous solution to the Calabi conjec-ture says that this equality of Chern numbers holds for surfaces of generaltype if and only if M = B / Γ for some torsion-free cocompact lattice Γ inPU(2 , B denotes the unit ball in C with its complex hyperbolicmetric [61]. It is a well-known consequence of Selberg’s lemma that anycocompact lattice Γ < PU(2 ,
1) contains a finite index subgroup Γ ′ so that B / Γ ′ is a manifold, hence there is no loss of generality in assuming this isthe case.In the noncompact setting, as described in detail in the proof of Theorem1.5, we can replace a lattice Γ < PU(2 ,
1) with a subgroup of finite indexso that M = B / Γ is a manifold of the form X r D , where X is a smoothprojective surface and D is a certain smooth divisor on X . In this setting,one has the logarithmic canonical divisor K X + D ∈ H ( X ), where K X isthe canonical divisor on X . To unify the two cases, when M is compact weconsider D to be the empty divisor and so X = M . Proof of Theorem 1.6.
An immersed totally geodesic complex hyperbolicsubmanifold of M then determines an irreducible complex curve C on M that compactifies to a curve C on X . Hirzebruch–H¨ofer relative propor-tionality [6, § B.3] is precisely the statement that a projective curve C on X determines a totally geodesic subspace of M if and only if Equation (1) holdswith respect to the divisor K X + D . See [44, Thm. 0.1] for our statement,which is slightly more general than the original. In particular, the theoremis an immediate consequence of Theorem 1.1 and Corollary 1.2. (cid:3) Remark 9.4.
Margulis asked whether arithmeticity is detected purely bythe topology of the locally symmetric space. Theorem 1.6 implies that if M contains a totally geodesic curve, then arithmeticity of M is completelydetermined by the restriction of the intersection pairing on H ( M ) to thecurves on M . In other words, arithmeticity is detected by the topology ofthe underlying variety.We now give references for where one can give a precise version of ourstatements in higher dimensions. For the equality of (logarithmic) Chern RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
1) 43 numbers that characterizes higher-dimensional complex hyperbolic mani-folds, see [56]. A general version of relative proportionality that uniquelydetermines complex hyperbolic totally geodesic submanifolds was given byM¨uller-Stach, Viehweg, and Zuo. See Theorem 2.3 and Addendum 2.4 in[44].This interpretation of our main results leads to the following question.
Question 9.5.
Can one classify the totally geodesic curves on nonarithmeticDeligne–Mostow orbifolds?
This seems particularly approachable in dimension two in the sense thatthe underlying spaces for these orbifolds are closely related to blowups of thecomplex projective plane. In particular, one can connect geodesic curves toclassical plane curves, where immersed geodesic curves will have singularitiesarising from self-intersections. One can then use relative proportionality todetect which curves are totally geodesic.9.3.
The proof of Theorem 1.7.
Proof of Theorem 1.7.
With notation as in the statement of the theorem,suppose that f : M → N is a surjective mapping so that f ( Z i ) is containedin a proper geodesic subspace of N for each i . By hypothesis, the inducedmap on fundamental groups induces a Zariski dense homomorphism ρ :Γ → PU( m, n,
1) associated with M and d = dim( N ). Let Λ < PU( d,
1) be the fundamental group of N .We now proceed exactly as in Proposition 8.7. Let ∆ i < Γ be the sub-group associated with Z i . The assumption on f implies that ρ (∆ i ) is nota Zariski dense subgroup of PU( d, Z i is a geodesic submanifold of M of thesame type with associated with the standard subgroup W of G and ρ (∆ i )has Zariski closure contained in some conjugate of a fixed proper, nontrivial,positive-dimensional J <
PU( d, S i be the closed subgroup of G associated with ∆ i by Lemma 8.2(3).The appropriate exterior power of the Lie algebra of H defines a vector space V such that for each i we can construct a measurable section from S i / ∆ i to ( G × P ( V )) / Γ in order to build a W -invariant, W -ergodic measure onthe bundle that projects to Haar measure on G/ Γ. This and Proposition 3.4allow us to apply Theorem 1.3(1) to conclude that ρ extends to a continuoushomomorphism from SU( n,
1) to PU( d, d = n and ρ (Γ) is alattice in PU( n, ρ (Γ) ≤ Λ, we see that ρ (Γ)is a finite index subgroup. It follows that f is homotopic to a cover. Thisproves the theorem. (cid:3) The proof of Theorem 1.8.
Proof of Theorem 1.8.
To prove the theorem we must take care to differen-tiate between Tr Ad for SL ( C ) and PO (3 , representations by Ad C and Ad R , respectively, as they denote the traces ofthe adjoint representation for sl ( C ) considered as a complex (resp. real)vector space.There are many known hyperbolic 3-manifolds M with nonintegral traces.See [33, § π ( M )from PSL ( C ) to SL ( C ) there is a γ ∈ Γ with a nonintegral trace for theassociated 2 × γ ) is not an algebraic integer. Theembedding of Γ in PSL ( C ) is the lattice embedding associated with thecomplete hyperbolic structure, and we can choose any lift to SL ( C ), as γ will have nonintegral trace for any chosen lift. In what follows, we identify γ with a matrix in SL ( C ).Let λ ± be the eigenvalues of γ . Then λ ± both have minimal polynomial t − Tr( γ ) t + 1, and it follows that λ and λ − also are not algebraic integers.A direct calculation gives:Tr(Ad C ( γ )) = λ + λ − + 1= Tr( γ ) − γ ) is not an algebraic integerthat Tr(Ad C ( γ )) is also not an algebraic integer.Now, suppose that M is a geodesic submanifold of a complex hyperbolic n -manifold. When n = 3, the inclusion Γ < π ( M ) givesΓ < PO (3 , < PU(3 , . When n >
3, we instead haveΓ < SO (3 , < PO ( n, < PU( n, , where we have chosen a lift of Γ from PO (3 ,
1) to SO (3 , su ( n, ( γ )) is not an algebraic integer. This willcontradict Theorem 1.5(3) and complete the proof of the theorem.For n = 3, we can conjugate and choose our hermitian form so that γ mapsto the image in PO (3 , < PU(3 ,
1) of the diagonal matrix with entries { λ , , , λ − } . A direct calculation from the root space decomposition for su (3 ,
1) gives:Tr(Ad su (3 , ( γ )) = 4 λ + 4 λ − + λ + λ − + 5= ( λ + λ − ) + 4( λ + λ − ) + 3Since the above expression is a monic integral polynomial in λ + λ − , whichis not an algebraic integer, we conclude that Tr(Ad su (3 , ( γ )) is also not analgebraic integer. We leave it to the reader to verify that the same proofworks for higher n where 3 is now a sum of roots of unity and 4 is changedto 2( n − (cid:3) We note that Theorem 1.8 does not apply for large families of hyperbolic3-manifolds. For example, if M is a finite-volume hyperbolic 3-manifold RITHMETICITY AND SUPERRIGIDITY FOR SU( n,
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