ARITHMETICITY, SUPERRIGIDITY, AND TOTALLY GEODESIC SUBMANIFOLDS

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1 ARITHMETICITY, SUPERRIGIDITY, AND TOTALLY GEODESIC SUBMANIFOLDS URI BADER, DAVID FISHER, NICHOLAS MILLER, AND MATTHEW STOVER Abstract. Let Γ be a lattice in SO 0 (n, 1) or SU(n, 1). We prove that if the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least 2, then Γ is arithmetic. This answers a question of Reid for hyperbolic n manifolds and, independently, McMullen for hyperbolic 3-manifolds. We prove these results by proving a superrigidity theorem for certain representations of lattices in SO 0 (n, 1) and SU(n, 1). 1. Introduction In this paper, a totally geodesic subspace of a finite volume real or complex hyperbolic manifold or orbifold will always mean a properly immersed, topologically closed, totally geodesic subspace. A totally geodesic subspace is maximal if it is not properly contained in another proper totally geodesic subspace. The main result of this paper is: Theorem 1.1. Let Γ be a lattice in SO 0 (n, 1) or SU(n, 1). If the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least 2, then Γ is arithmetic. This answers a question, first posed informally by Alan Reid in the mid-2000s. Independently, Curtis McMullen asked whether Theorem 1.1 is true in the setting of hyperbolic 3-manifolds (see [9, Qn. 7.6] or [20, Qn. 8.2]). This was very recently and independently proved by Margulis and Mohammadi in the case where M is a closed hyperbolic 3-manifold [16]. Their proof and ours both use a superrigidity theorem to prove arithmeticity, but the superrigidity theorems and their proofs are quite different. Combining Theorem 1.1 with a theorem of Reid [22] we obtain the following. Corollary 1.2. Let K be a knot in S 3 such that S 3 K admits a complete hyperbolic structure. Then S 3 K contains infinitely many immersed totally geodesic surfaces if and only if K is the figure-eight knot. In a prior paper with J.-F. Lafont, the latter three authors proved that a large class of nonarithmetic hyperbolic n-manifolds, including all the hybrids constructed by Gromov and Piatetski-Shapiro, have only finitely many maximal totally geodesic submanifolds [11]. This provided the first known examples of hyperbolic n-manifolds, n 3, for which the collection of totally geodesic hypersurfaces is finite and nonempty. Our approach to proving Theorem 1.1 is inspired by Margulis s superrigidity and arithmeticity theorems [17, 18]. Given a higher rank semisimple group G and a lattice Γ < G, superrigidity classifies representations ρ : Γ H into other algebraic 1

2 2 U BADER, D FISHER, N MILLER, AND M STOVER groups. In particular, it gives criteria for ρ to satisfy a strong dichotomy: either ρ extends to a continuous homomorphism from G to H or it has bounded image in H. The proof of arithmeticity follows from applying superrigidity to the so-called Galois conjugate representations, along with representations over p-adic fields that arise from nonarchimedean completions. Of course, superrigidity is well known to fail drastically for lattices in SO 0 (n, 1) and SU(n, 1). Our goal in this paper is to prove a strong enough superrigidity theorem to deduce our main results using Margulis s method of deducing arithmeticity. A famous example of this strategy is Margulis s proof of the arithmeticity of lattices with so-called dense commensurator. Unlike superrigidity, this theorem also holds in rank one and it is the full converse to a theorem of Borel [3]. Margulis proved arithmeticity of lattices with dense commensurator by classifying representations of lattices that extend to representations of some dense subgroup of G contained in the commensurator. Relating dense commensurators of arithmetic lattices back to the existence of infinitely many totally geodesic submanifolds, one can easily observe: Arithmetic geodesic submanifold dichotomy: For any 1 k n 1, an arithmetic hyperbolic n-manifold either contains no codimension k geodesic submanifolds, or it contains infinitely many and they are everywhere dense. This observation is one of the motivations for the question answered by Theorem 1.1 and was perhaps first made precise in dimension 3 by Maclachlan Reid and Reid [15, 23], who also exhibited the first hyperbolic 3-manifolds with no totally geodesic surfaces. Note that an analogous statement holds for any arithmetic locally symmetric space. See [11] for further discussion and examples. We now state the superrigidity theorem that allows us to prove Theorem 1.1. Let Γ be a lattice in G, which is SO 0 (n, 1) or SU(n, 1). We consider representations of Γ into algebraic groups over a local field k, and by algebraic group we always mean the k-points of a k-algebraic group. Let P be a minimal parabolic subgroup of G and N its unipotent radical. We call a k-algebraic group H with k-points H(k) compatible with G if, for every nontrivial k-subgroup B < H with k-points B, any continuous homomorphism α : P N H (B)/B restricts to the trivial homomorphism on N, where N H (B) is the normalizer of B in H (see 2.3). We now state our superrigidity theorem in the case of simple targets and refer to Theorem 3.2 for the statement in the semisimple case. Theorem 1.3. Let G be SO 0 (n, 1) for n 3, or SU(n, 1) for n 2, and Γ < G be a lattice. Fix W < G, a proper subgroup isomorphic to SO 0 (m, 1) or SU(m, 1) for some m 2. Let H be a connected, adjoint, simple k-algebraic group that is compatible with G, and let ρ : Γ H(k) be an unbounded homomorphism with Zariski dense image. If there is an irreducible H representation on a vector space V and a (left) W -invariant measure µ on (G P(V ))/Γ that descends to Haar measure on G/Γ, then ρ extends to a continuous homomorphism from G to H(k). Understanding invariant measures for dynamical systems that are not homogeneous plays an important role in other recent results in rigidity theory. For example, see work of Brown, Hurtado, and the second author on Zimmer s conjecture [4, 5]. In 3 we give a method for constructing a W -invariant measure on (G P(V ))/Γ projecting to Haar measure whenever there are infinitely many maximal W -orbit closures on G/Γ. In other words, we produce such a measure when the associated

3 TOTALLY GEODESIC SUBMANIFOLDS 3 locally symmetric space contains infinitely many maximal totally geodesic submanifolds. This W -invariant measure allows us to apply Theorems 1.3 and 3.2, and a standard application of Margulis s proof that superrigidity implies arithmeticity completes the proofs of Theorem 1.1 (see 2.5). Acknowledgments. Fisher was partially supported by NSF Grant DMS , the Institute for Advanced Studies, and the Simons Collaboration on Algorithms and Geometry. Miller was partially supported from U.S. National Science Foundation grants DMS , , RNMS: GEometric structures And Representation varieties (the GEAR Network). Stover was partially supported by Grant Number from the Simons Foundation/SFARI. Our approach to Theorem 1.1 owes a tremendous debt to the ideas of Gregory Margulis on superrigidity and arithmeticity [17, 18, 19]. The authors also thank Matt Bainbridge, Alex Eskin, and Alan Reid for helpful conversations. They particularly thank Hee Oh for detailed comments on an earlier draft, Jean-François Lafont for his participation in the early phases of this project, and Alex Furman for his inspiring work on superrigidity with the first author. 2. Preliminaries 2.1. Basic notation. We establish some notation. Let G be either the connected component SO 0 (n, 1) of the identity in SO(n, 1) for n 3 or SU(n, 1) for n 2. Fix a lattice Γ < G, acting on G by the right-action g γ = gγ 1. Let W < G be a proper subgroup isomorphic to either SO 0 (m, 1) or SU(m, 1), m 2. Let H n denote hyperbolic n-space and H n C be complex hyperbolic n-space. For the action of SO 0 (n, 1) on H n by isometries, we have that W preserves a totally geodesic subspace isometric to an embedded H m in H n. In the SU(n, 1) setting, W preserves a totally geodesic H m or H m C in Hn C, according to whether W is orthogonal or unitary. The full stabilizer of this totally geodesic subspace is a compact extension Ŵ of W, and there is a one-to-one correspondence between closed Ŵ - orbits on G/Γ and properly immersed totally geodesic subspaces of the orientable finite volume locally symmetric space X Γ = K \G/Γ, where K is the maximal compact subgroup of G. In this paper, we are interested in the setting where X Γ contains a totally geodesic subspace, i.e., where Ŵ Γ is a lattice in Ŵ. Fix a maximal R-split torus A < W. Since W and G are both R-rank one, A is also a maximal R-split torus of G. Fix a maximal unipotent subgroup N of G normalized by A and let M be the compact factor of the Levi decomposition of the centralizer of A. Then P = MAN is the Langlands decomposition of the maximal parabolic subgroup of G associated with the pair (A, N). These choices were made such that N = W N is a maximal unipotent subgroup of W. Remark 2.1. We consider G and all the subgroups introduced in this subsection as denoting the real points of a real algebraic group. This is important for our arithmeticity theorem, but is not used in the proofs of our superrigidity theorems Algebraic representations. In this subsection, we introduce the ideas from the work of Bader and Furman [1] used in the proof of our superrigidity theorem. Let k be a local field, fix a k-algebraic group H, and let H = H(k) denote the k-points of H. To start, let G be a locally compact second countable group, Γ < G be a lattice, and ρ : Γ H be a Zariski dense representation. Given a closed subgroup T < G, a T -algebraic representation of G consists of:

4 4 U BADER, D FISHER, N MILLER, AND M STOVER a k-algebraic group J, a k-(h J)-algebraic variety V, which is considered as a left H-space and a right J-space on which the J-action is faithful, a Zariski dense homomorphism τ : T J(k), an algebraic representation of G on V, i.e., an almost-everywhere defined measurable map φ : G V(k) such that φ(tgγ 1 ) = ρ(γ)φ(g)τ(t) 1 for every γ Γ, every t T, and almost every g G. We denote the data for a T -algebraic representation of G by J V, τ V, and φ V. A T -algebraic representation is called coset T -algebraic when V is the coset space H/B for some k-algebraic subgroup B of H, and J is a k-subgroup of N H (B)/B, where N H (B) denotes the normalizer of B in H. Given another T -algebraic representation U, let J U,V be the Zariski closure of (τ U τ V )(T ) in J U J V. Then a morphism π : U V is an (H J U,V )-equivariant map such that φ V agrees almost everywhere with π φ U. The proof of our superrigidity theorem uses the following. Theorem 2.2 (Thm. 4.3 [1]). The collection of T -algebraic representations of G forms a category. If the T -action on G/Γ is weakly mixing, then this category has an initial object, and this initial object is a coset T -algebraic representation. Recall that an initial object in a category is an object which has exactly one morphism to all other objects in the category. Though not stated explicitly, the following is also implicit in [1]. Lemma 2.3. Assume that the action of T on G/Γ is weakly mixing. Then T and N G (T ) have the same initial objects in their respective categories of T -algebraic representations. Moreover, T has the same initial objects as the iterated normalizer N G (N G (... (N G (T ))... )). Proof. For the first claim, the forward direction is the content of [1, Thm. 4.6]. For the backward direction, if φ : G H/B(k) is an initial object in the category of N(T )-algebraic representations with associated homomorphism τ from T to N H (B)/B(k), then φ and τ T form a T -algebraic representation and this representation must be initial by minimality. Indeed, otherwise another application of the forward direction contradicts minimality of B. The second claim follows immediately from the first Compatibility. Let G, N, and P = M AN be as in 2.1. Let k be a local field and H a k-algebraic group with H = H(k). Recall that H is compatible with G if for all nontrivial k-algebraic subgroups B < H with normalizer N H (B), N is in the kernel of every continuous homomorphism from P to the k-points of N H (B)/B. An easy observation is that if H is an algebraic group over a nonarchimedean local field, then G is compatible with H. The following lemma is key in applying Theorem 1.3 to prove Theorem 1.1. Lemma 2.4. If p + q = n + 1, then any simple factor of the R-algebraic group PO(p, q) is compatible with SO(n, 1), as is the C-algebraic group PO(n+1, C), along with the R-algebraic group PO (n + 1) for n odd. Similarly, SU(p, q) is compatible with SU(n, 1) whenever p + q = n + 1, as is SL n+1 (C).

5 TOTALLY GEODESIC SUBMANIFOLDS 5 Proof. We consider the case H = PO(p, q). The others are similar and left to the reader. Let B < H be the real points of a real algebraic subgroup and suppose that α : P N H (B)/B is a homomorphism with α(n) nontrivial. Note that M = S(O(n 1) O(1)) = SO(n 1), acts on N = R n 1 by the usual action of SO(n 1), which is transitive on lines. It follows that if M is in the kernel of α, then so is all of N. Conversely, we see that if α is nontrivial on M (implying that ker(α M ) is finite), then it is injective on N. In fact, we see that α is almost injective on all of MN, in the sense that the kernel is finite. First assume that n = 3, so any simple factor H of PO(p, q) is PO(2, 1) or PO(3, 1). It is immediate that the former case is impossible. For the latter case, notice that dim(n H (B)) 4, and considering the possible subgroups of H, α can only exist when B is the identity and N H (B) = H. We are therefore reduced n 4 and M almost simple. Considering the Levi decompositions of N H (B) and N H (B)/B, almost simplicity of M implies that we can lift α to obtain a homomorphism α : M H, where M is a finite cover of M. Note that H = PO(p, q) can only contain a group locally isomorphic to SO(n 1) when q {1, 2}. If q = 2, then M must map to an almostdirect factor of the maximal compact subgroup K = P(O(n 1) O(2)). There are no subgroups of H normalized by K except itself, but N H (B)/B is assumed to be nontrivial. Therefore B is trivial and N H (B) = H. Thus PO(n 1, 2) is compatible with G. This reduces us to the case PO(n, 1), where similar analysis shows that the only possible map α is the obvious one, i.e., B the identity and N H (B) = H. Remark 2.5. If we instead consider PO(n + 1, C) as an R-algebraic group, it is not compatible with SO(n, 1). However, suppose that ρ : Γ PO(n + 1, C) is a representation that is Zariski dense in the real Zariski topology. Then it is Zariski dense in the complex Zariski topology. In the proof of arithmeticity, a Galois conjugate isomorphic to PO(n + 1, C) is naturally given the real Zariski topology, but for the purposes of proving Theorem 1.3 we may instead consider it in the complex Zariski topology. The situation is similar for SL n+1 (C) and SU(n, 1) Orbit closures in the complex hyperbolic setting. In the case where G is SU(n, 1), there is a correspondence between orbit closures and totally geodesic real or complex hyperbolic subspaces analogous to the one described for SO 0 (n, 1) in [11, 5.3]. We do not give detailed proofs, but merely state two lemmas needed to easily adapt the proofs there. The first is a replacement for [11, Lem. 5.9], and the proof follows the proof of [12, Prop. A.6]. Lemma 2.6. Fix m 2. If Y is a subgroup of SU(n, 1) containing SO 0 (m, 1), then Y is either of the form S(O(l, 1) K) G, where l m and K < U(n l) or of the form S(U(l, 1) K) G, where l m and K < U(n l). Similarly, if Y is a closed subgroup of SU(n, 1) containing SU(m, 1), then Y is of the form S(U(l, 1) K) G, where l m and K < U(n l). The other lemma is a replacement for [11, Lem. 5.10]. Lemma 2.7. Fix 2 m n 1, let W be SO 0 (m, 1) or SU(m, 1) in G = SU(n, 1), and suppose Y is a subgroup satisfying W < Y < G. If W is not SO 0 (2, 1), then any subgroup Z of Y isomorphic to W is conjugate to W in Y. There are exactly two

6 6 U BADER, D FISHER, N MILLER, AND M STOVER conjugacy classes isomorphic to SO 0 (2, 1), namely it and SU(1, 1). Furthermore, Aut(W ) < G and Aut(W ) < Y unless Y is W K for some compact K Proof of Theorem 1.1. In this section, we describe how one deduces Theorem 1.1 from Theorems 1.3 and 3.2. This closely follows Margulis s proof of arithmeticity from superrigidity. For more details see [19, Ch. IX] or [26, Ch. 6]. Suppose Γ < G is a lattice and that H n /Γ or H n C /Γ contains infinitely many maximal totally geodesic submanifolds. Then it follows from work of Selberg, Calabi, Calabi Vesentini, and Garland that Γ is defined over a number field [6, 7, 13, 24]. This can also be deduced from our superrigidity theorems, see e.g. [26, Lem ]. More precisely, there is a number field l, a connected, semisimple l-algebraic group G, and an embedding w : l R = l w for which G(l w ) = G(l) w(l) R = G, and G(l) Γ is a finite index subgroup of Γ under this embedding. For the purposes of proving our main results, we can replace G with its connected adjoint form and replace Γ with G(l) Γ. Then Γ is arithmetic if and only if the image of Γ is precompact for every other embedding of l into a local completion G(l v ), v w. The embedding of Γ into G(l v ) can never extend to a continuous homomorphism of G(l w ), since Galois conjugation is everywhere discontinuous and nonarchimedean local fields are totally disconnected. See 3 for discussion of how the assumptions of Theorem 1.1 imply that Theorems 1.3 and 3.2 apply for the embedding of Γ into G(l v ), v w. It follows that Γ must be precompact under every such embedding, thus Γ is arithmetic. 3. Invariant measures To prove Theorem 1.1 from Theorem 1.3, we produce an invariant measure as in Theorem 1.3 from the hypotheses of Theorem 1.1. We also formulate the version of Theorem 1.3 needed when H is semisimple in Theorem 3.2, and produce the analogous measure there. We continue with the notation established in 2. By hypothesis, there are infinitely many closed maximal totally geodesic subspaces {C i } of X Γ = K \G/Γ. We can assume they all have the same dimension, say m, and that they are either all real hyperbolic or all complex hyperbolic. As in [11], there is a subgroup Ŵ of G locally isomorphic to SO(m, 1) SO(n m), SO(m, 1) SU(n m), or SU(m, 1) SU(n m) such that for each i there is a conjugate Ŵi of Ŵ and a closed Ŵi-orbit in G/Γ projecting to C i. See 2.4. The stabilizer of this orbit is a lattice Γ i = Ŵi Γ in Ŵi. We let W i be the Zariski closure of Γ i in Ŵi and note that W i is an l-defined subgroup of G, where l is the number field and G the l-algebraic group associated with Γ as in 2.5. Since Γ is nonarithmetic, we have a local completion k = l v of l for which the natural inclusion ρ : Γ H(k) = G(k) is not precompact. Since our representation into H(k) is defined by Galois conjugation or localization at a prime, it follows that ρ(γ i ) is contained in a proper k-algebraic subgroup L i (k) of H(k) that is isotypic to W i. Since W i could be reductive in general, L i (k) may not be semisimple. We therefore let L i (k) be the maximal semisimple subgroup of L i (k). There are at most finitely many possible isomorphism types for L i and so we can assume the isomorphism type L = L i is fixed. The variety Hom(L, H) is of

7 TOTALLY GEODESIC SUBMANIFOLDS 7 finite type, so there are only finitely many components of this variety, and hence only finitely many conjugacy classes of maps from L to H [14, Thm. XXIV.7.3.1(i)]. Therefore, we can assume that all the embeddings of L i (k) are conjugate in H(k). We now assume that H(k) is simple, which is automatic when G SO 0 (3, 1), and consider the semisimple case at the end of this section. Let m = dim(l) and consider the m th exterior power m : H(k) GL(V ) of the adjoint representation of H(k) on its Lie algebra h. The Lie algebra l i of L i (k) then determines a line l i in V, and all of these lines are in a single H(k)-orbit of some fixed line l. We replace V by an irreducible direct factor to which l (hence each l i ) projects nontrivially. Since the stabilizer of l is the normalizer of L(k) and hence a proper subgroup of H(k), this direct factor can be chosen to be nontrivial. The point stabilizer for the line l i contains L i (k) and hence Γ i. Given the closed W i -orbit W i /Γ i, note that l i is an invariant line bundle over W i /Γ i and therefore defines a measurable section: s i : W i /Γ i (W i P(V ))/Γ i Let ˆµ i be Haar measure on W i /Γ i and define µ i = s i ˆµ i. We can then construct a W -ergodic, W -invariant measure on (G P(V ))/Γ that projects to Haar measure on G/Γ by taking µ to be any weak- limit of the µ i on (G P(V ))/Γ. When Γ is cocompact it is clear these limits exist. In the noncocompact setting, existence requires the nondivergence of unipotent orbits and that W is generated by unipotent subgroups [8, 6]. From the analysis of orbit closures in [11, 5] and 2.4 and work of Mozes and Shah [21], we have: Lemma 3.1. The projection to G/Γ of the W -invariant measure µ defined above is Haar measure. For G = SO 0 (3, 1), the group H(k) need not be simple due to exceptional isomorphisms such as PO(2, 2) = PO(2, 1) PO(2, 1) or PO(4, k) = PGL(2, k) PGL(2, k) when k is p-adic or complex. To deduce Theorem 1.1 in this case requires a variant of Theorem 1.3. Since H is an adjoint semisimple group, it is a direct product H 1... H k and we write pr i for projection on the i th factor. The reformulation of Theorem 1.3 in this case is the following, where we define G, W, and Γ as above. Theorem 3.2. Let H be a connected, adjoint, semisimple k-algebraic group, and suppose that G is compatible with every simple factor of H(k). Let ρ : Γ H(k) be a homomorphism with Zariski dense image, and assume there exists an irreducible H(k) representation V and a W -invariant measure µ on (G P(V ))/Γ that projects to Haar measure on G/Γ. Then either there is a simple factor H i (k) of H(k) such that pr i ρ extends to a representation of G or ρ(γ) is precompact. To apply Theorem 3.2 in the proof of Theorem 1.1, we must find a representation V on which H(k) acts irreducibly and for which the above construction yields the necessary invariant measure. To this end, we need to consider cases when H(k) = PGL 2 (k) PGL 2 (k), where k is R, C, or a nonarchimedean local field of characteristic zero. Notice that the image ρ(γ i ) is not contained in a direct factor of H(k). If it were, then ρ(γ i ) would be normal in ρ(γ) hence, since ρ is injective, Γ i would be normal in Γ, which is impossible. Therefore ρ(γ i ) is contained in a conjugate of the diagonal (PGL 2 (k)) in PGL 2 (k) PGL 2 (k) for all i.

8 8 U BADER, D FISHER, N MILLER, AND M STOVER We take the adjoint representation of PGL 2 (k) PGL 2 (k) on k 6 and the diagonal three dimensional subspace U = (k 3 ) < k 3 k 3 stabilized by (PSL 2 (k)). A computation shows that 3 (k 3 k 3 ) splits as a direct sum of four irreducible representations of PGL 2 (k) PGL 2 (k), two that are trivial and two that are the representation V (3, 3) on k 3 k 3. One also checks that 3 (U) projects nontrivially to each V (3, 3) (in fact, to all four summands). Taking V = V (3, 3) and arguing as above, we also produce a W -invariant measure on (G P(V ))/Γ when G = SO 0 (3, 1) and H(k) is not simple. 4. Proof of Theorem 1.3 and Theorem 3.2 We assume for the entirety of this section that the notation from 2 is in place and that the hypotheses for Theorem 1.3 and Theorem 3.2 hold. In particular, we have a W -invariant measure µ on (G P(V ))/Γ that descends to Haar measure on G/Γ. Under the assumptions of Theorem 1.1, we proved in 3 that this does indeed hold. Recall that H is adjoint and connected From measures to measurable maps to varieties. The goal of this subsection is to prove: Proposition 4.1. There exists a proper k-algebraic subgroup L < H and a measurable W -invariant, 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). Proof. The W -invariant measure µ on (G P(V ))/Γ yields a W -invariant Γ-map φ : G P(P(V )), where P(P(V )) is the space of probability measures on P(V ). By [26, Cor ] and [26, Cor ], the image of this map lies in a single H(k)-orbit that can be identified with H(k)/ L for L a compact extension of the k-points of a k-algebraic subgroup of H(k). We claim that L is noncompact. That Γ < G is weakly cocompact in the sense of [19] follows from [19, Cor. III.1.10] when Γ is cocompact and from [2, Lem. 3] in the finite volume case. The rest of the hypotheses of [19, Thm. V.5.15] hold by assumption, so there is some element g W for which the Lyapunov exponent λ 1 (g, µ) is positive. If L were compact, then L would preserve a norm on V, and it is easy to extend this to a norm preserved by the W -action on (G P(V ))/Γ. This would imply that λ 1 (g, µ) = 0 for every g W, which is a contradiction. Let L be the Zariski closure of L. That L is a proper k-subgroup of H now follows from [26, Prop ]. This proves the proposition. In the semisimple case, L could contain a proper normal subgroup of H, so H may not act effectively on H/L. We consider L as a subgroup of H = H i and collect factors where L H i = H i. Choose any H j outside this collection and set H c j = i j H j. Consider the projections pr c : H H c j and pr j : H H j and define L to be ker(pr c L ). One checks that H c j \H/L = H j/l, where L = pr j (L), and that L is a proper subgroup of H j. By replacing H, L with H j, L, we will produce an extension of π j ρ to G, which proves Theorem 3.2. We therefore now assume that H is simple and compatible with G while completing the proof.

9 TOTALLY GEODESIC SUBMANIFOLDS From measurable maps to extension of homomorphisms. In this section, we complete the proof of Theorem 1.3. We continue with the notation and definitions from 2. We show that the existence of the map φ : G H/L(k) from Proposition 4.1 implies that the representation ρ of Γ extends to G. First consider the cases where either G is SO 0 (n, 1) and W is SO 0 (m, 1) or G is SU(n, 1) and W is SU(m, 1). This assumption implies that N = W N is normal in N. We will handle the case G = SU(n, 1) and W = SO 0 (m, 1) at the end of the proof. Consider an initial object in the category of N -algebraic representations of G. Applying Theorem 2.2, there is a k-algebraic subgroup L 1 of H such that this object is a measurable map Ψ : G H/L 1 (k) that is (N Γ)-equivariant for a continuous homomorphism τ 1 : N N H (L 1 )/L 1 (k). Since N is a normal subgroup of N, τ 1 extends to a representation from N to N H (L 1 )/L 1 (k) for which Ψ is (N Γ)-equivariant by [1, Thm. 4.6] and Lemma 2.3. We claim that τ 1 (N) is nontrivial. If not, since this is an initial object, then every morphism in the category of N-algebraic representations would also be N- invariant. In particular, this would be true for the N-algebraic representation φ : G H/L(k). Since this map is also W -invariant and N, W = G, then 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) is Zariski dense and L is a proper algebraic subgroup of the connected adjoint group H. We now repeat this argument with the normalizer P of N in G. Again [1, Thm. 4.6] and Lemma 2.3 imply that τ 1 extends to a continuous homomorphism τ 2 : P N H (L 1 )/L 1 (k) such that Ψ is P -equivariant. Since H is compatible with G, we see that L 1 is necessarily trivial. Since L 1 is trivial and A < P, the existing map Ψ : G H(k) is A-equivariant via the homomorphism τ 2 A, and therefore must be an initial object for the category of A-algebraic representations by Lemma 2.3. Once again, [1, Thm. 4.6] and Lemma 2.3 imply that τ 2 A extends to a homomorphism τ 3 : N(A) H(k) for which Ψ is N(A)-equivariant, where N(A) is the normalizer of A in G. Notice that N(A) contains a Weyl element w for A and hence P, N(A) = G. Since Ψ is equivariant for both P and N(A), using [1, Prop. 5.1] and following the end of the proof of [1, Thm. 1.3], we deduce that ρ : Γ H(k) extends to a continuous homomorphism ρ : G H(k). This proves the theorem in these cases. For the case SO 0 (m, 1) < SU(n, 1), there is one additional step to consider, as N is not normal in N. The normalizer N G (N ) is instead the subgroup N 0 of N generated by N and the center of N. However, notice that N G (N G (N )) = N G (N 0) = N, and therefore applying [1, Thm. 4.6] and Lemma 2.3 to N G (N G (N )) instead of N G (N ), we construct an initial object in the category of N-algebraic representations. The rest of the argument is unchanged, which completes the proof. References [1] U. Bader and A. Furman. An extension of Margulis super-rigidity theorem. Preprint: arxiv: [2] M. B. Bekka. On uniqueness of invariant means. Proc. Amer. Math. Soc., 126(2): , 1998.

10 10 U BADER, D FISHER, N MILLER, AND M STOVER [3] A. Borel. Density and maximality of arithmetic subgroups. J. Reine Angew. Math., 224:78 89, [4] A. Brown, D. Fisher, and S. Hurtado. Zimmer s conjecture: subexponential growth, measure rigidity, and strong property (T). Preprint: arxiv: , [5] A. Brown, D. Fisher, and S. Hurtado. Zimmer s conjecture for actions of SL(m, Z). Preprint: arxiv: , [6] E. Calabi. On compact, Riemannian manifolds with constant curvature. I. In Proc. Sympos. Pure Math., Vol. III, pages American Mathematical Society, Providence, R.I., [7] E. Calabi and E. Vesentini. On compact, locally symmetric Kähler manifolds. Ann. of Math. (2), 71: , [8] S. G. Dani and G. A. Margulis. Limit distributions of orbits of unipotent flows and values of quadratic forms. In I. M. Gel fand Seminar, volume 16 of Adv. Soviet Math., pages Amer. Math. Soc., [9] K. Delp, D. Hoffoss, and J. Manning. Problems in Groups, Geometry, and Three-Manifolds. Preprint: arxiv: [10] A. Eskin, S. Filip, and A. Wright. The algebraic hull of the Kontsevich Zorich cocycle. Ann. of Math. (2), 188(1): , [11] D. Fisher, J.-F. Lafont, N. Miller, and M. Stover. Finiteness of maximal geodesic submanifolds in hyperbolic hybrids. Preprint: arxiv: [12] D. Fisher and T. Nguyen. Quasi-isometric embeddings of non-uniform lattices. To appear in Commentarii Mathematici Helvetici, [13] H. Garland. On deformations of lattices in Lie groups. In Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), pages Amer. Math. Soc., Providence, R.I., [14] P. Gille and P. Polo, editors. Schémas en groupes (SGA 3). Tome I. Propriétés générales des schémas en groupes, volume 7 of Documents Mathématiques. Société Mathématique de France, Séminaire de Géométrie Algébrique du Bois Marie [15] C. Maclachlan and A. W. Reid. Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups. Math. Proc. Cambridge Philos. Soc., 102(2): , [16] G. Margulis and A. Mohammadi. Arithmeticity of hyperbolic 3-manifolds containing infinitely many totally geodesic surfaces. Preprint: arxiv: [17] G. A. Margulis. Discrete groups of motions of manifolds of nonpositive curvature. In Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 2, pages Canad. Math. Congress, Montreal, Que., [18] G. A. Margulis. Arithmeticity of the irreducible lattices in the semisimple groups of rank greater than 1. Invent. Math., 76(1):93 120, [19] G. A. Margulis. Discrete subgroups of semisimple Lie groups, volume 17 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer-Verlag, [20] D. B. McReynolds and A. W. Reid. The genus spectrum of hyperbolic 3 manifolds. Math. Res. Lett., 21: , [21] S. Mozes and N. Shah. On the space of ergodic invariant measures of unipotent flows. Ergodic Theory Dynam. Systems, 15(1): , [22] A. W. Reid. Arithmeticity of knot complements. J. London Math. Soc. (2), 43(1): , [23] A. W. Reid. Totally geodesic surfaces in hyperbolic 3-manifolds. Proc. Edinburgh Math. Soc. (2), 34(1):77 88, [24] A. Selberg. On discontinuous groups in higher-dimensional symmetric spaces. In Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960), pages Tata Institute of Fundamental Research, Bombay, [25] R. J. Zimmer. Strong rigidity for ergodic actions of semisimple Lie groups. Ann. of Math. (2), 112(3): , [26] R. J. Zimmer. Ergodic theory and semisimple groups, volume 81 of Monographs in Mathematics. Birkhäuser Verlag, Basel, 1984.

11 TOTALLY GEODESIC SUBMANIFOLDS 11 Weizmann Institute of Science address: Department of Mathematics, Indiana University, Bloomington, IN address: Department of Mathematics, Temple University, Philadelphia, PA address: