# Reading roadmap: Arithmeticity, superrigidity, and totally geodesic submanifolds

The proposed argument translates infinitely many maximal totally geodesic subspaces into homogeneous measures approaching Haar measure, lifts them to a projective bundle, and uses compatibility and algebraic representations to extend lattice representations. A trace-field/local-place criterion then yields arithmeticity; separate deductions address knot complements, geometric finiteness and Dehn fillings.

Source SHA-256: `6776902687fa24498e9bc0084edb02b1630009b90b7b06a7e2fba0e6d697397c`. Indexed 111,123 characters.

Edges point from prerequisite to consumer. Solid proof edges are attributed inferences, not formal certificates; references remain candidates.

## Geometry to homogeneous measures

Read the geometric and homogeneous formulations, then the closed-orbit and measure-convergence arguments.

Claims: thm:main, thm:main:homogeneous, prop:homogen, lem:clarifying, thm:MS

## Lift the measures

Follow how algebraic subgroups provide invariant lines and measures on a projective bundle. Keep the three-dimensional representation case separate.

Claims: prop:liftingmeasures

## Compatibility and superrigidity

Identify the allowed target groups, then follow initial algebraic representations, normalizer equivariance and the continuous extension.

Claims: lemma:compatible, thm:BF, lem:initobj, theorem:superrigiditydichotomy

## Arithmeticity and applications

Return to the adjoint trace field and its places, then read the knot, geometric-finiteness and Dehn-filling deductions separately.

Claims: thm:main, cor:knots, cor:kleinian, thm:dehnfilling

## Internal claim index

### Theorem 1.1 — `thm:main`

Source lines 392–396.

Let $\Gam$ be a lattice in $\SO_0(n, 1)$. If the associated locally symmetric space contains infinitely many maximal totally geodesic subspaces of dimension at least $2$, then $\Gam$ is arithmetic.

*Extractive statement; proof gist not independently reconstructed.*

- `thm:main:homogeneous` (proof_dependency; curated_source_check): We show in Proposition \ref{prop:homogen} that Theorem \ref{thm:main:homogeneous} implies Theorem \ref{thm:main}.
- `prop:homogen` (proof_dependency; curated_source_check): We show in Proposition \ref{prop:homogen} that Theorem \ref{thm:main:homogeneous} implies Theorem \ref{thm:main}.

### Corollary 1.2 — `cor:knots`

Source lines 403–407.

Let $K$ be a knot in $S^3$ such that $S^3 \ssm K$ admits a complete hyperbolic structure. Then $S^3 \ssm K$ contains infinitely many immersed totally geodesic surfaces if and only if $K$ is the figure-eight knot.

*Extractive statement; proof gist not independently reconstructed.*

### Corollary 1.3 — `cor:kleinian`

Source lines 412–423.

{\ } \item If $M$ is a geometrically finite hyperbolic $3$-manifold containing infinitely many totally geodesic surfaces with finite area, then $M$ has finite volume and $\pi_1(M)$ is arithmetic. \item If $M$ is a convex cocompact hyperbolic $n$-manifold containing infinitely many maximal totally geodesic surfaces with finite area, then $M$ is compact and $\pi_1(M)$ is arithmetic. \item If $n \ge 4$ is even and $M$ is a finite volume hyperbolic $n$-manifold, then $M$ is arithmetic if and only if it contains infinitely many totally geodesic hype

*Extractive statement; proof gist not independently reconstructed.*

### Theorem 1.4 — `thm:dehnfilling`

Source lines 430–434.

Let $M$ be a cusped hyperbolic $3$-manifold of finite volume with at least one torus cusp, and $N$ a hyperbolic $3$-manifold obtained by Dehn filling on some nonempty subset of the torus cusps of $M$. Then only finitely many totally geodesic surfaces in $N$ are isotopic to the image of a totally geodesic surface in $M$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Given $M$ and $N$ as in the statement of the theorem, let $\rho : \Gam \to \SO_0(3,1)$ be the representation defined above. We will prove that if infinitely many totally geodesic surfaces in $N$ are the images of totally geodesic surfaces in $M$, then $\rho$ must extend to a homomorphism $\SO_0(3,1) \to \PO_0(3,1)$. Since $\rho$ has a nontrivial kernel, this is impossible. We let $G$ denote $\SO_0(3,1)$ containing $\Gam$ as a lattice and $H$ denote $\PO_0(3,1)$ as the target for $\rho$. Note that $\rho$ has unbounded and Zariski dense image. To apply Theorem \ref{theorem:superrigiditydichotomy}, we take $W = \SO_0(2,1)$ and then must produce an $H$-representation $V$ and a $W$-invariant meas


Candidate references: `theorem:superrigiditydichotomy` (candidate_proof_dependency); `thm:main` (candidate_proof_dependency); `prop:liftingmeasures` (candidate_proof_dependency).

### Theorem 1.5 — `thm:main:homogeneous`

Source lines 463–467.

If there exists an infinite sequence $\{\mu_i\}$ of $W$-invariant, ergodic measures with proper support for which Haar measure on $G/\Gamma$ is a weak-$*$ limit of the $\mu_i$, then $\Gamma$ is arithmetic.

*Extractive statement; proof gist not independently reconstructed.*

- `theorem:superrigiditydichotomy` (proof_dependency; curated_source_check): The existence of the desired extension $G\to \bfH(\ell_v)$ will follow from Theorem~\ref{theorem:superrigiditydichotomy}
- `lemma:compatible` (proof_dependency; curated_source_check): Our proof will be complete once we show that the pair consisting of $\ell_v$ and $\bfH$ is compatible with $G$. This will be done in \S\ref{ssec:Compatible}.
- `prop:liftingmeasures` (proof_dependency; curated_source_check): in \S\ref{Ssec:buildingmeasures} we will produce an $\ell_v$-vector space $V$, endowed with a faithful irreducible representation of $\bfH(\ell_v)$, and a $W$-invariant measure on $(G\times\bbP(V))/\Gamma$

### Theorem 1.6 — `theorem:superrigiditydichotomy`

Source lines 478–482.

Let $G$ be $\SO_0(n, 1)$ for $n \ge 3$, $W<G$ be a noncompact simple subgroup, and $\Gamma < G$ be a lattice. Suppose that $k$ is a local field and $\bH$ is a connected $k$-algebraic group such that the pair consisting of $k$ and $\bfH$ is compatible with $G$. Finally, let $\rho: \Gamma \rightarrow \bH(k)$ be a homomorphism with unbounded, Zariski dense image. If there exist a $k$-rational faithful irreducible representation $\bfH \to \SL(V)$ on a $k$-vector space $V$ and a $W$-invariant measure $\nu$ on $(G\times \mathbb{P}(V))/\Gamma$ that pr

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Observe that the action of $G$ on $G/\Gamma$ is mixing by the Howe--Moore theorem. In particular, the action of each noncompact subgroup of $G$ is weakly mixing on $G/\Gamma$. This allows us to apply freely the discussion and results of \S\ref{ssec:AlgRep} regarding $T$-algebraic representations of $G$ for an arbitrary noncompact closed subgroup $T$ of $G$. Recall our setting from \S\ref{sec:FN}, and first consider $T=U'=U\cap W$, which is noncompact. Given an initial object in the category of $U^\prime$-algebraic representations of $G$, Theorem \ref{thm:BF} implies that there is a $k$-algebraic subgroup $\bfJ$ of $\bH$ such that this object is a measurable map $\Psi:G \rightarrow \bH/\bfJ(k

- `thm:BF` (proof_dependency; curated_source_check):  object in the category of $U^\prime$-algebraic representations of $G$, Theorem \ref{thm:BF} implies that there is a $k$-algebraic subgroup $\bfJ$ of $\bH$ such that this object is a measurable map $\Psi:G \rightarrow \bH/\bfJ(k)$ that is $(U
- `lem:initobj` (proof_dependency; curated_source_check): fJ)/\bfJ(k)$. Since $U^\prime$ is normal in $U$ and $U$ is normal in $P$, Lemma \ref{lem:initobj} implies that $\tau$ extends to a continuous homomorphism $\tau:P \to N_{\bfH}(\bfJ)/\bfJ(k)$ making the map $\Psi$ an initial object in the category 

Candidate references: `ssec:AlgRep` (candidate_proof_dependency); `sec:FN` (candidate_proof_dependency); `thm:BF` (candidate_proof_dependency); `lem:initobj` (candidate_proof_dependency); `external:BaderFurman` (external_reference).

### Remark 1.7 — `rem:SU`

Source lines 485–488.

We state the theorem for $G=\SO_0(n,1)$ for simplicity, but the same theorem holds, with practically the same proof, for every connected simple $\bbR$-rank one Lie group. In particular, there is an analogue of Theorem \ref{theorem:superrigiditydichotomy} for lattices in $\SU(n,1)$.

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `theorem:superrigiditydichotomy` (internal_reference).

### Proposition 3.1 — `prop:homogen`

Source lines 526–535.

For the real hyperbolic space $X_\Gamma = K \backslash G /\Gamma$ the following are equivalent: \item $X_\Gamma$ contains infinitely many maximal totally geodesic subspaces of dimension two or higher, \item for some $1<m<n$, there exists an infinite sequence $\{\mu_i\}$ of $W_m$-invariant, ergodic measures with proper support for which Haar measure on $G/\Gamma$ is a weak-$*$ limit of the $\mu_i$, \item for some $1<m<n$, there exists an infinite sequence of homogeneous, $W_m$-ergodic measures $\{\mu_i\}$ for which Haar measure on $G/\Gamma$ is 

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: As noted immediately after the statement of the proposition, it is enough to show that (1) and (3) are equivalent. We begin with the easier implication, namely that (3) implies (1). Fix $1<m<n$, and suppose that $\{\mu_i\}$ is a sequence of homogeneous, $W_m$-ergodic measures for which Haar measure on $G/\Gamma$ is a weak-$*$ limit. Let $\bar{\mu}_i=\pi_*\mu_i$ be the push-forward measures. By Lemma~\ref{lem:clarifying}\eqref{lem:clarifying1}, each measure $\bar{\mu}_i$ is supported on a closed totally geodesic subspace of $X_\Gamma$ and we let $Z_i$ be a maximal totally geodesic subspace of $X_\Gamma$ containing it. Since Haar measure of $G/\Gamma$ is by hypothesis a weak-$*$ limit of the s


Candidate references: `lem:clarifying` (candidate_proof_dependency); `lem:clarifying1` (candidate_proof_dependency); `lem:clarifying3` (candidate_proof_dependency); `thm:MS` (candidate_proof_dependency); `lem:clarifying2` (candidate_proof_dependency).

### Lemma 3.2 — `lem:clarifying`

Source lines 542–556.

Fix $1< m \le n$ and $X_\Gamma=K\backslash G/\Gamma$. Then the following hold. \item Let $S\le G$ be a closed subgroup containing $W_m$ and $h \in G$ be such that $Sh\Gamma/\Gamma \subset G/\Gamma$ is a closed $S$-orbit. Then the subspace $Z=\pi(Sh\Gamma/\Gamma)$ of $X_\Gamma$ is a closed totally geodesic $m^\prime$-dimensional subspace for some $m^\prime \geq m$ and, up to normalization, the $m^\prime$-volume of $Z$ is the push-forward of the corresponding homogeneous measure on $G / \Gamma$. \item Furthermore, under the assumption above, $m^\

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: We start by observing that for each $m^\prime$, the image of $W_{m^\prime}$ in $K\backslash G$, namely $K\backslash KW_{m^\prime}$, is an $m^\prime$-dimensional closed totally geodesic subspace. As $G$ acts transitively on the collection of $m^\prime$-dimensional closed totally geodesic subspaces of $K\backslash G$, any such subspace is of the form $K\backslash KW_{m^\prime}g$ for some $g\in G$. Since $N_{m^\prime}$ is contained in $KW_{m^\prime}$, every intermediate subgroup $W_{m^\prime} \le S_1 \le N_{m^\prime}$ has the property that $K\backslash KS_1g=K\backslash KW_{m^\prime}g$ is an $m^\prime$-dimensional closed totally geodesic subspace of $K\backslash G$, and the push-forward of the 


Candidate references: `lem:clarifying1` (candidate_proof_dependency); `lem:clarifying2` (candidate_proof_dependency); `lem:clarifying3` (candidate_proof_dependency); `external:Gelander-Levit` (external_reference).

### Theorem 3.3 — `thm:MS`

Source lines 626–629.

Suppose that $W< G$ is a closed connected semisimple subgroup that is generated by unipotent elements. Let $\{\mu_i\}$ be a sequence of homogeneous, $W$-ergodic probability measures on $G/\Gamma$ that weak-$*$ converges in the space of all finite Radon measures to a measure $\mu$. Then $\mu$ is a homogeneous, $W$-ergodic probability measure on $G/\Gamma$ and there exists a sequence $\{g_i\}$ in $G$ and a natural number $i_0$ such that, for each $i\geq i_0$, the measure $g_i\mu$ is a homogeneous, $W$-ergodic probability measure on $G/\Gamma$ who

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: We first claim that $\mu$ is not the zero measure. This is trivial if $G/\Gamma$ is compact. As this claim is conjugation invariant, we will assume as we may that $W=W_m$ for some $m>1$. Using \cite[Lem.~5.13]{FLMS} we fix a compact set $C_1$ in $X_\Gamma=K\backslash G/\Gamma$ that meets every closed totally geodesic subspace of dimension at least $2$. Then, choose a compact set $C_2$ that contains $C_1$ in its interior and consider its preimage in $G / \Gam$, i.e., the compact subset $F=\pi^{-1}(C_2)$ of $G/\Gamma$. Fix a 1-parameter unipotent subgroup $\{u_t\}$ in $ W$ and set $\epsilon=1/2$. Applying \cite[Thm.~6.1]{DaniMargulis} we find a compact subset $F^\prime\subset G/\Gamma$ such th


Candidate references: `lem:clarifying` (candidate_proof_dependency); `lem:clarifying1` (candidate_proof_dependency); `eq:DM` (candidate_proof_dependency); `external:FLMS` (external_reference); `external:DaniMargulis` (external_reference); `external:MS` (external_reference).

### Proposition 3.4 — `prop:liftingmeasures`

Source lines 718–722.

Under the hypotheses of Theorem \ref{thm:main:homogeneous} or Theorem \ref{thm:main}, there is a $k$-rational faithful irreducible representation $\bfH \to \SL(V)$ on a $k$-vector space $V$ and a $W$-invariant measure on $(G\times\bbP(V))/\Gamma$ that projects to Haar measure on $G/\Gamma$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: We retain all notation from prior subsections. We note that, as $\mathbf{H}$ is semisimple, each of its $k$-rational representation is into $\SL(V) < \GL(V)$. We will construct a faithful irreducible representation supporting a $W$ invariant measure as required. By Proposition \ref{prop:homogen} we can assume that we have $W$-invariant, $W$-homogeneous measures $\mu_i$ on $G/\Gamma$ converging to Haar measure. As in the proof of Proposition \ref{prop:homogen} we know that each $\mu_i$ is Haar measure on some $x_i(S_i/\Gamma_i)$, where $S_i \in \calS$, $x_i \in G$, and $\Gamma_i= S_i \cap \Gamma$ is Zariski dense in the closed subgroup $S_i$. Thus $S_i$ is an $\ell$-defined subgroup of $\bfG$

- `prop:homogen` (proof_dependency; curated_source_check): By Proposition \ref{prop:homogen} we can assume that we have $W$-invariant, $W$-homogeneous measures $\mu_i$ on $G/\Gamma$ converging to Haar measure.

Candidate references: `thm:main:homogeneous` (internal_reference); `thm:main` (internal_reference); `prop:homogen` (candidate_proof_dependency).

### Lemma 3.5 — `lemma:compatible`

Source lines 764–768.

We retain the setting of \S\ref{sec:FN} and let $\bfH$ be the adjoint group of $\bfG(k)$ for $k$ a local field and $\ell \to k$ a field embedding. Then the pair consisting of $k$ and $\bfH$ is compatible with $G$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: As mentioned above, we can and hence will assume that $k$ is archimedean and, passing to a finite extension, in fact assume that $k=\mathbb{C}$. In particular we get that $\bfH$ is split and we identify it with $\PO(n+1, \bbC)$. In what follows, we identify algebraic subgroups of $\bfH$ with their complex points. Assume we have a nontrivial algebraic subgroup $\bfJ<\PO(n+1, \bbC)$ and a continuous homomorphism $\tau:P\to \bfN/\bfJ$, where we denote $\bfN=\bfN_\bfH(\bfJ)$. Fix a nontrivial subgroup $U'<U$. Letting $\bfU,\bfU',\bfM,\bfA$ and $\bfP$ be the Zariski closures of the images of $U,U',M,A$ and $P$ in $\bfN/\bfJ$ correspondingly, we must show that $\bfU'=\bfU$. We can obviously assume


Candidate references: `sec:FN` (internal_reference); `external:BorelTits` (external_reference).

### Remark 3.6 — `rem:line:812`

Source lines 812–815.

Note that $\PO(n+1, \bbC)$ can also be viewed as an algebraic group over $\bbR$ and for $k=\bbR$ it is not compatible with $G = \SO_0(n,1)$. In the proof of arithmeticity, a Galois conjugate isomorphic to $\PO(n+1, \bbC)$ is naturally given the real Zariski topology, but for the purposes of proving Theorem \ref{theorem:superrigiditydichotomy} we may instead consider it in the complex Zariski topology.

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `theorem:superrigiditydichotomy` (internal_reference).

### Proposition 4.1 — `lemma:properness`

Source lines 833–837.

There exists a proper $k$-algebraic subgroup $\bfL < \bH$ and a measurable $W$-invariant, $\Gamma$-equivariant map $\phi: G \rightarrow \bH/ \bfL(k)$. We can also view $\phi$ as a measurable $\Gamma$-map from $W \backslash G$ to $\bH/\bfL(k)$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Via disintegration, the $W$-invariant measure $\nu$ on $(G\times \mathbb{P}(V))/\Gamma$ yields a $W$-invariant $\Gamma$-map \[ \tilde{\phi}: G \rightarrow \mathcal{P}(\mathbb{P}(V)), \] where $\mathcal{P}(\bbP(V))$ is the space of probability measures on $\bbP(V)$. By \cite[Cor.\ 3.2.12 and Thm.\ 3.2.4]{Zimmer}, the image of this map lies in a single $\bH(k)$-orbit that can be identified with $\bH(k)/\tilde{L}$ for $\wt{L}$ a compact extension of the $k$-points of a $k$-algebraic subgroup of $\bH(k)$. We thus get a $\Gamma$-equivariant map $W\backslash G\to \bH(k)/\tilde{L}$. We claim that $\tilde{L}$ is not compact. If it were, we could find an $\bH(k)$-invariant metric on $\bH(k)/\tilde{L}


Candidate references: `external:Zimmer` (external_reference); `external:Bader-Gelander` (external_reference).

### Remark 4.2 — `rem:line:855`

Source lines 855–858.

One can also prove the group $\tilde{L}$ is noncompact by showing nontriviality of the Lyapunov spectrum of the $W$-action on the bundle $(G\times \mathbb{P}(V))/\Gamma$ using \cite[Thm.\ V.5.15]{MargulisBook}. This is delicate when $\Gamma<G$ is nonuniform, relying on its weak cocompactness and integrability of the standard cocycle $\alpha: G \times G/\Gamma \rightarrow \Gamma$.

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `external:MargulisBook` (external_reference).

### Theorem 4.3 — `thm:BF`

Source lines 893–896.

Thm.\ 4.3 \cite{BaderFurman}

The collection of $T$-algebraic representations of $G$ forms a category. If the $T$-action on $G / \Gam$ is weakly mixing, then this category has an initial object and this initial object is a coset $T$-algebraic representation.

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `external:BaderFurman` (external_reference).

### Lemma 4.4 — `lem:initobj`

Source lines 903–906.

Assume that the action of $T$ on $G/\Gam$ is weakly mixing. Then initial objects for $T$ and $N_G(T)$ have the same map. Moreover, the initial object for $T$ and for the iterated normalizer $N_G(N_G(\dots (N_G(T))\dots))$ have the same map.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: For the first claim, the forward direction is the content of \cite[Thm.\ 4.6]{BaderFurman}. For the backward direction, if $\phi : G \to \bfH/\bfJ(k)$ is an initial object in the category of $N_G(T)$-algebraic representations with associated homomorphism $\tau$ from $T$ to $N_\bfH(\bfJ)/\bfJ(k)$, then $\phi$ and $\tau\vert_T$ form a $T$-algebraic representation and this representation must be initial by minimality. Indeed, otherwise another application of the forward direction contradicts minimality of $\bfJ$. The second claim follows immediately from the first.


Candidate references: `external:BaderFurman` (external_reference).

### Remark 5.2 — `rem:line:1050`

Source lines 1050–1053.

We also note that it is frequently the case that a $\pi_1$-injective surface in a $3$-manifold remains $\pi_1$-injective under Dehn filling (e.g., see \cite{CooperLong}). Therefore, infinitely many totally geodesic surfaces in $M$ may descend to $\pi_1$-injective surfaces in $N$. Our results say that these surfaces are very rarely totally geodesic.

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `external:CooperLong` (external_reference).

### Remark 5.6 — `rem:line:1086`

Source lines 1086–1089.

In part (3) of Corollary \ref{cor:kleinian}, we note that having infinitely many totally geodesic hypersurfaces gives a geometric characterization of arithmeticity in even dimensions. For $n \neq 3,7$ odd, there is a similar statement. In this case, every arithmetic hyperbolic manifold either contains maximal totally geodesic submanifold of codimension $1$ or $2$ (see \cite{Meyer}), hence having infinitely many such submanifolds again characterizes arithmeticity. There are arithmetic and nonarithmetic hyperbolic $3$-manifolds that contain no to

*Extractive statement; proof gist not independently reconstructed.*


Candidate references: `cor:kleinian` (internal_reference); `external:Meyer` (external_reference); `external:FLMS` (external_reference).

## External inputs and references

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Source lines 1138–1143.

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Candidate references: `section:proofs` (external_reference); `thm:BF` (external_reference); `lem:initobj` (external_reference); `theorem:superrigiditydichotomy` (external_reference).

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Source lines 1143–1148.

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Candidate references: `sec:Intro` (external_reference).

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Source lines 1148–1153.

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Candidate references: `lemma:properness` (external_reference).

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Source lines 1153–1158.

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Source lines 1158–1163.

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Source lines 1163–1169.

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Candidate references: `lemma:compatible` (external_reference).

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Source lines 1169–1175.

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Candidate references: `sec:Intro` (external_reference).

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Source lines 1175–1180.

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Candidate references: `sec:Intro` (external_reference).

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Source lines 1180–1186.

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Candidate references: `sec:FN` (external_reference).

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Source lines 1186–1191.

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Candidate references: `rem:line:1050` (external_reference).

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Source lines 1191–1198.

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Candidate references: `section:measuresandarithmeticity` (external_reference); `thm:MS` (external_reference).

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Source lines 1198–1203.

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Source lines 1203–1208.

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Candidate references: `section:measuresandarithmeticity` (external_reference).

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Source lines 1208–1213.

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Source lines 1213–1220.

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Source lines 1220–1225.

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Source lines 1225–1230.

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Candidate references: `sec:Intro` (external_reference).

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Source lines 1230–1236.

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Candidate references: `sec:Intro` (external_reference).

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Source lines 1236–1242.

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Source lines 1242–1248.

C.~Maclachlan and A.~W. Reid. \newblock {\em The arithmetic of hyperbolic 3-manifolds}, volume 219 of {\em Graduate Texts in Mathematics}. \newblock Springer-Verlag, 2003.


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A.~Marden. \newblock {\em Outer circles}. \newblock Cambridge University Press, 2007. \newblock An introduction to hyperbolic 3-manifolds.

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G.~Margulis and A.~Mohammadi. \newblock Arithmeticity of hyperbolic 3-manifolds containing infinitely many totally geodesic surfaces. \newblock Preprint: arXiv:1902.07267.


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Source lines 1260–1267.

G.~A. Margulis. \newblock Discrete groups of motions of manifolds of nonpositive curvature. \newblock In {\em Proceedings of the {I}nternational {C}ongress of {M}athematicians ({V}ancouver, {B}.{C}., 1974), {V}ol. 2}, pages 21--34. Canad. Math. Congress, Montreal, Que., 1975.


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G.~A. Margulis. \newblock Arithmeticity of the irreducible lattices in the semisimple groups of rank greater than {$1$}. \newblock {\em Invent. Math.}, 76(1):93--120, 1984.


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G.~A. Margulis. \newblock {\em Discrete subgroups of semisimple Lie groups}, volume~17 of {\em Ergebnisse der Mathematik und ihrer Grenzgebiete (3)}. \newblock Springer-Verlag, 1991.


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C.~T. McMullen, A.~Mohammadi, and H.~Oh. \newblock Geodesic planes in hyperbolic 3-manifolds. \newblock {\em Invent. Math.}, 209(2):425--461, 2017.


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C.~T. McMullen, A.~Mohammadi, and H.~Oh. \newblock Geodesic planes in the convex core of an acylindrical 3-manifold. \newblock Preprint: arXiv:1802.03853, 2018.


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D.~B. McReynolds and A.~W. Reid. \newblock The genus spectrum of hyperbolic 3--manifolds. \newblock {\em Math. Res. Lett.}, 21:169--185, 2014.


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J.~S. Meyer. \newblock Totally geodesic spectra of arithmetic hyperbolic spaces. \newblock {\em Trans. Amer. Math. Soc.}, 369(11):7549--7588, 2017.


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M.~Ratner. \newblock Raghunathan's topological conjecture and distributions of unipotent flows. \newblock {\em Duke Math. J.}, 63(1):235--280, 1991.


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A.~W. Reid. \newblock Arithmeticity of knot complements. \newblock {\em J. London Math. Soc. (2)}, 43(1):171--184, 1991.


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A.~W. Reid. \newblock Totally geodesic surfaces in hyperbolic {$3$}-manifolds. \newblock {\em Proc. Edinburgh Math. Soc. (2)}, 34(1):77--88, 1991.


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A.~Selberg. \newblock On discontinuous groups in higher-dimensional symmetric spaces. \newblock In {\em Contributions to function theory (internat. {C}olloq. {F}unction {T}heory, {B}ombay, 1960)}, pages 147--164. Tata Institute of Fundamental Research, Bombay, 1960.


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Source lines 1343–1348.

W.~Thurston. \newblock Geometry and topology of three-manifolds. \newblock http://http://library.msri.org/books/gt3m/.


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R.~J. Zimmer. \newblock Strong rigidity for ergodic actions of semisimple {L}ie groups. \newblock {\em Ann. of Math. (2)}, 112(3):511--529, 1980.


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Source lines 1358–1364.

R.~J. Zimmer. \newblock {\em Ergodic theory and semisimple groups}, volume~81 of {\em Monographs in Mathematics}. \newblock Birkh\"auser Verlag, Basel, 1984.


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## Coverage and limits

{"mode": "native_latex_index", "claims": 23, "sections": 5, "external_references": 41, "source_bibliography_entries": 41, "unresolved_bibliography": 0, "unresolved": 0, "numbering": "Shared native TeX counters; journal numbering must be checked separately. Source environment aliases normalized only in graph metadata."}

- Reference edges are not certified proof dependencies. Edge direction is prerequisite → consumer.
- Numbers reflect simple source counters; labels and line anchors are authoritative.
- Extractive statements and proof openings are marked; they are not reconstructed proof gists.
- Macros, included files and custom counters require an expanded source or annotated fallback.
- Primary model source is arXiv 1903.08467v5, 23 pages, dated 26 April 2020. Published Annals PDF is separately frozen: 25 article pages plus one JSTOR cover sheet. Figure pixels and external proof texts are not supplied to native-TeX packets. Full edition equivalence is not asserted.
- The dated adjudication adds a partial set of coordinator-audited proof edges. Missing dependencies and unchecked nodes remain unresolved. A sink is not necessarily a headline theorem.