Posted by: matheuscmss | September 6, 2018

Benoist’s minicourse “Arithmeticity of discrete groups”. Lecture II: Closedness of the L-orbits

As it was announced in the end of the first post of this series, we will discuss today the first half of the proof of the following result:

Theorem 1 Let {G:=SL(2p,\mathbb{R})}, {p\geq 2}, and {U := \left\{\left(\begin{array}{cc} I & B \\ 0 & I\end{array}\right)\in G: B\in M(p,\mathbb{R})\right\}}. Suppose that {\Gamma} is a discrete and Zariski dense subgroup of {G} such that {\Gamma\cap U} is cocompact. Then, {\Gamma} is commensurable to some {\mathbb{Z}}-form {G_{\mathbb{Z}}} of {G}.

Remark 1 This statement is originally due to Hee Oh, but the proof below is a particular case of Benoist–Miquel’s arguments. In particular, our subsequent discussions can be generalized to obtain the statement of Theorem 1 of the previous post in full generality.

Remark 2 Theorem 1 is not true without the higher rank assumption {p\geq 2} (i.e., {\textrm{rank}_{\mathbb{R}} G = 2p-1\geq 2}): indeed, {\Gamma = \left\langle\left(\begin{array}{cc} 1 & 3 \\ 0 & 1\end{array}\right), \left(\begin{array}{cc} 1 & 0 \\ 3 & 1\end{array}\right)\right\rangle} has infinite index in {SL(2,\mathbb{Z})}.

Our task is to construct a {\mathbb{Z}}-form {G_{\mathbb{Z}}} satisfying the conclusions of Theorem 1. This is not very easy because it must cover all possible cases of {\mathbb{Z}}-forms such as:

Example 1

  • {G_{\mathbb{Z}} = SL(2p,\mathbb{Z})};
  • {G_{\mathbb{Z}} = SL(2s,D_{\mathbb{Z}})} where {D_{\mathbb{Z}}} are integers in a division algebra over {\mathbb{Q}};
  • {G_{\mathbb{Z}} = SU(2p,\mathbb{Z}[\sqrt{2}])}”, i.e., {G_{\mathbb{Z}} = \left\{g\in SL(2p,\mathbb{Z}[\sqrt{2}]): g^{\sigma} = {}^Tg^{-1}\right\}} where {\sigma} is Galois conjugation (and {{}^Tg} is the transpose of {g}).

Before trying to construct adequate {\mathbb{Z}}-forms, let us make some preliminary reductions.

We denote by {P=N_G(U)} the parabolic subgroup normalizing of {U} in {G}: more concretely,

\displaystyle P = \left\{g\in G: g=\left(\begin{array}{cc} A & B \\ 0 & D\end{array}\right) \right\} = \{g\in G: g(\mathbb{W}) = \mathbb{W}\}

where {\mathbb{W}:=\mathbb{R}^p\times\{0\}\subset\mathbb{R}^{2p}}.

Next, we consider {U^- = \left\{\left(\begin{array}{cc} I & 0 \\ C & I\end{array}\right)\in G\right\}}. In the literature, {U^-} is called an opposite horospherical subgroup to {U}.

Since {\Gamma} is Zariski dense in {G}, there exists {\gamma_0 = \left(\begin{array}{cc} A_0 & B_0 \\ C_0 & D_0\end{array}\right)\in \Gamma} such that {\gamma_0(\mathbb{W})\oplus \mathbb{W} = \mathbb{R}^{2p}} (i.e., {\textrm{det}(C_0)\neq 0}). By taking a basis of {\mathbb{R}^{2p}} such that {\gamma_0(\mathbb{W}) = \{0\}\times\mathbb{R}^{p}}, we have that

\displaystyle \gamma_0 U\gamma_0^{-1} = U^- \ \ \ \ \ (1)

In particular, {U^-\cap\Gamma} is cocompact in {U^-} (thanks to the assumptions of Theorem 1).

Remark 3 This is the one of the few places in Benoist–Miquel argument where the Zariski denseness of {\Gamma} is used.

Remark 4 In general, the argument above works when {U} is reflexive, that is, {U} is conjugated to an opposite horospherical subgroup {U^-}.

We denote by {P^-=N_G(U^-) = \left\{g\in G: g=\left(\begin{array}{cc} A & 0 \\ C & D\end{array}\right)\in G \right\}} an opposite parabolic subgroup, and

\displaystyle L=P\cap P^- = \left\{g\in G: g=\left(\begin{array}{cc} A & 0 \\ 0 & D\end{array}\right)\in G \right\}

the common Levi subgroup of {P} and {P^-}. In particular, we have decompositions (in semi-direct products)

\displaystyle P=L\,U \quad \textrm{and} \quad P^- = L\,U^-

Let {\mathfrak{u}=\textrm{Lie}(U)} and {\mathfrak{u}^- = \textrm{Lie}(U^-)} be the Lie algebras of {U} and {U^-}. Note that {\Lambda:=\log(\Gamma\cap U)} and {\Lambda^-:=\log(\Gamma\cap U^-)} (resp.) are lattices in {\mathfrak{u}} and {\mathfrak{u}^-} (resp.). In other terms,

\displaystyle \Lambda\in X_{\mathfrak{u}} \textrm{ and } \Lambda^-\in X_{\mathfrak{u}^-}

where {X_{\ast}} is the space of lattices in {\ast\in\{\mathfrak{u}, \mathfrak{u}^-\}}.

Remark 5 Note that {\log g = g-\textrm{Id}_{2p\times 2p}} for all {g\in U} in the context of the example {G=SL(2p,\mathbb{R})} and {U := \left\{\left(\begin{array}{cc} I & B \\ 0 & I\end{array}\right)\in G: B\in M(p,\mathbb{R})\right\}}.

Observe that {L} is the intersection of the normalizers of {U} and {U^-}. Therefore, {L} acts on the spaces of lattices {X_{\mathfrak{u}}} and {X_{\mathfrak{u}^-}} via the adjoint map {\textrm{Ad}_L} (i.e., by conjugation).

As it turns out, the key step towards the proof of Theorem 1 consists in showing that the {\textrm{Ad}_{L}}-orbits of {\Lambda\in X_{\mathfrak{u}}} and {(\Lambda, \Lambda^-)\in X_{\mathfrak{u}}\times X_{\mathfrak{u}^-}} are closed. In other terms, the proof of Theorem 1 can be divided into two parts:

  • closedness of the {\textrm{Ad}_L}-orbits of {\Lambda\in X_{\mathfrak{u}}} and {(\Lambda, \Lambda^-)\in X_{\mathfrak{u}}\times X_{\mathfrak{u}^-}};
  • construction of the {\mathbb{Z}}-form {G_{\mathbb{Z}}} based on the closedness of the {\textrm{Ad}_L}-orbits above.

In the remainder of this post, we shall establish the closedness of relevant {\textrm{Ad}_L}-orbits. Then, the next post of this series will be dedicated to obtain an adequate {\mathbb{Z}}-form {G_{\mathbb{Z}}} (i.e., arithmeticity) from this closedness property.

Remark 6 Hee Oh’s original argument used Ratner’s theory for the semi-simple part of {L} to derive the desired closedness property. The drawback of this strategy is the fact that it doesn’t allow to treat some cases (such as  {G=SO(2,m)}), and, for this reason, Benoist and Miquel are forced to proceed along the lines below.

1. Closedness of the {\textrm{Ad}_L}-orbit of {\Lambda}

Consider Bruhat’s decomposition {\mathfrak{g} = \mathfrak{u}\oplus\mathfrak{l}\oplus\mathfrak{u}^-} (where {\mathfrak{l}=\textrm{Lie}(L)} is the Lie algebra of {L}) and the corresponding projection {\pi:\mathfrak{g}\rightarrow\mathfrak{u}}.

Given {g\in G}, set {M(g)=\pi \circ \textrm{Ad}_g \circ \pi\in \textrm{End}(\mathfrak{u})}, i.e.,

\displaystyle \textrm{Ad}_g = \left(\begin{array}{ccc} M(g) & \ast & \ast \\ \ast & \ast & \ast \\ \ast & \ast & \ast\end{array}\right),

and consider the Zariski open set

\displaystyle \Omega = U^- \, P = \left\{\left(\begin{array}{cc} A & B \\ C & D \end{array}\right)\in G: \textrm{det} A\neq 0\right\}.

Our first step towards the closedness of the {\textrm{Ad}_L}-orbit of {\Lambda} is to exploit the discreteness of {\Gamma} and the commutativity of {U} in order to get that the actions of the matrices {M(g)}, {g\in \Gamma\cap\Omega}, on the vectors {X} of the lattice {\Lambda} do not produce arbitrarily short vectors:

Proposition 2 The set {\{M(g) X: g\in\Gamma\cap\Omega, X\in\Lambda\}} is closed and discrete in {\mathfrak{u}}.

Proof: Let {g_n = v_n \ell_n u_n\in\Gamma} with {v_n\in U^-}, {\ell_n\in L}, {u_n\in U}, and {X_n\in \Lambda} such that {X_n' :=M(g_n)X_n \rightarrow X_{\infty}'\in \mathfrak{u}}.

Our task is to show that {X_{\infty}' = X_n'} for all {n} sufficiently large.

For this sake, note that a direct calculation reveals that {\textrm{Ad}_u\circ \pi = \pi = \pi\circ \textrm{Ad}_v} for all {u\in U} and {v\in U^-}. In particular, {X_n' = M(g_n) X_n= \textrm{Ad}(\ell_n)X_n}. Now, we use the cocompactness of {\Gamma\cap U^-} to write {v_n = \delta_n^{-1} v_n'} with {\delta_n\in \Gamma\cap U^-} and {v_n'\rightarrow v_{\infty}'} (modulo taking subsequences).

By definition, {\gamma_n:= \delta_n g_n \exp(X_n) g_n^{-1}\delta_n^{-1} = v_n'\ell_n u_n \exp(X_n) u_n^{-1} \ell_n^{-1} v_n'^{-1}\in \Gamma}. Since {U} is commutative, {u_n\exp(X_n)u_n^{-1} = \exp(X_n)} and hence

\displaystyle \Gamma\ni \gamma_n = v_n' \ell_n\exp(X_n)\ell_n^{-1} v_n'^{-1} = v_n' \exp(X_n') v_n'^{-1} \rightarrow v_{\infty}' \exp(X_{\infty}') v_{\infty}'^{-1}

as {n\rightarrow \infty}. Because {\Gamma} is discrete, it follows that {\textrm{Ad}_{v_n'}(X_n') = \textrm{Ad}_{v_{\infty}'}(X_{\infty}')} for all {n} sufficiently large. Therefore, {X_n'=\pi\circ \textrm{Ad}_{v_n'}(X_n') = \pi\circ\textrm{Ad}_{v_{\infty}'}(X_{\infty}') = X_{\infty}'} for all {n} sufficiently large. This completes the proof of the proposition. \Box

Remark 7 The fact that {U} is commutative plays a key role in the proof of this proposition.

Next, we shall combine this proposition with Mahler’s compactness criterion to study the set of determinants of the matrices {M(g)} for {g\in\Gamma}.

Proposition 3 Let {\Phi(g) = \textrm{det}_{\mathfrak{u}} M(g) = \textrm{det}(A)^{2p}} for {g=\left(\begin{array}{cc} A & B \\ C & D \end{array}\right)\in G}. Then, the set {\{\Phi(g): g\in \Gamma\}} is closed and discrete in {\mathbb{R}}.

Proof: Given {g_n\in\Gamma} such that {\Phi(g_n)\rightarrow\Phi_{\infty}}, we want to show that {\Phi(g_n) = \Phi_{\infty}} for all {n} sufficiently large.

By contradiction, let us assume that this is not the case. In particular, there is a subsequence {g_{n_k}} with {\Phi(g_{n_k})\neq 0}, i.e., {g_{n_k}\in\Gamma\cap\Omega}, and also {\Phi(g_{n_k})\neq \Phi_{\infty}} for all {k}. Note that, by definition, {\Phi(g_{n_k})} is the covolume of {M(g_{n_k})\Lambda}.

By Proposition 2, {M(g_{n_k})\Lambda} doesn’t have small vectors. Since these lattices also have bounded covolumes (because {\Phi(g_{n_k})\rightarrow\Phi_{\infty}}), we can invoke Mahler’s compactness criterion to extract a subsequence {g_{n_c}} such that {M(g_{n_c})\Lambda\rightarrow \Lambda_{\infty}}. In this setting, Proposition 2says that we must have {M(g_{n_c})\Lambda = \Lambda_{\infty}} for all {c} sufficiently large, so that {\Phi(g_{n_c}) = \Phi_{\infty}} for all {c} sufficiently large, a contradiction. \Box

Now, we shall modify {\Phi} to obtain a polynomial function {F} on {\mathfrak{u}}. For this sake, we recall the element {\gamma_0 = \left(\begin{array}{cc} A_0 & B_0 \\ C_0 & D_0 \end{array}\right)\in \Gamma} introduced in (1) (conjugating {U} and {U^-}). In this context,

\displaystyle F(X) := \Phi(\exp(X)\gamma_0) = (\textrm{det}(C_0))^{2p} (\textrm{det}(B))^{2p}

is a polynomial function of {X= \left(\begin{array}{cc} 0 & B \\ 0 & 0 \end{array}\right)\in \mathfrak{u}}. Moreover, an immediate consequence of Proposition 3 is:

Corollary 4 {F(\Lambda)} is a closed and discrete subset of {\mathbb{R}}.

The polynomial {F} is relevant to our discussion because it is intimately connected to the action of {\textrm{Ad}_L}:

Remark 8

  • on one hand, a straighforward calculation reveals that {F\circ \textrm{Ad}_{\ell}} is proportional to {F} for all {\ell\in L} (i.e., {F\circ \textrm{Ad}_{\ell} = \lambda_{\ell} F} for some [explicit] {\lambda_{\ell}\in \mathbb{R}}): see Lemma 3.12 of Benoist–Miquel paper;
  • on the other hand, some purely algebraic considerations show that {\textrm{Ad}_L} is the virtual stabilizer of the proportionality class of {F}, i.e., {\textrm{Ad}_L} is a finite-index subgroup of {\{\varphi\in \textrm{Aut}(\mathfrak{u}): F\circ \varphi \textrm{ is proportional to } F\}}: see Proposition 3.13 of Benoist–Miquel paper.

At this stage, we are ready to establish the closedness of the {\textrm{Ad}_L}-orbit of {\Lambda}:

Theorem 5 The {\textrm{Ad}_L}-orbit of {\Lambda} is closed in {X_{\mathfrak{u}}}.

Proof: Given {\ell_n\in L} such that {\textrm{Ad}_{\ell_n}\Lambda\rightarrow\Lambda_{\infty}}, we write {\textrm{Ad}_{\ell_n}\Lambda = \varphi_n(\Lambda_{\infty})} with {\varphi_n\in \textrm{Aut}(\mathfrak{u})} converging to the identity element {e\in\textrm{Aut}(\mathfrak{u})}.

Our task is reduce to show that {\varphi_{n}\in\textrm{Ad}_L} for all {n} sufficiently large. For this sake, it suffices to find {\varphi_{n_k}} stabilizing the proportionality class of the polynomial {F} (thanks to Remark 8).

In this direction, we take {X\in\Lambda_{\infty}}. By definition, {\varphi_n(X)\in \textrm{Ad}_{\ell_n}\Lambda}. Also, by Remark 8, we know that {F(\textrm{Ad}_{\ell_n}\Lambda) = \lambda_{\ell_n} F(\Lambda)} for some constant {\lambda_{\ell_n}} depending on the covolume {\Phi(\ell_n)} of {\textrm{Ad}_{\ell_n}\Lambda}. In particular, {F(\varphi_n X)\in\lambda_{\ell_n} F(\Lambda)}.

As it turns out, since the lattices {\textrm{Ad}_{\ell_n}\Lambda} converge to {\Lambda_{\infty}}, one can check that the quantities {\lambda_{\ell_n}} converge to some {\lambda_{\infty}\neq 0} related to the covolume of {\Lambda_{\infty}}. Moreover, {\lambda_{\ell_n} F(\Lambda)\ni F(\varphi_n X)\rightarrow F(X)} because {\varphi_n\rightarrow e}. Hence, we can apply Corollary 4 to deduce that {F(X)\in\lambda_{\infty} F(\Lambda)} and

\displaystyle F(\varphi_n X) = (\lambda_{\infty}/\lambda_{\ell_n}) F(X)

for all {n} sufficiently large depending on {X\in\Lambda}, say {n\geq n(X)}.

At this point, we observe that the degrees of the polynomials {F\circ\varphi_n - (\lambda_{\infty}/\lambda_{\ell_n}) F} are uniformly bounded and {\Lambda_{\infty}} is Zariski-dense in {\mathfrak{u}}. Thus, we can choose {n_0\in\mathbb{N}} such that

\displaystyle F(\varphi_n X) = (\lambda_{\infty}/\lambda_{\ell_n}) F(X)

for all {n\geq n_0} and {X\in\mathfrak{u}}.

In other terms, {\varphi_n\in\textrm{Aut}(\mathfrak{u})} stabilizes the proportionality class of {F} for all {n\geq n_0}. It follows from Remark 8 that {\varphi_n\in\textrm{Ad}_L} for all {n} sufficiently large. This completes the argument. \Box

2. Closedness of the {\textrm{Ad}_L}-orbit of {(\Lambda, \Lambda^-)}

The proof of the fact that the {\textrm{Ad}_L}-orbit of {(\Lambda,\Lambda')} is closed in {X_{\mathfrak{u}} \times X_{\mathfrak{u}^-}} follows the same ideas from the previous section: one introduces the polynomial {G(X,Y)=\Phi(\exp(X)\exp(Y))} for {(X,Y)\in \mathfrak{u}\times \mathfrak{u}^-}, one shows that {G(\Lambda\times\Lambda^-)} is closed and discrete in {\mathbb{R}}, and one exploits this information to get the desired conclusion.

In particular, our discussion of the first half of the proof of Theorem 1 is complete. Next time, we will see how this information can be used to derive the arithmeticity of {\Gamma}. We end this post with the following remark:

Remark 9 Roughly speaking, we covered Section 3 of Benoist–Miquel article (and the reader is invited to consult it for more details about all results mentioned above).Finally, a closer inspection of the arguments shows that the statements are true in greater generality provided {U} is reflexive and commutative (cf. Remarks 4 and 7).


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