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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