1.1 Background

Let $H<G$ be Lie groups and $\Gamma $ be a lattice of G. Consider the quotient space $X=G/\Gamma $ , then the group multiplication induces a left group action of H on X. If X is non-compact, in many cases, it is particularly important to understand whether or not an H-orbit intersects a given compact subset of X. For example, let $H=\{u_t:t\in \mathbb {R}\}$ be a one-parameter unipotent subgroup of G. In their fundamental work [Reference Dani and Margulis8], Dani and Margulis proved that given any $x\in X$ , there exists a compact set $C\subset X$ depending on x such that $\{t\in \mathbb {R}:u_t x \in C\}$ is unbounded. In particular, if this unipotent subgroup satisfies a certain algebraic condition, then this compact set C can be chosen uniformly for all $x\in X$ . This result was crucially used in Ratner’s proof of uniform distribution of trajectories of unipotent subgroups on homogeneous spaces [Reference Ratner26]. However, let G be an algebraic group defined over $\mathbb {Q}$ and $\Gamma $ be an arithmetic subgroup of G. Let $H=T$ be a maximal $\mathbb {R}$ -split torus of G and $X=G/\Gamma $ . Tomanov and Weiss [Reference Tomanov and Weiss34] proved that there exists a fixed compact set C of X intersecting every T-orbit. This result enables them to classify all T-closed orbits on X. Motivated by these results, in this article, we study the following non-divergence property of group action on homogeneous spaces.

Now, we take X to be a homogeneous space of the form $G/\Gamma $ , where G is the connected component of the real points of a connected semisimple algebraic group $\boldsymbol {G}$ defined over $\mathbb {Q}$ and $\Gamma $ is an arithmetic lattice in G. Moreover, we take H to be a closed subgroup of G. The question that we wish to answer is the following.

Note that this property only depends on the G-conjugacy class of H. There are several different approaches toward this question depending on the group H.

1. (1) When H is generated by unipotent flows, one may make use of the polynomial nature of H-orbits. See [Reference Dani and Margulis8].

2. (2) When H is a finitely generated Zariski dense subgroup of a semisimple subgroup without compact factors, one can study the random walk generated by a set of generators of this subgroup. See [Reference Benoist and Quint3, Reference Eskin and Margulis12].

3. (3) When H is the real points of an $\mathbb {R}$ -diagonalizable algebraic torus, one of the most successful approaches relies on some tools from algebraic topology initiated by McMullen [Reference McMullen22], and refined by Solan and Tamam [Reference Solan28, Reference Solan and Tamam29] (cf. [Reference Levin, Shapira and Weiss20, Reference Shapira and Weiss27] for similar ideas). See [Reference Tomanov31, Reference Tomanov and Weiss34, Reference Weiss36, Reference Weiss37] for a different approach in this case.

These approaches indicate that certain algebraic obstruction is the only obstruction to uniform non-divergence, which we explain now. In the following, we assume that $\boldsymbol {H}$ is an $\mathbb {R}$ -algebraic subgroup of $\boldsymbol {G}$ and $H=\boldsymbol {H}(\mathbb {R})$ .

When $H=\{\mathrm {id} \}$ , the uniform non-divergence holds if and only if $G/\Gamma $ is compact, which holds if and only if $\boldsymbol {G}$ has no proper $\mathbb {Q}$ -parabolic subgroups. This is a hint that the general case may have to do with parabolic subgroups, and the formulation we find is linked with $\mathbb {Q}$ -quasiparabolic subgroups (see [Reference Bac and Thang1, Definition 1.1]).

Assume that there exists an absolutely irreducible $\mathbb {Q}$ -representation $\rho : \boldsymbol {G}\to \mathrm {GL}_n(\boldsymbol {V})$ and a non-zero rational vector $v\in \boldsymbol {V}(\mathbb {Q})$ such that:

• • $\boldsymbol {H}$ fixes v;

• • v is a highest weight vector.

In this case, we say that $\boldsymbol {H}$ is contained in a proper $\mathbb {Q}$ -quasiparabolic subgroup of $\boldsymbol {G}$ . Now, we claim that the H-action on $G/\Gamma $ is not uniformly non-divergent. This can be seen as follows. Let $\delta : \boldsymbol {G}_m \to \boldsymbol {G}$ be a $\mathbb {Q}$ -cocharacter that stabilizes the line spanned by v and satisfies $\lim _{t\to 0} \rho (\delta (t)) \cdot v=0$ . Let $G_v$ be the stabilizer of v in G. By Mahler’s criterion,

$$ \begin{align*} H \delta_t \Gamma /\Gamma \subset G_v \delta(t) \Gamma /\Gamma = \delta(t) G_v \Gamma /\Gamma \quad \text{diverges as } t \text{ tends to }0, \end{align*} $$

which implies the claim.

The approaches mentioned above are all able to show that up to G-conjugacy, the converse holds in each case (namely, assume that H is unipotent, finitely generated and Zariski dense in a connected semisimple group without compact factors, or an $\mathbb {R}$ -split algebraic torus). In the case of a torus, [Reference Solan28, Reference Solan and Tamam29] did not prove this exactly, but we will later explain how the result follows. In light of this, the following conjecture seems quite plausible.

Recall that an $\mathbb {R}$ -algebraic group $\boldsymbol {H}$ is said to be $\mathbb {R}$ -isotropic if and only if in a Levi decomposition, its reductive part is an almost direct product of an $\mathbb {R}$ -split torus and a semisimple group without compact factors.

To allow for a more general class of closed subgroups, one should allow in item (2) above a ‘compact modification’. Moreover, Conjecture 1.4 is wrong without the algebraicity assumption, see Question 1.10 below.

One may wish to compare this non-divergence criterion with the one obtained in [Reference Zhang39, Theorem 1.7] (cf. [Reference Eskin, Mozes and Shah13]) in a related but different context.

Unfortunately, combining approaches mentioned above in a naive way does not seem to yield a proof of the conjecture. Partial progress toward Conjecture 1.4 has been made in our previous work [Reference Zhang and Zhang38] and an application of this partial result has been found to obtain a finiteness result of totally geodesic submanifolds with bounded volume [Reference Zhang and Zhang38, Theorem 1.5]. Nevertheless, in this paper, we settle the conjecture in the affirmative when $\boldsymbol {H}$ is reductive (see Theorem 1.5).

We remark that Conjecture 1.4 is also settled when $\boldsymbol {G}$ is $\mathbb {Q}$ -split, since it reduces to the reductive case. To see this, let us assume that the unipotent radical of $\boldsymbol {H}$ is non-trivial and it suffices to explain why statement (2) implies statement (1) in the conjecture above. Let $\boldsymbol {H'}$ be the observable hull of $\boldsymbol {H}$ in $\boldsymbol {G}$ , that is, the smallest subgroup of $\boldsymbol {G}$ containing $\boldsymbol {H}$ with the property that

$$ \begin{align*} v \text{ is fixed by }\rho(\boldsymbol{H}) \implies v \text{ is fixed by }\rho(\boldsymbol{H'}) \end{align*} $$

for every finite-dimensional representation $\rho $ of $\boldsymbol {G}$ and every vector v. If $\boldsymbol {H'}$ is equal to $\boldsymbol {G}$ , then it is called epimorphic and by [Reference Weiss35, Theorem 9], the action of $H'$ is minimal and, hence, uniformly non-divergent. So let us assume that $\boldsymbol {H'}$ is not equal to $\boldsymbol {G}$ . Then, by appealing to Sukhanov’s theorem [Reference Sukhanov30] (see [Reference Bac and Thang1, Theorem B], the $(3)\iff (4)$ part) and note that our $\boldsymbol {G}$ is assumed to be $\mathbb {Q}$ -split, $\boldsymbol {H}'$ is contained in a proper $\mathbb {Q}$ -quasiparabolic subgroup of $\boldsymbol {G}$ , which violates statement (2) in the conjecture.

In this article, we also study the case where $\Gamma $ is not an arithmetic lattice of $\boldsymbol {G}$ (see Theorem 1.14). By the Margulis arithmeticity theorem, it is necessary that the algebraic group $\boldsymbol {G}$ is of real rank 1. It turns out that in this case, a maximal $\mathbb {R}$ -split torus of $\boldsymbol {G}$ plays a crucial role in the uniform non-divergence property of subgroup actions.

One may also consider a stronger set-intersection property replacing a compact set C by a ‘deformation retract’ of the whole space. For a small sample of such research, see [Reference Levin, Shapira and Weiss20, Reference McMullen22, Reference Shapira and Weiss27–Reference Solan and Tamam29].

1.2 Notation

We will use the following conventions throughout.

• • Capitalized boldface letters $\boldsymbol {A},\boldsymbol {B},\ldots, $ and so on are often reserved for algebraic groups. The corresponding uppercase Roman letters $A,B,\ldots $ denote their real points (if they are defined over $\mathbb {R}$ ). Additionally, lowercase Gothic letters $\mathfrak {a},\mathfrak {b},\ldots $ are used for their (real) Lie algebras.

• • For an algebraic group $\boldsymbol {A}$ , let $\mathrm {X}(\boldsymbol {A})$ denote the character group of $\boldsymbol {A}$ consisting of algebraic group morphisms from $\boldsymbol {A}$ to $\mathbb {C}^{\times }$ over $\mathbb {C}$ . Assume that $\boldsymbol {A}$ is contained in another algebraic group $\boldsymbol {B}$ . For every $g\in \boldsymbol {B}$ and $\chi \in \mathrm {X}(g^{-1}\boldsymbol {A}g)$ , we define a character $g(\chi )\in \mathrm {X}(\boldsymbol {A})$ by $g(\chi )(a):=\chi (g^{-1}a g)$ for every $a\in \boldsymbol {A}$ .

• • For an algebraic group $\boldsymbol {A}$ (respectively a Lie group A), we let $\boldsymbol {A}^{\circ }$ (respectively $A^{\circ }$ ) denote its identity component with respect to the Zariski topology (respectively analytic topology).

• • For two groups living in some ambient group, $\mathrm {N}_{A}(B)$ (respectively $\mathrm {Z}_{A}(B)$ ) denotes the normalizer (respectively centralizer) of B in A. If $\boldsymbol {A}$ and $\boldsymbol {B}$ are algebraic groups, we write $\boldsymbol {\mathrm {N}}_{\boldsymbol {A}}(\boldsymbol {B})$ or $\boldsymbol {\mathrm {Z}}_{\boldsymbol {A}}(\boldsymbol {B})$ . The notation $\mathrm {Z}(A)$ (respectively $\boldsymbol {\mathrm {Z}}(\boldsymbol {A})$ ) denotes the center of a group A (respectively an algebraic group $\boldsymbol {A}$ ).

This notation should cause little confusion since for a connected algebraic group $\boldsymbol {B}$ over $\mathbb {R}$ , the real points of $\boldsymbol {\mathrm {N}}_{\boldsymbol {A}}(\boldsymbol {B})$ or $\boldsymbol {\mathrm {Z}}_{\boldsymbol {A}}(\boldsymbol {B})$ coincide with $\mathrm {N}_{A}(B)$ or $\mathrm {Z}_{A}(B)$ , respectively. This is because the real points of a connected real algebraic group is Zariski dense. Same remarks apply to the center operation.

Let us now fix the playground.

• • Throughout the paper, we let $\boldsymbol {G}$ be a connected semisimple $\mathbb {Q}$ -algebraic group and $\Gamma \subset G$ be an arithmetic lattice.

In addition to this, we also fix the following data associated with $\boldsymbol {G}$ .

• • Let $\boldsymbol {T}$ be a maximal $\mathbb {R}$ -split torus of $\boldsymbol {G}$ containing a maximal $\mathbb {Q}$ -split torus $\boldsymbol {S}$ .

• • Fix an ordering of $\mathbb {Q}$ -simple roots. Let $r:=\mathrm {rank}_{\mathbb {Q}}(\boldsymbol {G})$ . Let $\boldsymbol {P}_1,\ldots ,\boldsymbol {P}_r$ be the standard maximal parabolic $\mathbb {Q}$ -subgroups of $\boldsymbol {G}$ and $\chi _1,\ldots ,\chi _r$ be the corresponding $\mathbb {Q}$ -fundamental weights. Each $\chi _i$ may be viewed as a character on $\boldsymbol {P}_i$ or $\boldsymbol {T}$ for $1\leq i\leq r$ (for details, see §2).

• • Let $\mathrm {W}(G)\cong \mathrm {N}_G(T)/\mathrm {Z}_G(T)$ be an $\mathbb {R}$ -Weyl group of $\boldsymbol {G}$ .

• • We fix a Cartan involution $\tau :\boldsymbol {G}\to \boldsymbol {G}$ such that $\tau (a)=a^{-1}$ for any $a\in \boldsymbol {T}$ .

Later, we will consider an algebraic group $\boldsymbol {M}$ and a maximal $\mathbb {R}$ -split torus $\boldsymbol {D}$ in $\boldsymbol {\mathrm {Z}_G(M)}$ . In this case, we let $\mathrm {W}(\mathrm {Z}_G(M))\cong \mathrm {N}_{\mathrm {Z}_G(M)}(D)/\mathrm {Z}_{\mathrm {Z}_G(M)}(D)$ be an $\mathbb {R}$ -Weyl group of $\boldsymbol {\mathrm {Z}}_{\boldsymbol {G}}(\boldsymbol {M})$ .

1.3 Main results

One of our main results, which confirms special cases of the Conjecture 1.4, is the following theorem.

Let us briefly mention some previous results in the setting of Theorem 1.5. Tomanov and Weiss [Reference Tomanov and Weiss34] proved that if H is any torus containing a maximal $\mathbb {R}$ -split torus of G, then the action of H on $G/\Gamma $ is uniformly non-divergent. In our earlier work [Reference Zhang and Zhang38], the uniform non-divergence property was established for that reductive group H with no compact factors satisfying that $\boldsymbol {\mathrm {Z}_G{(H)}}/\boldsymbol {\mathrm {Z}(H)}$ is $\mathbb {R}$ -anisotropic. Both of these above-mentioned results fall into the scope of Theorem 1.5.

Theorem 1.5 needs to assume $\boldsymbol {A}$ to be algebraic. We have the following, more general, Theorem 1.7 dropping the algebraicity assumption on $\boldsymbol {A}$ , whose item (2) implies that Theorem 1.5 does not hold without assuming $\boldsymbol {A}$ to be algebraic. This is because a set of linear functionals independent over $\mathbb {Z}$ is not necessarily independent over $\mathbb {R}$ (see Remark 1.8(2)). Indeed, Theorem 1.5 will be deduced from Theorem 1.7 in the next subsection.

Recall that we have fixed a Cartan involution $\tau :\boldsymbol {G}\to \boldsymbol {G}$ such that $\tau (a)=a^{-1}$ for any $a\in \boldsymbol {T}$ .

We observe that uniform non-divergence property of H as in Theorem 1.7 is preserved if the semisimple part of H is replaced by a Zariski dense finitely generated subgroup.

Proof. The direct implication is immediate since $H'\subset H$ . For the converse, the proof follows from [Reference Benoist and Quint3, Remark 5.2, Proposition 5.3]. Let us complete the details below.

Let $B \subset G/\Gamma $ be a bounded set. It suffices to show that there exists a possibly larger bounded set $B'$ of $G/\Gamma $ such that for every $x\in G/\Gamma $ ,

$$ \begin{align*} M\cdot x \cap B \neq \emptyset \implies \Lambda\cdot x \cap B' \neq \emptyset. \end{align*} $$

Without loss of generality, we assume that ${G}=\mathrm {SL}_n(\mathbb {R})$ and $\Gamma =\mathrm {SL}_n(\mathbb {Z})$ for some n. For $\varepsilon>0$ , let $f_{\varepsilon }: G/\Gamma \to [0,\infty ]$ be a proper function as in [Reference Benoist and Quint3, (5.1)] ( $H^{\text {nc}}$ there should be replaced by our M). By [Reference Benoist and Quint3, Remark 5.2] and Mahler’s criterion, we find $\varepsilon _0>0$ such that

$$ \begin{align*} M\cdot x \cap B \neq \emptyset \implies f_{\varepsilon_0}(x) < \infty. \end{align*} $$

Then, [Reference Benoist and Quint3, Proposition 5.3] implies that there exists some $C_0>1$ such that for $x\in G/\Gamma $ satisfying $M\cdot x\cap B\neq \emptyset $ , there exists $\gamma _x \in \Lambda $ such that

(1.1) $$ \begin{align} f_{\varepsilon_0}(\gamma_x\cdot x) < C_0. \end{align} $$

As $f_{\varepsilon _0}$ is a proper function,

$$ \begin{align*} B':= \{y\in G/\Gamma,\; f_{\varepsilon_0}(y)< C_0\} \end{align*} $$

is the desired bounded set.

In light of Theorem 1.7 and Conjecture 1.4, it is curious to ask the following question.

By Proposition 2.4, it is clear that item (2) implies item (1). Additionally, Theorem 1.7 gives an affirmative answer to the above question in the special case where $H=AM$ , with $\boldsymbol {M}$ semisimple and $A\subset \mathrm {Z}_G(M)$ .

We note the following immediate consequences of Theorem 1.7.

To make our investigation complete, we also study the uniform non-divergence property of subgroup action on quotient $G/\Gamma $ , where $\mathrm {rank}_{\mathbb {R}}(\boldsymbol {G})=1$ . By the Margulis arithmeticity theorem, in this case, $\Gamma $ could be a non-arithmetic lattice. The proof of Theorem 1.7 crucially uses the arithmetic structure of $\Gamma $ , which is not available when $\Gamma $ is non-arithmetic. Nevertheless, the reduction theory of Garland and Raghunathan [Reference Garland and Raghunathan15] allows us to establish the following theorem in the rank one case. The proof of this theorem will be given in §6.

Theorem 1.14 allows us to give an alternative proof of the following ‘compact core’ lemma [Reference Fisher, Lafont, Miller and Stover14, Lemma 5.13], which plays an essential role in the analysis of dynamics in non-compact rank one locally symmetric spaces [Reference Bader, Fisher, Miller and Stover2, Reference Fisher, Lafont, Miller and Stover14].

1.4 Proof of Theorem 1.5 assuming Theorem 1.7

(i) $\implies $ (ii).

Conjugating by some $g\in G$ , we assume that $\boldsymbol {D}\subset \boldsymbol {T}$ .

If the action of H on $G/\Gamma $ is not uniformly non-divergent, then by Theorem 1.7, there exist $w\in \mathrm {W}(G)$ , $w^{\prime }\in \mathrm {W}(\mathrm {Z}_G(M))$ , and a non-empty subset $I\subset \{1,\ldots , r\}$ such that $w^{-1}\boldsymbol {M} w\subset \bigcap _{i\in I} \boldsymbol {P}_{i} $ and $\{w^{\prime }w(\chi _{i}):i\in I\}$ are linearly dependent as linear functionals on $\mathrm {Lie}(A)$ . Since $\boldsymbol {A}$ is $\mathbb {R}$ -algebraic and $A=\boldsymbol {A}(\mathbb {R})$ , by Corollary 4.6, $\{w^{\prime }w(\chi _{i}):i\in I\}$ are linearly dependent as (algebraic) characters on $\boldsymbol {A}$ .

Let

$$ \begin{align*} \boldsymbol{H}':= w^{-1}w^{\prime-1}\boldsymbol{H}w' w,\; \boldsymbol{A}':= w^{-1}w^{\prime-1}\boldsymbol{A}w' w,\; \boldsymbol{M}':= w^{-1}w^{\prime-1}\boldsymbol{M}w' w. \end{align*} $$

Also, let $\boldsymbol {D}':=w^{-1}w^{\prime -1}\boldsymbol {D}w' w = w^{-1}\boldsymbol {D} w$ , which is a maximal $\mathbb {R}$ -split torus in $\boldsymbol {\mathrm {Z}}_{\boldsymbol {G}}(\boldsymbol {M}')$ . Then, $\boldsymbol {M}' \subset \boldsymbol {P}_i$ for every $i\in I$ , $\boldsymbol {A}' \subset \boldsymbol {D}' \subset \boldsymbol {T}$ , and $\{\chi _i\}_{i\in I}$ are linearly dependent as characters on $\boldsymbol {A}'$ . Since each $\chi _i$ is trivial restricted to the semisimple $\boldsymbol {M}'$ , we have that $\{ \chi _i\}_{i\in I}$ are linearly dependent as characters on $\boldsymbol {H}'$ . So we are done.

(ii) $\implies $ (iii).

Replacing $g^{-1}\boldsymbol {H}g$ by $\boldsymbol {H}$ , we assume that $g=\mathrm {id}$ in item (iii). Write $\boldsymbol {P}_I= \boldsymbol {L}_I \ltimes \boldsymbol {U}_I$ , where $\boldsymbol {L}_I $ is a Levi group defined over $\mathbb {Q}$ containing $\boldsymbol {T}$ and $\boldsymbol {U}_I$ is the unipotent radical of $\boldsymbol {P}_I$ . Then, there exists $u\in U_I$ such that $u\boldsymbol {H}u^{-1}\subset \boldsymbol {L}_I$ . Replacing $\boldsymbol {H}$ by $u\boldsymbol {H}u^{-1}$ , we may assume that $\boldsymbol {H}$ is contained in $\boldsymbol {L}_I$ .

By assumption, there are integers $\{l_i\}_{i\in I}$ such that $\prod _{i\in I} \chi _i^{l_i} = 1$ when restricted to $\boldsymbol {H}$ . Thus,

$$ \begin{align*} \boldsymbol{H} \subset \boldsymbol{L} :=\bigg\{l\in \boldsymbol{L}_I \,\bigg\vert\, \prod_{i\in I} \chi_i^{l_i}(l)=1\bigg\}^{\circ}. \end{align*} $$

Note that $\boldsymbol {L}$ is a connected reductive $\mathbb {Q}$ -subgroup. Hence, it suffices to prove that $\boldsymbol {\mathrm {Z}}_{\boldsymbol {G}}(\boldsymbol {L})/\boldsymbol {\mathrm {Z}}(\boldsymbol {L})$ is not $\mathbb {Q}$ -anisotropic, which holds if there is a $\mathbb {Q}$ -cocharacter whose image centralizes $\boldsymbol {L}$ and yet is not contained in $\boldsymbol {L}$ .

Indeed, $\{\chi _i\}_{i\in I}$ are linearly independent when restricted to $\boldsymbol {\mathrm {Z}}^{\text {spl}}(\boldsymbol {L}_I)$ , the $\mathbb {Q}$ -split part of the central torus of $\boldsymbol {L}_I$ . Hence, there exists a cocharacter $\delta : \mathbb {G}_m \to \boldsymbol {\mathrm {Z}}^{\text {spl}}(\boldsymbol {L}_I)$ , which is automatically defined over $\mathbb {Q}$ , such that

$$ \begin{align*} \prod_{i\in I} \chi_i^{l_i}\circ \delta \neq 1. \end{align*} $$

Thus, the image of $\delta $ centralizes $\boldsymbol {L}$ and is not contained in $\boldsymbol {L}$ . So we are done.

Note that in the case where $\boldsymbol {A}=\boldsymbol {D}$ , and so $\boldsymbol {H}=\boldsymbol {D}\boldsymbol {M}$ , if there exist $w\in \mathrm {W}(G)$ and a non-empty subset $I\subset \{1,\ldots ,r\}$ such that $w^{-1}\boldsymbol {H} w\subset \bigcup _{i\in I}\boldsymbol {P}_i$ , then $\{w(\chi _i):i\in I\}$ are linearly independent as characters on $\boldsymbol {D}$ . Otherwise, it would contradict the fact that $\boldsymbol {Z_G(H)}/\boldsymbol {Z(H)}$ is $\mathbb {R}$ -anisotropic.

1.5 Examples

Example 1.17. Let $\mathbb {K}$ be a totally real field extension of $\mathbb {Q}$ with $[\mathbb {K}:\mathbb {Q}]=m$ . Let $\boldsymbol {G}=\mathrm {Res}_{\mathbb {K}/\mathbb {Q}}(\boldsymbol {\mathrm {SL}}_n)$ , where $\mathrm {Res}$ denotes Weil’s restriction of scalar operator (see e.g. [Reference Platonov and Rapinchuk24, Ch. 2]). Denote $G=\boldsymbol {G}(\mathbb {R})$ , $\Gamma =\boldsymbol {G}(\mathbb {Z})$ , and

$$ \begin{align*} A=\bigg\{\mathrm{diag}(e^{t_1},\ldots,e^{t_n}):\sum_{i=1}^n t_i=0\bigg\}\subset \mathrm{SL}_n(\mathbb{R}). \end{align*} $$

Let $\Delta :\mathrm {SL}_n(\mathbb {R})\to G$ be the diagonal embedding. Then, the identity component of the real points of a maximal $\mathbb {Q}$ -split torus $\boldsymbol {S}$ of $\boldsymbol {G}$ is $S=\Delta (A)$ . Let W be the Weyl group of $\mathrm {SL}_n(\mathbb {R})$ defined by $W\cong N_{\mathrm {SL}_n(\mathbb {R})}(A)/Z_{\mathrm {SL}_n(\mathbb {R})}(A)\cong S_n$ , where $S_n$ is the usual symmetric group. Then, the Weyl group $\mathrm {W}(G)=W\times \cdots \times W$ , that is, $\mathrm {W}(G)$ , is the product of m copies of W.

The maximal $\mathbb {Q}$ -torus $\boldsymbol {T}$ containing $\boldsymbol {S}$ can be decomposed uniquely into its $\mathbb {Q}$ -anisotropic part and its $\mathbb {Q}$ -split part $\boldsymbol {S}$ . Thus, we have a projection from $\mathrm {Lie}(T)$ to $\mathrm {Lie}(S)$ . Note that $\mathbb {Q}$ -fundamental weights $\{\chi _1,\ldots ,\chi _r\}$ on $\mathrm {Lie}(T)$ factor through its projection to $\mathrm {Lie}(S)$ . Thus, if for some $w \in \mathrm {W}(G)$ the projection from $\mathrm {Ad}(w)\mathrm {Lie}(S)$ to $\mathrm {Lie}(S)$ is trivial, then $\{w(\chi _1),\ldots ,w(\chi _r)\}$ becomes trivial, and hence linearly dependent, on $\mathrm {Lie}(S)$ . However, if the projection is surjective, then $\{w(\chi _1),\ldots ,w(\chi _r)\}$ is linearly independent on $\mathrm {Lie}(S)$ .

Assume that $n=2$ and, hence, $\mathrm {rank}_{\mathbb {Q}}(\boldsymbol {G})=1$ . It is easy to verify that when m is even, there exists $w\in \mathrm {W}(G)$ such that $\mathrm {Ad}(w) \mathrm {Lie}(S)$ projects trivially to $\mathrm {Lie}(S)$ , where the projection is with respect to the Killing form on $\mathrm {Lie}(T)$ . Also, when m is odd, one can verify that for any $w\in \mathrm {W}(G)$ , $\mathrm {Ad}(w)\mathrm {Lie}(S)$ projects onto $\mathrm {Lie}(S)$ . By Corollary 1.13, we conclude that the action of S on $G/\Gamma $ is uniformly non-divergent if and only if m is an odd number (cf. [Reference Tomanov32, Reference Tomanov33] for a study on different problems in the similar setting).

When $n\geq 3$ and $m\geq 2$ , one can always find $w\in \mathrm {W}(G)$ such that $\{w(\chi _1),\ldots ,w(\chi _r)\}$ are linearly dependent on $\mathrm {Lie}(S)$ . We conclude that when $n\geq 3$ , the S action on $G/\Gamma $ is uniformly non-divergent if and only if $m=1$ or $\mathbb {K}=\mathbb {Q}$ .

For instance, when $n=3$ and $m=2$ , the projection $\mathrm {Lie}(T)\to \mathrm {Lie}(S)$ can be written as

$$ \begin{align*} \left( \begin{bmatrix} t_1 & & \\ & t_2& \\ && -t_1-t_2 \end{bmatrix}\, \begin{bmatrix} s_1 & & \\ & s_2& \\ && -s_1-s_2 \end{bmatrix}\right) \end{align*} $$

mapped to the diagonal embedding of

$$ \begin{align*} \begin{bmatrix} (t_1+s_1)/2 & & \\ & (t_2+s_2)/2& \\ && (-t_1-t_2-s_1-s_2)/2 \end{bmatrix}. \end{align*} $$

One can check that there does not exist $w\in \mathrm {W}(G)$ such that this projection becomes trivial on $\mathrm {Ad}(w)\mathrm {Lie}(S)$ .

However, if w denotes the Weyl element which fixes the first coordinate, but swaps the first and the last entry in the second coordinate, then the two fundamental weights become, after applying w, linearly dependent on $\mathrm {Lie}(S)$ . So, we can still conclude that the action is not uniformly non-divergent by Corollary 1.13.

Example 1.18. Let $\mathbb {K}$ be a totally real field extension of $\mathbb {Q}$ with $[\mathbb {K}:\mathbb {Q}]=2$ . Let $\boldsymbol {G}=\mathrm {Res}_{\mathbb {K}/\mathbb {Q}}(\boldsymbol {\mathrm {SL}}_4)$ . Let $\Gamma =\boldsymbol {G}(\mathbb {Z})$ . Then, $G=\mathrm {SL}_4(\mathbb {R})\times \mathrm {SL}_4(\mathbb {R})$ . Consider the semisimple $\mathbb {R}$ -algebraic group $\boldsymbol {M}\subset \boldsymbol {G}$ defined by

$$ \begin{align*} \boldsymbol{M}= \begin{bmatrix} 1 & \\ & \boldsymbol{\mathrm{SO}(2,1)} \end{bmatrix} \times \begin{bmatrix} 1 & & & \\ & 1 & &\\ & & 1 & \\ & & & 1 \end{bmatrix}, \end{align*} $$

where $\boldsymbol {\mathrm {SO}(2,1)}$ is viewed as a subgroup of $\boldsymbol {\mathrm {SL}}_3$ , which is embedded in the lower right block in $\boldsymbol {\mathrm {SL}}_4$ . Note that a maximal $\mathbb {R}$ -split torus $\boldsymbol {T}$ of $\boldsymbol {G}$ is the product of two maximal $\mathbb {R}$ -split tori $\boldsymbol {T}'$ of $\boldsymbol {\mathrm {SL}}_4$ , which are full diagonal tori. So, $\mathrm {Lie}(T)=\{(x_1,x_2):x_i\in \mathrm {Lie}(T')\}$ .

A maximal $\mathbb {R}$ -split torus $\boldsymbol {D}$ of $\boldsymbol {\mathrm {Z}_G(M)}$ has real points

$$ \begin{align*} D^{\circ}=\left\{\begin{bmatrix} e^{3t} & & &\\ & e^{-t} & &\\ & & e^{-t} &\\ & & & e^{-t} \end{bmatrix}: t\in \mathbb{R} \right\}\times (T')^{\circ}. \end{align*} $$

Let $\{\unicode{x3bb} _1,\unicode{x3bb} _2,\unicode{x3bb} _3\}$ be the set of all fundamental $\mathbb {Q}$ -weights in $\boldsymbol {\mathrm {SL}}_4$ with respect to the full diagonal subgroup of $\boldsymbol {\mathrm {SL}}_4$ . Then, $\{\chi _1,\chi _2,\chi _3\}$ is the set of all fundamental $\mathbb {Q}$ -weights in $\boldsymbol {G}$ , where for any $v=(x_1,x_2)\in \mathrm {Lie}(T)$ ,

For $1\leq i\leq 3$ , let $v_i\in \mathrm {Lie}(T)$ be such that $\chi _i(v)=(v_i|v)$ for any $v\in \mathrm {Lie}(T)$ , where $(\cdot |\cdot )$ is the Killing form on $\mathrm {Lie}(T)$ . Denote $\mathrm {W}(G)\cong \mathrm {N}_G(T)/\mathrm {Z}_G(T)$ and $\mathrm {W}(\mathrm {Z}_G(M))\cong \mathrm {N}_{\mathrm {Z}_G(M)}(D)/\mathrm {Z}_{\mathrm {Z}_G(M)}(D)$ . For any $w\in \mathrm {W}(G)$ , $w'\in \mathrm {W}(\mathrm {Z}_G(M))$ (implicitly, we are fixing a choice of representatives of these Weyl groups with the understanding that the choice would not affect the discussion below), and non-empty $I\subset \{1,2,3\}$ , define a linear subspace

$$ \begin{align*} U(w,w',I):=\mathrm{Span}_{\mathbb{R}}\{w'w(v_i):i\in I\}\subset\mathrm{Lie}(T). \end{align*} $$

Let

$$ \begin{align*} B:=\{U(w,w',I): w^{-1}\boldsymbol{M}w\subset \boldsymbol{P}_I, w'\in W(Z_G(M))\}, \end{align*} $$

then B is a finite collection of linear subspaces of $\mathrm {Lie}(T)$ . For a linear subspace $V\subset \mathrm {Lie}(D)$ satisfying $\pi _{U(w,w',I)}(V)=U(w,w',I)$ for any $U(w,w',I)\in B$ , let $A=\exp (V)\subset D$ (since $\dim D=4$ and there are at most three linear functionals in $U(w,w',I)$ , generic three-dimensional subspaces V would meet this requirement). Then, by Theorem 1.7 and Corollary 4.4, the action of $H=AM$ is uniformly non-divergent on $G/\Gamma $ . We also note that one can choose $V\subset \mathrm {Lie}(D)$ such that $A=\exp (V)$ is non-algebraic.

1.6 Overview of the proof strategy

Here, we indicate the strategy showing that statements (1) and (2) are equivalent in Theorem 1.7.

First, we assume statement (2) does not hold. Let H be as in Theorem 1.7. Denote $\pi :G \to G/\Gamma $ . Note that $G/\Gamma =\bigcup _{\eta>0} X_{\eta }$ such that $\{X_{\eta }:\eta>0\}$ is a nested collection of compact sets and, for any compact set $C\subset G/\Gamma $ , there exists $\eta _C>0$ with $C\subset X_{\eta _C}$ . We want to prove that every H-orbit intersects a fixed compact set $X_{\eta _0}$ for some suitably chosen $\eta _0>0$ .

Note that $H=AM$ , where M is a semisimple group and A is an $\mathbb {R}$ -diagonalizable subgroup of $Z_G(M)$ . Given $g\in G$ , for any $h\in H$ , as $h\pi (g)=am\pi (g)$ , it is important to understand the structure of the set of elements $g\in G$ such that M fails to bring $\pi (g)$ back to K.

For any $\eta>0$ , consider the subset

$$ \begin{align*} G^M_{\eta}:=\{g\in G: M\pi(g)\cap X_{\eta}\neq \emptyset\}. \end{align*} $$

Then, $G/\Gamma - \pi (G^M_{\eta })$ is the collection of elements in $G/\Gamma $ which cannot be brought back by M to the compact set $X_{\eta }$ .

By the work of Daw et al [Reference Daw, Gorodnik and Ullmo11], $G/\Gamma -\pi (G_{\eta }^M)=\bigcup _{\boldsymbol {P}} \Sigma ^M_{\eta ,\boldsymbol {P}}$ , where the union is taken over all $\mathbb {Q}$ -parabolic subgroup $\boldsymbol {P}$ satisfying certain conditions determined by M, and $\Sigma ^M_{\eta ,\boldsymbol {P}}$ is a certain generalized Siegel set (for a precise description, see the paragraph above Proposition 3.5). We wish to bring back any given element in $G/\Gamma -\pi (G_{\eta }^M)$ using subgroup A.

It turns out that the structure of $\bigcup _{\boldsymbol {P}} \Sigma ^M_{\eta ,\boldsymbol {P}}$ leads to an open cover of A and, thus, an open cover on $\mathrm {Lie}(A)\cong \mathbb {R}^n$ . However, this open cover is not good enough for us to apply a topological covering theorem of Euclidean spaces (Theorem 3.4), which essentially says that one cannot cover an Euclidean space using a family of open sets with ‘low multiplicity’. Thanks to a compactness criterion of Tomanov and Weiss (Proposition 2.3) and the fundamental result of Dani and Margulis on quantitative non-divergence of unipotent orbits (Theorem 3.10), we are able to construct a good cover of $\mathrm {Lie}(A)$ (Lemma 3.13) to which the topological covering theorem applies. To finish the argument, we assume that statement (1) holds and prove by contradiction. For suitably chosen $\eta _0>0$ , assume that there exists $x\in G/\Gamma -\pi (G_{\eta _0}^M)$ such that $Ax\cap X_{\eta _0}=\emptyset $ . By some linear algebra argument combined with the covering theorem, this will lead to a contradiction.

To see that statement (2) implies statement (1), we first find a suitable Cartan involution (see Proposition 3.14) and again make use of some simple linear algebra argument (see §4) to finish the proof. This part of proof is inspired by [Reference Tomanov and Weiss34, Example 1].

1.7 Outline of the paper

The rest of this article will be mainly devoted to proving Theorem 1.7. In §2, we will recall some basic notions and theorems from algebraic groups. In §3, we study properties of certain topological cover of the group ${G}$ and the $\mathbb {R}$ -diagonalizable group A, which is constructed using the work in [Reference Daw, Gorodnik and Ullmo10, Reference Daw, Gorodnik and Ullmo11]. Certain good properties of the cover are ensured by the fundamental result of Dani and Margulis on quantitative non-divergence of unipotent orbits [Reference Dani and Margulis8]. In §4, we prove some simple but useful linear algebra lemmas, which enable us to deal with the situation when A is not necessarily algebraic. In §5, we finish the proof of Theorem 1.7. There, we make use of a topological covering theorem, which is initially introduced by McMullen in [Reference McMullen22], and later developed by Solan and Tamam [Reference Solan28, Reference Solan and Tamam29]. In §6, we recall the reduction theory for real rank one quotient by Garland and Raghunathan, and give the proof of Theorem 1.14.