2.1 Cartan projection

Recall the Cartan decomposition $G = KA^+ K$ , which means that for every $g\in G$ , there exists a unique element $\mu(g)\in \mathfrak a^+$ such that $g\in K \exp \mu(g) K$ . The map $G\to \mathfrak a^+$ given by $g\mapsto \mu(g)$ is called the Cartan projection. We have

(2.5) \begin{equation}\mu(g^{-1})={\rm i} (\mu(g))\quad\text{for all $g\in G$. }\end{equation}

Let $X = G/K$ be the associated Riemannian symmetric space and set $o = [K] \in X$ . Fix a K-invariant norm $\| \cdot \|$ on $\mathfrak g$ and a Riemannian distance d on X, induced from the Killing form on $\mathfrak g$ , so that

$$d(go, ho)=\|\mu(g^{-1}h)\|$$

for any $g, h \in G$ . For $p \in X$ and $R \gt 0$ , let B(p, R) denote the metric ball $ \{x \in X : d(x, p) \lt R\}$ .

Lemma 2.1 [Reference BenoistBen97, Lemma 4.6]. For any compact subset $Q \subset G$ , there exists a constant $C=C_Q\gt 0$ such that for all $g \in G$ ,

$$\sup_{q_1, q_2\in Q} \| \mu(q_1gq_2) -\mu(g)\| \lt C .$$

We then write

$$\mu_{\theta} := p_{\theta} \circ \mu :G\to \mathfrak a_\theta^+.$$

In view of (2.3), we have $\psi\circ \mu_\theta=\psi \circ \mu$ for all $\psi\in \mathfrak a_\theta^*$ .

2.2 The $\theta$ -boundary $\mathcal F_\theta$

We set

$$\mathcal F_\theta=G/P_{\theta} \quad\text{and}\quad \mathcal F=G/P.$$

Let

$$ \pi_\theta:\mathcal F\to \mathcal F_\theta$$

denote the canonical projection map given by $gP\mapsto gP_\theta$ , $g\in G$ . We set

(2.6) \begin{equation} \xi_\theta=[P_\theta] \in \mathcal F_{\theta}.\end{equation}

By the Iwasawa decomposition $G=KP=KAN$ , the subgroup K acts transitively on $\mathcal F_\theta$ , and hence $\mathcal F_\theta\simeq K/ M_\theta.$

We consider the following notion of convergence of a sequence in G (or in X) to an element of $\mathcal F_\theta$ . For a sequence $g_i \in G$ , we say $g_i \to \infty$ $\theta$ -regularly if $\min_{\alpha\in \theta} \alpha(\mu(g_i)) \to \infty$ as $i \to \infty$ .

• – $g_i \to \infty$ $\theta$ -regularly; and

• – $\lim_{i\to\infty} \kappa_{i}\xi_\theta= \xi$ in $\mathcal F_\theta$ for some $\kappa_{i}\in K$ such that $g_i\in \kappa_{i} A^+ K$ .

2.3 Points in general position

Let $P_\theta^+$ be the standard parabolic subgroup of G opposite to $P_\theta$ such that $P_\theta\cap P_\theta^+=L_\theta$ . We have $P_\theta^+ =w_0 P_{{\rm i}(\theta)}w_0^{-1}$ and hence

$$\mathcal F_{{\rm i}(\theta)}=G/P_\theta^+.$$

For $g\in G$ , we set

$$g_\theta^+ := gP_{\theta}\quad \text{and}\quad g_\theta^- := g w_0 P_{{\rm i}(\theta)};$$

as we fix $\theta$ in the entire paper, we write $g^{\pm}=g_\theta^{\pm}$ for simplicity when there is no room for confusion. Hence, for the identity $e\in G$ , $(e^+, e^-)=(P_\theta, P_\theta^+)=(\xi_\theta, w_0\xi_{{\rm i}(\theta)})$ . The G-orbit of $(e^+, e^-)$ is the unique open G-orbit in $G/P_\theta\times G/P_\theta^+$ under the diagonal G-action. We set

(2.7) \begin{equation} \mathcal F_{\theta}^{(2)}= \{(g_\theta^+, g_\theta^-): g\in G\}.\end{equation}

Two elements $\xi\in \mathcal F_\theta$ and $\eta\in \mathcal F_{{\rm i}(\theta)}$ are said to be in general position (or antipodal) if $(\xi, \eta)\in \mathcal F_\theta^{(2)}$ .

2.4 Busemann maps and Gromov products

The $\mathfrak a$ -valued Busemann map $\beta: \mathcal F\times G \times G \to\mathfrak a $ is defined as follows: for $\xi\in \mathcal F$ and $g, h\in G$ ,

$$ \beta_\xi ( g, h):=\sigma (g^{-1}, \xi)-\sigma(h^{-1}, \xi)$$

where $\sigma(g^{-1},\xi)\in \mathfrak a$ is the unique element such that $g^{-1}k \in K \exp (\sigma(g^{-1}, \xi)) N$ for any $k\in K$ with $\xi=kP$ . For $(\xi,g,h)\in \mathcal F_\theta\times G\times G$ , we define

(2.8) \begin{equation} \beta_{\xi}^\theta (g, h): =p_\theta ( \beta_{\xi_0} (g, h)) \quad\text{for $\xi_0\in \pi_\theta^{-1}(\xi)$};\end{equation}

this is well-defined independent of the choice of $\xi_0$ (see [Reference QuintQui02b, Lemma 6.1]). We also have $\|\beta_\xi^{\theta}(g, h)\|\le d(go, ho)$ for all $g, h\in G$ (see [Reference QuintQui02b, Lemma 8.9]). The Busemann map has the following properties: for all $\xi \in \mathcal F_{\theta}$ and $g_1, g_2, g_3\in G$ ,

$$\begin{aligned} \text{(Invariance)} \quad & \beta_{\xi}^{\theta}(g_1, g_2) = \beta_{g_3 \xi}^{\theta}(g_3 g_1, g_3 g_2);\\ \text{(Cocycle property)} \quad & \beta_{\xi}^{\theta}(g_1, g_2) = \beta_{\xi}^{\theta}(g_1, g_3) + \beta_{\xi}^{\theta}(g_3, g_2).\end{aligned}$$

For $p, q \in X$ and $\xi \in \mathcal F_{\theta}$ , we set $\beta_{\xi}^{\theta}(p, q) := \beta_{\xi}^{\theta}(g, h)$ where $g, h \in G$ satisfies $go = p$ and $ho=q$ . It is easy to check that this is well-defined.

Definition 2.3. For $(\xi, \eta) \in \mathcal F_{\theta}^{(2)}$ , we define the $\theta$ -Gromov product as

$$\mathcal G^{\theta}(\xi, \eta) = \tfrac{1}{2} ( \beta_{\xi}^{\theta}(e, g) + {\rm i} (\beta_{\eta}^{{\rm i}(\theta)}(e, g))),$$

where $g \in G$ satisfies $(g^+, g^-) = (\xi, \eta)$ . This does not depend on the choice of g (see [Reference Kim, Oh and WangKOW25b, Lemma 9.11]).

3.1 Growth indicators

We say that $\Gamma$ is $\theta$ -discrete if the restriction $\mu_\theta|_{\Gamma}:\Gamma \to \mathfrak a_\theta^+$ is proper. The $\theta$ -discreteness of $\Gamma$ implies that $\mu_\theta(\Gamma)$ is a closed discrete subset of $\mathfrak a_\theta^+$ . Indeed, $\Gamma$ is $\theta$ -discrete if and only if the counting measure on $\mu_\theta(\Gamma)$ weighted with multiplicity is a Radon measure on $\mathfrak a_\theta^+$ .

Definition 3.1 ( $\theta$ -growth [Reference QuintQui02a, Reference Kim, Oh and WangKOW25b]). For a $\theta$ -discrete subgroup $\Gamma\lt G$ , the $\theta$ -growth indicator $\psi_\Gamma^{\theta}:\mathfrak a_\theta\to [-\infty, \infty] $ is defined as follows: if $u \in \mathfrak a_\theta$ is non-zero,

(3.1) \begin{equation} \psi_\Gamma^{\theta}(u)=\|u\| \inf_{u\in \mathcal C}\tau^\theta_{\mathcal C},\end{equation}

where $\mathcal C\subset \mathfrak a_\theta$ ranges over all open cones containing u, and $\psi_{\Gamma}^{\theta}(0) = 0$ . Here, $-\infty\le \tau^{\theta}_{\mathcal C}\le \infty$ denotes the abscissa of convergence of the series ${\mathcal P}^{\theta}_{\mathcal C}(s)=\sum_{\gamma\in \Gamma, \mu_\theta(\gamma)\in \mathcal C} e^{-s\|\mu_\theta(\gamma)\|}$ . As mentioned, we simply write $\psi_{\Gamma} := \psi_{\Gamma}^{\Pi}$ .

This definition is independent of the choice of a norm on $\mathfrak a_\theta$ . It has been proved in [Reference QuintQui02a, Theorem 1.1.1] and [Reference Kim, Oh and WangKOW25b, Theorem 3.3] that

$$\psi_{\Gamma}^{\theta} \lt \infty, \quad \mathcal L_{\theta} = \{\psi_{\Gamma}^{\theta} \ge 0\}\quad \text{and } \quad \psi_{\Gamma}^{\theta} \gt 0\; \text{on}\operatorname{{\rm int}} \mathcal L_{\theta},$$

where $\operatorname{{\rm int}}\mathcal L_\theta$ denotes the interior of $\mathcal L_\theta$ in the relative topology of $\mathfrak a_{\theta}$ . Moreover, $\psi_\Gamma^\theta$ is upper semi-continuous and concave. When $\theta={\rm i}(\theta)$ , it follows from (2.5) that $\psi_\Gamma^\theta$ is i-invariant.

We say that a linear form $\psi$ is tangent to $\psi_\Gamma^\theta$ (at $u\in \mathfrak a_\theta-\{0\}$ ) if $\psi\ge \psi_\Gamma^\theta$ and $\psi(u)=\psi_\Gamma^\theta(u)$ . For any $u\in \operatorname{{\rm int}} \mathcal L_\theta$ , there exists $\psi\in \mathfrak a_\theta^*$ tangent to $\psi_\Gamma^\theta$ at u. Moreover, for any $\psi\in \mathfrak a_\theta^*$ tangent to $\psi_\Gamma^\theta$ at an interior direction of $\mathfrak a_\theta^+$ , there exists a $(\Gamma, \psi)$ -Patterson–Sullivan measure (see [Reference QuintQui02b, Theorem 8.4] and [Reference Kim, Oh and WangKOW25b, Proposition 5.9]).

For $\theta$ -Anosov subgroups, we have a more precise classification of Patterson–Sullivan measures in terms of tangent forms.

Definition 3.2 A finitely generated subgroup $\Gamma\lt G$ is $\theta$ -Anosov if there exists a constant $C \gt 1$ such that for all $\alpha \in \theta$ and $\gamma\in \Gamma$ , we have

(3.2) \begin{equation} \alpha(\mu(\gamma)) \ge C^{-1} |\gamma| - C,\end{equation}

where $|\cdot|$ denotes a fixed word metric on $\Gamma$ .

We recall that all $\theta$ -Anosov subgroups are assumed to be non-elementary in this paper. Define

(3.3) \begin{equation} \mathscr T_\Gamma=\{\psi\in \mathfrak a_\theta^*:\text{$\psi$ is tangent to $\psi_\Gamma^\theta$} \} .\end{equation}

By the duality lemma (see [Reference QuintQui03a, § 4] and [Reference SambarinoSam14, Lemma 4.8]), the following can be deduced from [Reference SambarinoSam24, Theorem A] (see [Reference Kim, Oh and WangKOW25b, Theorem 13.2]).

1. (1) We then have that $\psi_\Gamma^\theta$ is analytic on $\operatorname{{\rm int}} \mathcal L_{\theta}$ , strictly concave on $\mathcal L_{\theta}$ , and vertically tangent.Footnote ⁵

2. (2) For any $\psi \in \mathscr T_\Gamma$ , there exists a unique unit vector $u=u_\psi\in \operatorname{{\rm int}} \mathcal L_\theta$ such that $\psi(u)=\psi_\Gamma^\theta(u)$ . If $\Gamma$ is Zariski dense, the map $\psi\mapsto u_\psi$ is a bijection between $\mathscr T_\Gamma$ and $\{u\in \operatorname{{\rm int}} \mathcal L_{\theta}:\|u\|=1\}$ .

The following theorem has been proved in [Reference Lee and OhLO23, Theorem 1.3] for $\theta=\Pi$ and $\Gamma$ Zariski dense. The general case follows from [Reference SambarinoSam24, Theorem A] (for a more general discussion on conformal measures, see [Reference Kim, Oh and WangKOW25b, Corollary 1.13]).

Theorem 3.4. Let $\Gamma$ be a $\theta$ -Anosov subgroup. For any $\psi\in \mathscr T_\Gamma$ , there exists a unique $(\Gamma, \psi)$ -Patterson–Sullivan measure on $\Lambda_\theta$ , which we denote by $\nu_\psi=\nu_{\psi, \theta}$ . The map

$$\psi\mapsto \nu_\psi$$

is a surjection from $\mathscr T_\Gamma$ to the space of all $\Gamma$ -Patterson–Sullivan measures. If $\Gamma$ is Zariski dense, then the map $\psi \mapsto \nu_{\psi}$ is bijective. Moreover, if $\psi_1\ne \psi_2$ in $\mathscr T_\Gamma$ , then $\nu_{\psi_1}$ and $\nu_{\psi_2}$ are mutually singular to each other.

When $\psi\in \mathfrak a_\theta^*$ is positive on $\mathcal L_{\theta}-\{0\}$ , the abscissa of convergence of the $\psi$ -Poincaré series

$$s \mapsto \sum_{\gamma \in \Gamma} e^{-s \psi(\mu(\gamma))}$$

is a well-defined positive number, which we denote by $\delta_\psi$ (see [Reference Kim, Oh and WangKOW25b, Lemma 4.3]). Equivalently, $\delta_\psi$ is also given by (1.3).

Lemma 3.6 [Reference Kim, Oh and WangKOW25b, Lemma 4.5]. If $\psi\in \mathfrak a_\theta^*$ is positive on $\mathcal L_\theta-\{0\}$ , then

$$\delta_\psi \psi \in \mathscr T_\Gamma .$$

In particular, $\psi \in \mathscr T_{\Gamma}$ if and only if $\delta_{\psi} = 1$ .

Since $\mu(g^{-1}) = {\rm i}(\mu(g))$ for all $g \in G$ , we have that $\Gamma$ is $\theta$ -Anosov if and only if $\Gamma$ is $\theta \cup {\rm i}(\theta)$ -Anosov. If $\Gamma$ is $\theta$ -Anosov, then the canonical projection map $p:\Lambda_{\theta \cup {\rm i}(\theta)} \to \Lambda_{\theta}$ is a $\Gamma$ -equivariant homeomorphism. Recalling that $\mathfrak a_\theta^*$ can be considered as a subset of $\mathfrak a_{\theta\cup{\rm i}(\theta)}^*$ from (2.4), we recall the following, which will be of use.

3.2 Gromov hyperbolic space and quasi-isometry

We collect a few basic facts about $\theta$ -Anosov subgroups that will be used repeatedly.

Recall that a geodesic metric spaceFootnote ⁶ $(Z, d_Z)$ is called a Gromov hyperbolic space if it satisfies a uniformly thin-triangle property; that is, there exists $T \gt 0$ such that for any geodesic triangle in Z, one side of the triangle is contained in the T-neighborhood of the union of two other sides. We denote by $\partial Z$ the Gromov boundary of Z, which is the space of equivalence classes of geodesic rays in Z. For any $z_1\ne z_2\in Z\cup \partial Z$ , there may be more than one geodesic connecting $z_1$ and $z_2$ . By the notation $[z_1, z_2]$ , we mean ‘a’ geodesic in Z connecting $z_1$ to $z_2$ . For $w\in Z$ , the nearest-point projection of w to a geodesic $[z_1,z_2]$ is any point $w'\in [z_1, z_2]$ satisfying $d_Z(w,w')=\inf \{d_Z(w, z):z\in [z_1, z_2]\}$ . This is coarsely well-defined. We refer to [Reference Bridson and HaefligerBH99] for basics on Gromov hyperbolic spaces. Recall that d denotes the Riemannian distance function on $X=G/K$ .

1. (1) We have that $(\Gamma, \mathsf{d}_\Gamma)$ is a Gromov hyperbolic space.Footnote ⁷

2. (2) Then, $\mathcal L_\theta-\{0\}$ is contained in the relative interior of $\mathfrak a_\theta^+$ in $\mathfrak a_\theta$ .

3. (3) The orbit map $ (\Gamma,\mathsf{d}_\Gamma) \to (\Gamma o, d)$ given by $ \gamma \mapsto \gamma o $ is a quasi-isometric embedding, i.e., there exist $Q=Q_\Gamma\ge 1$ such that for all $\gamma_1, \gamma_2\in \Gamma$ ,

   $$ Q^{-1}\cdot \mathsf{d}_{\Gamma}(\gamma_1, \gamma_2) -Q \le d(\gamma_1 o, \gamma_2 o) \le Q \cdot \mathsf{d}_{\Gamma}(\gamma_1, \gamma_2) +Q .$$

4. (4) The orbit map $\Gamma \to \Gamma o$ uniquely extends to a $\Gamma$ -equivariant continuous map $f:\Gamma \cup \partial \Gamma \to \Gamma o \cup \Lambda_{\theta}$ and $f|_{\partial \Gamma}$ is a homeomorphism onto $\Lambda_\theta$ . For $\theta={\rm i}(\theta)$ , f maps two distinct points of $\partial \Gamma$ to points in general position.

We henceforth identity $\partial \Gamma$ and $\Lambda_\theta$ using f. For any $\xi\ne \eta \in \Gamma \cup \partial \Gamma$ , note that $f([\xi, \eta])=[\xi, \eta] o$ is the image of $[\xi, \eta]$ under the orbit map.

4.1 Coarse triangle inequalities for uniformly regular quasi-isometric embeddings

We set $\mathcal W_\theta$ to be the set of all Weyl elements that fix $\mathfrak a_\theta$ pointwise. We define a closed cone $\mathcal C$ in $\mathfrak a^+$ to be $\theta$ -admissible if the following three conditions hold:

1. (1) $\mathcal C$ is i-invariant, so ${\rm i}(\mathcal C)=\mathcal C$ ;

2. (2) $\mathcal W_\theta \cdot \mathcal C=\bigcup_{w\in \mathcal W_\theta} \operatorname{{\rm Ad}}_w \mathcal C $ is convex; and

3. (3) $\mathcal C\cap (\bigcup_{\alpha\in \theta} \ker \alpha) =\{0\}$ .

For a $\theta$ -admissible cone $\mathcal C$ , we say that an ordered pair $(x_1, x_2)$ of distinct points in X is $\mathcal C$ -regular if for $g_1, g_2 \in G$ such that $g_1 o = x_1$ and $g_2 o = x_2$ , we have

$$\mu(g_1^{-1} g_2) \in \mathcal C.$$

In this case, $x_2 = g_2 o \in g_1 K (\exp \mathcal C) o$ and hence for some $g \in g_1 K$ , $x_1 = g_1 o = go$ and $x_2 \in g (\exp \mathcal C) o$ . Note that if $(x_1, x_2)$ is $\mathcal C$ -regular, then $(x_2, x_1)$ is ${\rm i}(\mathcal C)$ -regular and hence $\mathcal C$ -regular by the i-invariance of $\mathcal C$ .

Theorem 4.1 will be deduced as a special case of the following theorem: we write $\mathcal C_\theta=p_\theta(\mathcal C)$ .

Theorem 4.3. Let Z be a geodesic metric space and $\mathcal C\subset \mathfrak a^+$ a $\theta$ -admissible cone. Let $f: Z\to X$ be a $\mathcal C$ -regular quasi-isometric embedding. ^{Footnote 8} If $\psi \in \mathfrak a_\theta^*$ is positive on $\mathcal C -\{0\}$ , then there exists a constant $D=D_\psi \ge 0$ such that for all $x_1,x_2,x_3\in f(Z)$ ,

\[ \mathsf{d}_\psi (x_1,x_3) \le \mathsf{d}_\psi (x_1,x_2) + \mathsf{d}_\psi (x_2,x_3) + D. \]

We continue to use notation $\operatorname{{\rm int}}\mathfrak a^+$ and $\operatorname{{\rm int}}\mathfrak a_\theta^+$ for relative interiors in the topology of $\mathfrak a$ and $\mathfrak a_\theta$ , respectively. Unless mentioned otherwise, for any proper cone $\mathcal C$ in $\mathfrak a^+$ (respectively, $\mathfrak a_\theta^+$ ), we denote by $\operatorname{{\rm int}} \mathcal C$ the interior of $\mathcal C$ in the relative topology of $\mathfrak a^+$ (respectively, $\mathfrak a_\theta^+$ ).

4.2 Proof of Theorem 4.1 assuming Theorem 4.3

Let $\psi\in \mathfrak a_\theta^*$ be such that $\psi\gt 0$ on $\mathcal L_\theta-\{0\}$ . We will construct a $\theta$ -admissible cone $\mathcal C\subset \mathfrak a^+$ such that $\mathcal L-\{0\}\subset \operatorname{{\rm int}} \mathcal C $ and $\psi$ is positive on $\mathcal C -\{0\}$ .

Since $\theta={\rm i}(\theta)$ by hypothesis, it follows from (2.5) that ${\rm i}|_{\mathfrak a_\theta}$ is an involution preserving $\mathcal L_\theta$ . Since $\psi$ is positive on $\mathcal L_\theta-\{0\}$ and $\mathcal L_{\theta} - \{0\} \subset \operatorname{{\rm int}} \mathfrak a_{\theta}^+$ (Theorem 3.8(2)), we can choose a closed convex cone $\mathcal C_0 \subset \mathrm{int}\, \mathfrak a_{\theta}^+ \cup \{0\}$ satisfying:

1. (1) $\mathcal L_{\theta}-\{0\} \subset \mathrm{int}\, \mathcal C_0$ ;

2. (2) ${\rm i}(\mathcal C_0)=\mathcal C_0$ ; and

3. (3) $\psi\gt 0$ on $\mathcal C_0 - \{0\}$ .

We observe that $\mathcal W_\theta \cdot \mathfrak a^+ $ is equal to the union of all Weyl chambers containing $\mathfrak a_\theta^+$ , and hence is a convex cone by [Reference Kapovich, Leeb and PortiKLP17, Lemma 2.6].

Let $\alpha\in \theta$ . It follows from (3.2) that $ \mathcal L\cap \ker \alpha =\{0\}$ . Since $\ker \alpha \cap \mathfrak a^+$ is contained in the boundary of $\mathcal W_\theta \cdot \mathfrak a^+$ , it follows from the convexity of $\mathcal W_\theta\cdot \mathfrak a^+$ that $\mathcal W_\theta \cdot \mathfrak a^+$ is contained in the half-space $\{\alpha\ge 0\}$ . Hence, both $\operatorname{{\rm int}} \mathfrak a_\theta^+$ and $\mathcal W_\theta \cdot \mathcal L -\{0\}$ are contained in the open half-space $\{\alpha\gt 0\}$ . Therefore, we can find a linear form $h_\alpha\in \mathfrak a^*$ such that

$$(\ker \alpha\cap \mathfrak a^+ ) -\{0\} \subset \{h_\alpha\lt 0\} \quad\text{and}\quad(\mathcal C_0\cup \mathcal W_\theta \cdot \mathcal L) -\{0\}\subset\{h_\alpha \gt 0\} .$$

Now set $H:=\bigcap_{\alpha\in \theta, w\in \mathcal W_\theta} \{h_\alpha\circ \operatorname{{\rm Ad}}_{w} \ge 0\}$ , which is clearly a $\mathcal W_\theta$ -invariant convex cone. By our choice of $h_\alpha$ , $\operatorname{{\rm int}} H$ contains $\mathcal L-\{0\}$ . Since $\theta={\rm i}(\theta)$ and hence $\mathcal W_\theta =\mathcal W_{{\rm i}(\theta)}$ , we have that ${\rm i}(H)$ is also a $\mathcal W_\theta$ -invariant convex cone whose interior contains $\mathcal L-\{0\}={\rm i}(\mathcal L)-\{0\}$ .

Define

\[ \mathcal C := p_\theta^{-1}(\mathcal C_0) \cap \mathfrak a^+\cap H \cap {\rm i}(H). \]

By construction, we have $ \mathcal C \cap (\bigcup_{\alpha \in \theta} \ker \alpha )= \{0\}$ . In particular, $\mathcal C$ is a proper closed cone in $\mathfrak a^+$ . Then, $\operatorname{{\rm int}} \mathcal C$ contains $\mathcal L-\{0\}$ . Since $\psi\gt 0$ on $\mathcal C_0$ and $\psi$ is $p_\theta$ -invariant, $\psi\gt 0$ on $\mathcal C$ . Since ${\rm i}(\mathcal C_0)=\mathcal C_0$ , we have ${\rm i}(\mathcal C)=\mathcal C$ . Using the fact that $p_\theta : \mathfrak a \to \mathfrak a_\theta$ is $\mathcal W_\theta$ -equivariant, we have that

$$\mathcal W_\theta \cdot \mathcal C = \mathcal W_\theta \cdot (p_\theta^{-1}(\mathcal C_0) \cap \mathfrak a^+) \cap H \cap {\rm i}(H) = p_\theta^{-1}(\mathcal C_0) \cap (\mathcal W_\theta \cdot \mathfrak a^+)\cap H\cap {\rm i}(H).$$

Since $p_{\theta}^{-1}(\mathcal C_0)$ , $\mathcal W_{\theta} \cdot \mathfrak a^+$ , H, and ${\rm i}(H)$ are convex, it follows that $\mathcal W_\theta\cdot \mathcal C $ is convex.

Therefore, $\mathcal C$ is $\theta$ -admissible. Since the orbit map $ (\Gamma,\mathsf{d}_\Gamma) \to (X, d)$ , $\gamma \mapsto \gamma o $ , is a quasi-isometric embedding by Theorem 3.8(3) and any open cone containing $\mathcal L$ contains $\mu(\Gamma)$ except for finitely many points, Theorem 4.1 follows from Theorem 4.3 once we prove that the orbit map is a $\mathcal C$ -regular embedding, as below.

The rest of this section is devoted to the proof of Theorem 4.3. We begin by recalling the following theorem; in particular, the metric space Z in Theorem 4.3 is always Gromov hyperbolic.

Theorem 4.5 [Reference Kapovich, Leeb and PortiKLP18, Theorem 1.4]. Let Z and $ f:Z\to X$ be as in Theorem 4.3. Then Z is Gromov hyperbolic. If Z is proper in addition, then f continuously extends to

$$ f : \bar Z \to X \cup \mathcal F_{\theta},$$

where $\bar Z = Z \cup \partial Z$ is the Gromov compactification and f maps two distinct points in $\partial Z$ to points in general position.

4.3 Diamonds

The notion of diamonds in X, due to Kapovich, Leeb, and Porti, plays a key role in the proof of Theorem 4.3. We fix

$$\text{a $\theta$-admissible cone $\mathcal C \subset \mathfrak a^+$ }$$

in the following. For a $\mathcal C$ -regular pair $(x_1,x_2)$ of points in X, define the $\mathcal C$ -cone with the tip at $x_1$ containing $x_2$ to be

\[ V_{\mathcal C}(x_1,x_2) = gM_{\theta}(\exp \mathcal C) o, \]

where $g=g(x_1, x_2) \in G$ is any element such that $x_1 = go$ and $x_2 \in g (\exp \mathcal C) o$ ; it is easy to check such g always exists and this definition is independent of the choice of g. For any $h\in G$ , we have $ hV_{\mathcal C}(x_1,x_2) = V_{\mathcal C}( hx_1, hx_2) $ .

Definition 4.6 (Diamonds) For a $\mathcal C$ -regular pair $(x_1,x_2)$ of points in X, the $\mathcal C$ -diamond with tips at $x_1$ and $x_2$ is defined as

\[ \Diamond_{\mathcal C}(x_1,x_2) = V_{\mathcal C}(x_1,x_2) \cap V_{\mathcal C}(x_2,x_1). \]

The $\mathcal C$ -cones and $\mathcal C$ -diamonds are convex subsets of X (see [Reference Kapovich, Leeb and PortiKLP17, Propositions 2.10 and 2.13]). Note also the equivariance property that for $h\in G$ , $ h \Diamond_{\mathcal C}(x_1,x_2) = \Diamond_{\mathcal C}(hx_1, hx_2) $ . It follows that for any $\mathcal C$ -regular pair $(x_1, x_2)$ , the diamond $\Diamond_{\mathcal C}(x_1,x_2)$ is of the form $h \Diamond_{\mathcal C}(o,ao)$ for some $a\in \exp \mathcal C$ and $h\in G$ . Therefore, the following example describes all diamonds up to translations.

Example 4.7. For $a \in \exp \mathcal C$ , the diamond $\Diamond_{\mathcal C}(o, a o)$ can be explicitly described as follows. First note that as we can take $g(o, ao) =e$ , we have $V_{\mathcal C}(o, a o) = M_{\theta}(\exp \mathcal C) o.$ Recalling that ${\rm i}=-\operatorname{{\rm Ad}}_{w_0}$ , we also have $a o = a w_0 o $ and $ o = (aw_0) (w_0^{-1} a^{-1} w_0) o \in aw_0 (\exp \mathcal C) o .$ So we can take $g(ao, o)=aw_0$ . Since $w_0 M_{\theta} w_0^{-1} = M_{\theta}$ and $w_0 (\exp \mathcal C) w_0^{-1} = \exp (-\mathcal C)$ , we have $V_{\mathcal C}(a o, o) = a w_0 M_{\theta} (\exp \mathcal C) o = a M_{\theta} \exp (- \mathcal C) o$ . Therefore,

$$ \Diamond_{\mathcal C}(o, a o) = M_{\theta} (\exp \mathcal C) o \cap a M_{\theta} \exp(-\mathcal C) o$$

(see Figure 4.7).

Figure 1.

A diamond drawn in $\mathfrak a$ .

Lemma 4.8 (Simultaneous nesting property) If $(x_1, x_2)$ is $\mathcal C$ -regular, then for any $x \in \Diamond_{\mathcal C}(x_1, x_2)$ , there exist $g \in G$ and $a \in \exp \mathcal C$ such that

$$x_1 = go, \quad x = ga o ,\quad x_2 \in g M_{\theta} (\exp \mathcal C) o \cap g a (M_{\theta} \exp \mathcal C) o.$$

Proof. We may first assume that $x_1 = o$ . By the $\mathcal C$ -regularity of the pair $(x_1, x_2)$ , we have $x_2 \in K (\exp \mathcal C) o$ . By multiplying an element of K to $x_1$ and $x_2$ , we may also assume that $x_1 = o$ and $x_2 \in (\exp \mathcal C) o$ , and hence $x \in M_{\theta} (\exp \mathcal C) o$ . We again multiply an element of $M_{\theta}$ to $x_1, x_2$ , and x if necessary so that we have $x_1 = o$ , $x_2 \in M_{\theta}(\exp \mathcal C) o$ , and $x = ao$ for some $a \in \exp \mathcal C$ . Then it suffices to show that $x_2 = a k a' o$ for some $k \in M_{\theta}$ and $a' \in \exp \mathcal C$ . We write $x_2 = m a_0 o$ for $m \in M_{\theta}$ and $a_0 \in \exp \mathcal C$ . We then have $ o \in V_{\mathcal C}(x_2, o) = m a_0 k_0 M_{\theta} (\exp \mathcal C) o$ for some $k_0 \in K$ . Hence, we have

$$ k_0^{-1} w_0^{-1} (w_0 a_0^{-1} w_0^{-1}) \in M_{\theta} (\exp \mathcal C) K. $$

This implies that $k_0^{-1} \in M_{\theta} w_0$ and hence $k_0 \in w_0 M_{\theta}$ . Since $a o \in V_{\mathcal C}(x_2, o)$ as well, we now have $a o \in m a_0 w_0 M_{\theta} (\exp \mathcal C) o$ . Then, for some $k \in K$ , we have

$$ a k \in m a_0 w_0 M_{\theta} \exp \mathcal C w_0^{-1} = m a_0 M_{\theta} \exp (- \mathcal C). $$

Hence, for some $a' \in \exp \mathcal C$ , we have

$$a k a' \in m a_0 M_{\theta}.$$

Looking at $G/P_{\theta}$ , we have $ k P_{\theta} = a^{-1} m a_0 M_{\theta} a'^{-1} P_{\theta} = P_{\theta}$ . Therefore, $k \in M_{\theta}$ . Since $x_2 = m a_0 o = aka' o$ , the claim follows.

Lemma 4.9. For any $\mathcal C$ -regular pair $(g_1 o, g_2 o)$ with $g_1, g_2\in G$ and for any $ go \in \Diamond_{\mathcal C}(g_1 o, g_2 o)$ with $g\in G$ ,

(4.2) \begin{equation} \mu_{\theta}(g_1^{-1}g) + \mu_{\theta}(g^{-1} g_2) = \mu_{\theta}(g_1^{-1}g_2).\end{equation}

We write

$$ a = a_1 a_2 \in A_{\theta}^+B_{\theta}^+ \quad \text{and }\quad a' = a_1' a_2' \in A_{\theta}^+ B_{\theta}^+. $$

We then have

$$aka' = a_2 k a_2' (a_1a_1').$$

Since $a_2 k a_2' \in S_{\theta}$ , we can write its Cartan decomposition $a_2 k a_2' = m b m' \in M_{\theta}B_{\theta}^+M_{\theta}$ , and hence

$$aka' = m (b a_1 a_1')m'.$$

Let $w \in \mathcal W$ be a Weyl element such that $b a_1 a_1' \in w A^+ w^{-1}$ . Since $\tilde{a} = \exp \mu(aka')$ , we must have $b a_1 a_1' = w \tilde{a} w^{-1}$ . Hence we have

$$g_2 o = aka' o = m w \tilde{a} o.$$

On the other hand, we also have $g_2 o = \tilde{k} \tilde{a} o,$ where $\tilde{k} \in M_{\theta}$ . This implies that $mw \in M_{\theta}$ ; and, in particular, $w \in M_{\theta}$ . Therefore $\tilde{a} = w^{-1} b a_1 a_1' w = (w^{-1} b w)(a_1 a_1') \in B_{\theta}A_{\theta}^+$ , which implies that

(4.3) \begin{equation} p_{\theta}(\log \tilde{a}) = \log a_1 + \log a_1' = p_{\theta}(\log a) + p_{\theta}(\log a').\end{equation}

Since

$$ \mu_{\theta}(g_1^{-1} g) = p_{\theta}(\log a), \quad \mu_{\theta}(g^{-1} g_2) = p_{\theta}(\log a'),\quad \text{and} \quad \mu_{\theta}(g_1^{-1}g_2) = p_{\theta}(\log \tilde{a}),$$

this finishes the proof.

As an immediate corollary, we obtain that $\mathsf{d}_{\psi}$ is additive on each diamond for any $\psi\in \mathfrak a_\theta^*$ .

Lemma 4.10 (Additivity of $\mathsf{d}_\psi$ on diamonds) Let $\psi \in \mathfrak a_{\theta}^*$ . For any $\mathcal C$ -regular pair $(x_1, x_2)$ and for any $x\in \Diamond_{\mathcal C}(x_1, x_2)$ , we have

$$\mathsf{d}_\psi(x_1,x)+ \mathsf{d}_{\psi}(x,x_2) = \mathsf{d}_\psi(x_1,x_2).$$

4.4 KLP Morse lemma

The Morse lemma due to Kapovich, Leeb, and Porti, which we will call the KLP Morse lemma, is stated as follows [Reference Kapovich, Leeb and PortiKLP18, Theorem 5.16, Corollary 5.28]: the image of an interval in $\mathbb R $ under a Q-quasi-isometry is called a Q-quasi-geodesic.

Theorem 4.11 (KLP Morse lemma) Let $\mathcal C,\mathcal C' \subset \mathfrak a^+$ be $\theta$ -admissible closed cones such that $\operatorname{{\rm int}} \mathcal C'$ contains $ \mathcal C-\{0\}$ (see Figure 2). Let $Q, B\ge 1$ be constants. There exists a constant $D_0 = D_0(\mathcal C,\mathcal C', Q, B) \ge 0$ so that the following holds: let $I \subset \mathbb R$ be an interval and $c : I \to X$ a $(\mathcal C, B)$ -regular Q-quasi-geodesic.

Figure 2.

The choice of $\mathcal C'$ viewed on the unit sphere of $\mathfrak a^+$ .

1. (1) If $I = [a, b]$ with $b - a \ge B$ , then the image c(I) is contained in the $D_0$ -neighborhood of the diamond $\Diamond_{\mathcal C'}(c(a),c(b))$ .

2. (2) If $I = [a, \infty)$ for some $a \in \mathbb R$ , then c(I) is contained in the $D_0$ -neighborhood of the cone $g M_{\theta}(\exp \mathcal C') o,$ where $g \in G$ is such that $g o = c(a)$ and $g^+ = c(\infty) \in \mathcal F_{\theta}$ .

3. (3) If $I = \mathbb R$ , then c(I) is contained in the $D_0$ -neighborhood of the parallel set $g M_{\theta} A o,$ where $g \in G$ is such that $g^{\pm} = c(\pm \infty) \in \mathcal F_{\theta}$ .

We note that the above applies for an interval in $\mathbb Z$ , as any $\mathcal C$ -regular quasi-isometric embedding $c:I\cap \mathbb Z \to X$ can be extended to a $\mathcal C$ -regular quasi-geodesic $I\to X$ simply by setting $c(t) := c( \lfloor t \rfloor)$ where $\lfloor t \rfloor$ is the largest integer not bigger than t.

As an application of Theorem 4.11, we obtain the following.

Corollary 4.12 Given $\mathcal C, \mathcal C', Q, B\ge 1$ as in Theorem 4.11 and $\psi\in \mathfrak a_\theta^*$ , there exists a constant $D_1 = D_1(\mathcal C,\mathcal C', Q, B,\psi) \ge 0$ so that the following holds: let $I \subset \mathbb R$ be an interval and $c : I \to X$ a $(\mathcal C, B)$ -regular Q-quasi-geodesic. Then, for all $a \le t \le b$ in I, we have

(4.4) \begin{equation} |\mathsf{d}_\psi( c(a), c(b)) - \mathsf{d}_\psi( c(a), c(t)) - \mathsf{d}_\psi(c(t), c(b))| \le D_1. \end{equation}

Proof. Suppose that $b - a \ge B$ . Since c is $(\mathcal C, B)$ -regular, the pair (c(a), c(b)) is $\mathcal C$ -regular. Applying Theorem 4.11, we obtain that the image of $ c : [a,b] \to X$ lies in the $D_0$ -neighborhood of $\Diamond_{\mathcal C'}(c(a), c(b))$ . For each $a \lt t \lt b$ , choose $x_t\in \Diamond_{\mathcal C'}(c(a), c(b))$ so that $d(x_t, c(t)) \le D_0$ . Hence, by Lemma 4.10,

(4.5) \begin{equation} \mathsf{d}_\psi(c(a),x_t)+ \mathsf{d}_{\psi}(x_t, c(b)) = \mathsf{d}_\psi(c(a),c(b)). \end{equation}

For each $a \le t \le b$ , write $c(t)=g_t o$ and $x_t=h_t o$ for $g_t, h_t\in G$ . We then have $\| \mu(h_t^{-1} g_t)\|\le D_0$ . By applying Lemma 2.1 to a compact subset $\{g \in G : \|\mu(g)\|\le D_0\}$ , we have, for all $a\lt t\lt b$ ,

$$|\mathsf{d}_{\psi}(c(a), x_t) - \mathsf{d}_{\psi}(c(a) , c(t)) | = | \psi( \mu(g_a^{-1} h_t) - \mu(g_a^{-1} g_t )) | \le C,$$

where $C\gt 0$ is a uniform constant depending only on $\psi$ and $D_0$ . Similarly, we have $|\mathsf{d}_\psi (x_t, c(b)) - \mathsf{d}_\psi (c(t), c(b)) |\le C $ . By (4.5), this implies that

$$ |\mathsf{d}_\psi( c(a), c(b)) - \mathsf{d}_\psi( c(a), c(t)) - \mathsf{d}_\psi(c(t), c(b))| \le 2 C. $$

Setting $D_1= 2C + 3 \|\psi\|(QB + Q)$ , where $\|\psi\|$ is the operator norm of $\psi$ , we have shown that (4.4) holds whenever $b-a\ge B$ . If $b - a \lt B$ , then the image of c([a, b]) has diameter smaller than $Q(b-a) + Q \lt QB + Q$ . Then,

$$\mathsf{d}_{\psi}(c(t_1), c(t_2)) \lt \|\psi\| (QB + Q)$$

for all $t_1, t_2 \in [a, b]$ , and hence the left-hand side of (4.4) is bounded above by $ 3\|\psi\| (QB + Q)\le D_1$ . This completes the proof.

We are ready to give the following proof.

4.5 Proof of Theorem 4.3

Let $f : Z \to X$ be as in Theorem 4.3. Let $\psi \in \mathfrak a_{\theta}^*$ be such that $\psi \gt 0$ on $ \mathcal C - \{0\}$ . Choose a $\theta$ -admissible cone $\mathcal C' \subset \mathfrak a^+$ such that $\operatorname{{\rm int}} \mathcal C'$ contains $\mathcal C-\{0\}$ and such that $\psi\gt 0$ on $\mathcal C'-\{0\}$ . Let $x_1, x_2, x_3 \in f(Z)$ be a triple of distinct points. We choose $z_1, z_2, z_3 \in Z$ such that $x_i = f(z_i)$ for $i = 1,2, 3$ . Choose geodesics $c_1$ and $c_2$ in Z connecting $z_1$ to $z_2$ and $z_2$ to $z_3$ , respectively. By Theorem 4.5, $(Z, d_Z)$ is Gromov hyperbolic. We denote by z the nearest-point projection of $z_2$ to a geodesic segment connecting $z_1$ and $z_3$ . Then, by the Gromov hyperbolicity of $(Z, d_Z)$ , there exists a uniform constant $\delta \gt 0$ so that the $\delta$ -neighborhood of z intersects both geodesics $c_1$ and $c_2$ . We choose two points $y_1 \in c_1$ and $y_2 \in c_2$ that are $\delta$ -close to z. We concatenate the segment of $c_1$ connecting $z_1$ and $y_1$ , a geodesic connecting $y_1$ and $y_2$ , and the segment of $c_2$ connecting $y_2$ and $z_3$ , and denote the concatenated path by c. We can parameterize $c : [0, b] \to Z$ so that c is a q-quasi-geodesic for some $b \gt 0$ and uniform $q \ge 1$ by the Gromov hyperbolicity of $(Z, d_Z)$ and the choice of $y_1$ and $y_2$ .

Since f is a $\mathcal C$ -regular quasi-isometric embedding, so is $f \circ c$ . Hence, we obtain

(4.6) \begin{equation} \mathsf{d}_\psi(x_1, x_3) \le \mathsf{d}_\psi(x_1, f(y_1)) + \mathsf{d}_\psi( f(y_1) ,x_3) +D_1,\end{equation}

where $D_1$ is the constant given by Corollary 4.12. Applying Corollary 4.12 to the restriction of $f \circ c$ to the interval $[c^{-1}(y_1), b]$ again, we have

(4.7) \begin{equation} \mathsf{d}_\psi( f(y_1) ,x_3) \le \mathsf{d}_\psi(f(y_1), f(y_2)) + \mathsf{d}_{\psi}( f(y_2), x_3) + D_1.\end{equation}

Since $d_Z(y_1, y_2) \le 2 \delta$ , combining (4.6) and (4.7) yields

(4.8) \begin{equation} \mathsf{d}_{\psi}(x_1, x_3) \le \mathsf{d}_\psi(x_1, f(y_1)) + \mathsf{d}_{\psi}( f(y_2), x_3) + D_1',\end{equation}

where $D_1':= \sup \{\mathsf{d}_\psi(f(w_1), f(w_2)) :{d_Z(w_1, w_2) \le 2 \delta} \} + 2 D_1\lt \infty $ .

Since f is $\mathcal C$ -regular and $\psi \gt 0$ on $\mathcal C - \{0\}$ , there exists $D_2 \gt 0$ such that

(4.9) \begin{equation} \mathsf{d}_{\psi}(f(w_1), f(w_2)) \ge - D_2\end{equation}

for all $w_1, w_2 \in Z$ ; indeed, if f is $(\mathcal C, B)$ -regular for some $B\ge0$ , then $\mathsf{d}_{\psi}(f(w_1), f(w_2)) \ge 0$ whenever $d_Z(w_1, w_2)\ge B$ , and $\sup \{|\mathsf{d}_\psi (f(w_1), f(w_2))| :d_Z(w_1, w_2)\lt B\} $ is bounded by a uniform constant depending only on B, the quasi-isometry constant of f, and $\|\psi\|$ .

Hence, applying (4.9) and Corollary 4.12 to $f ( c_1 )$ , we have

(4.10) \begin{align} \mathsf{d}_\psi (x_1, f(y_1)) & \le \mathsf{d}_\psi (x_1, f(y_1)) +\mathsf{d}_\psi(f(y_1),x_2) + D_2\nonumber\\ & \le \mathsf{d}_\psi(x_1,x_2) +D_1 + D_2.\end{align}

Similarly, we also obtain

(4.11) \begin{equation} \mathsf{d}_\psi(f(y_2),x_3) \le \mathsf{d}_\psi(x_2,x_3)+D_1 + D_2. \end{equation}

Combining (4.8), (4.10), and (4.11), we obtain

$$ \mathsf{d}_{\psi}(x_1, x_3) \le \mathsf{d}_{\psi}(x_1, x_2) + \mathsf{d}_{\psi}(x_2, x_3) + D_1' + 2( D_1 + D_2). $$

This completes the proof of Theorem 4.3.

We state the following consequence of the KLP Morse lemma applied to Anosov subgroups.

1. (1) If $\xi, \eta \in \Gamma$ and $\mathsf{d}_{\Gamma}(\xi, \eta) \ge B$ , then $f([\xi, \eta])$ is contained in the $D_0$ -neighborhood of the diamond $\Diamond_{\mathcal C}(f(\xi), f(\eta))$ .

2. (2) If $\xi \in \Gamma$ and $\eta \in \partial \Gamma$ , then $f([\xi, \eta])$ is contained in the $D_0$ -neighborhood of $g M_{\theta} (\exp \mathcal C) o$ , where $g \in G$ is such that $g o = \xi$ and $g P_{\theta} = f(\eta)$ .

3. (3) If $\xi, \eta \in \partial \Gamma$ , then $f([\xi, \eta])$ is contained in the $D_0$ -neighborhood of $g M_{\theta} A o$ , where $g \in G$ is such that $g P_{\theta} = f(\xi)$ and $g w_0 P_{\theta} = f(\eta)$ .

Moreover, the cone $\mathcal C$ can be taken arbitrarily close to $\mathcal L$ as long as its interior contains $\mathcal L - \{0\}$ .

Now note that any geodesic $[\xi,\eta]$ in $(\Gamma, \mathsf{d}_\Gamma)$ can be written as $[\xi, \eta]=\{\gamma_i:i\in I\}$ for an interval I in $\mathbb Z$ , and $\iota: i\mapsto \gamma_i$ is an isometry between I and $[\xi, \eta]$ . Since $c:=f \circ \iota $ is a $(\mathcal C, B)$ -regular Q-quasi-geodesic, we can apply Theorem 4.11, which implies the above claims (1)–(3), where the cone $\mathcal C$ in the statement is given by $\mathcal C'$ in this proof. Note from the proof of Theorem 4.1 that the cone $\mathcal C'$ can be taken arbitrarily close to the limit cone $\mathcal L$ of $\Gamma$ as long as $\operatorname{{\rm int}} \mathcal C'$ contains $\mathcal L - \{0\}$ .