Non-concentration property of Patterson–Sullivan measures for Anosov subgroups

Abstract

Let G be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma <G$ with respect to a parabolic subgroup $P_\theta $, we prove that any $\Gamma $-Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $. More generally, we prove that for a Zariski dense $\theta $-transverse subgroup $\Gamma <G$, any $(\Gamma , \psi )$-Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $, provided the $\psi $-Poincaré series of $\Gamma $ diverges at its abscissa of convergence. In particular, our result also applies to relatively Anosov subgroups.

1. Introduction

Let G be a connected semisimple real algebraic group and $\mathfrak g=\operatorname {Lie} G$ . Let A be a maximal real split torus of G and set $\mathfrak {a} = \operatorname {Lie} A$ . Fix a positive Weyl chamber $\mathfrak a^+<\mathfrak a$ and a maximal compact subgroup $K< G$ such that the Cartan decomposition $G=K (\exp \mathfrak a^+) K$ holds. We denote by $\mu (g)\in \mathfrak a$ the Cartan projection of $g\in G$ , that is, the unique element of $\mathfrak a^+$ such that $g\in K \exp (\mu (g))K$ . Let $\Pi $ be the set of all simple roots for $(\mathfrak g, \mathfrak a^+)$ and fix a non-empty subset $\theta \subset \Pi $ . Let $P_\theta $ be the standard parabolic subgroup of G corresponding to $\theta $ and set

$$ \begin{align*}\mathcal F_\theta=G/P_\theta.\end{align*} $$

Let $\Gamma <G$ be a Zariski dense discrete subgroup. Denote by $\Lambda _\theta \subset \mathcal F_{\theta }$ the limit set of $\Gamma $ , which is the unique $\Gamma $ -minimal subset of $\mathcal F_\theta $ [1]. Let $\mathfrak {a}_\theta =\bigcap _{\alpha \in \Pi - \theta } \ker \alpha $ . For a linear form $\psi \in \mathfrak a_\theta ^*$ , a Borel probability measure $\nu $ on $\mathcal {F}_\theta $ is called a $(\Gamma , \psi )$ -conformal measure if

$$ \begin{align*}\frac{d \gamma_*\nu}{d\nu}(\xi)=e^{\psi(\beta_\xi^\theta(e,\gamma))} \quad \text{for all}\ \gamma \in \Gamma\ \text{and}\ \xi \in \mathcal{F}_\theta, \end{align*} $$

where $\gamma _* \nu (B) = \nu (\gamma ^{-1}B)$ for any Borel subset $B\subset \mathcal F_\theta $ and $\beta _\xi ^\theta $ denotes the $\mathfrak a_\theta $ -valued Busemann map defined in equation (2.2). By a $\Gamma $ -Patterson–Sullivan measure on $\mathcal F_\theta $ , we mean a $(\Gamma ,\psi )$ -conformal measure supported on $\Lambda _\theta $ for some $\psi \in \mathfrak a_{\theta }^*$ .

Patterson–Sullivan measures play a fundamental role in the study of geometry and dynamics for $\Gamma $ -actions. For G of rank one, they were constructed by Patterson and Sullivan for any non-elementary discrete subgroup $\Gamma $ of G [17, 22], and hence the name. Their construction was generalized by Quint for any Zariski dense subgroup of a semisimple real algebraic group [19].

A finitely generated subgroup $\Gamma <G$ is called a $\theta $ -Anosov subgroup if there exist $C_1, C_2>0$ such that for all $\gamma \in \Gamma $ and $\alpha \in \theta $ ,

$$ \begin{align*}\alpha(\mu(\gamma))\ge C_1|\gamma| -C_2,\end{align*} $$

where $|\gamma |$ denotes the word length of $\gamma $ with respect to a fixed finite generating set of $\Gamma $ . A $\theta $ -Anosov subgroup is necessarily a word hyperbolic group [11, Theorem 1.5, Corollary 1.6]. The notion of Anosov subgroups was first introduced by Labourie for surface groups [15], and was extended to general word hyperbolic groups by Guichard and Wienhard [8]. Several equivalent characterizations have been established, one of which is the above definition (see [7, 9–11]). Anosov subgroups are regarded as natural generalizations of convex cocompact subgroups of rank one groups, and include the images of Hitchin representations and of maximal representations as well as higher rank Schottky subgroups; see [12, 23].

A special case of our main theorem is the following non-concentration property of Patterson–Sullivan measures for $\theta $ -Anosov subgroups.

Theorem 1.1. Let $\Gamma <G$ be a Zariski dense $\theta $ -Anosov subgroup. For any $\Gamma $ -Patterson–Sullivan measure $\nu $ on $\mathcal F_\theta $ , we have

$$ \begin{align*}\nu(S)=0\end{align*} $$

for any proper subvariety S of $ \mathcal F_\theta $ .

Remark 1.2. This was proved by Flaminio and Spatzier [6] for $G=\operatorname {SO}(n,1)$ , $n\ge 2$ , and by Edwards, Lee, and Oh [5] when $\theta =\Pi $ and the opposition involution of G is trivial in equation (2.1).

Indeed, we work with a more general class of discrete subgroups, called $\theta $ -transverse subgroups. Denote by $\operatorname {i}$ the opposition involution of G (see equation (2.1)).

Definition 1.3. A discrete subgroup $\Gamma < G$ is called $\theta $ -transverse if:

• • it is $\theta $ -regular, that is, $ \liminf _{\gamma \in \Gamma } \alpha (\mu ({\gamma }))=\infty $ for all $\alpha \in \theta $ ; and

• • it is $\theta $ -antipodal, that is, any two distinct $\xi , \eta \in \Lambda _{\theta \cup \operatorname {i}(\theta )}$ are in general position.

Since $\operatorname {i} (\mu ( g))=\mu (g^{-1}) $ for all $g\in G$ , it follows that $\Gamma $ is $\theta $ -transverse if and only if $\Gamma $ is $\operatorname {i}(\theta )$ -transverse. The class of $\theta $ -transverse subgroups includes all discrete subgroups of rank one Lie groups, $\theta $ -Anosov subgroups, and relatively $\theta $ -Anosov subgroups.

Let $p_{\theta } : \mathfrak a \to \mathfrak a_{\theta }$ be the projection which is invariant under all Weyl elements fixing $\mathfrak a_{\theta }$ pointwise, and set $\mu _\theta =p_\theta \circ \mu $ . A linear form $\psi \in \mathfrak a_{\theta }^*$ is said to be $(\Gamma , \theta )$ -proper if the composition $\psi \circ \mu _\theta : \Gamma \to [-\varepsilon , \infty )$ is a proper map for some $\varepsilon> 0$ . The following is our main theorem from which Theorem 1.1 is deduced by applying Selberg’s lemma [21].

Theorem 1.4. Let $\Gamma <G$ be a Zariski dense virtually torsion-free $\theta $ -transverse subgroup. Let $\psi \in \mathfrak a_{\theta }^*$ be a $(\Gamma , \theta )$ -proper linear form such that $\sum _{\gamma \in \Gamma } e^{-\psi (\mu _{\theta }(\gamma ))} = \infty $ . For any $(\Gamma , \psi )$ -Patterson–Sullivan measure $\nu $ on $\mathcal F_{\theta }$ , we have

$$ \begin{align*}\nu(S)=0\end{align*} $$

for any proper subvariety S of $ \mathcal F_\theta $ .

For a $\theta $ -Anosov $\Gamma $ , the existence of a $(\Gamma , \psi )$ -Patterson–Sullivan measure implies that $\psi $ is $(\Gamma , \theta )$ -proper and $\sum _{\gamma \in \Gamma } e^{-\psi (\mu _{\theta }(\gamma ))} = \infty $ [16, 20]. Therefore, Theorem 1.1 is a special case of Theorem 1.4.

The following is added to the proof. The growth indicator $\psi _\Gamma ^\theta $ is a higher rank version of the classical critical exponent of $\Gamma $ [13, 18]. For a Zariski dense $\theta $ -transverse subgroup and a $(\Gamma , \theta )$ -proper $\psi \in \mathfrak a_{\theta }^*$ , the existence of a $(\Gamma , \psi )$ -conformal measure implies that $\psi $ is bounded from below by $\psi _\Gamma ^\theta $ ([19, Theorem 8.1] for $\theta = \Pi $ , [13, Theorem 1.4] for a general $\theta $ ). When $\Gamma $ is relatively $\theta $ -Anosov and $\psi $ is tangent to $\psi _\Gamma ^\theta $ , the abscissa of convergence of the series $s \mapsto \sum _{\gamma \in \Gamma } e^{-s\psi (\mu _{\theta }(\gamma ))}$ is equal to $1$ , and a recent work [3, Theorem 1.1] shows that $\sum _{\gamma \in \Gamma } e^{-\psi (\mu _{\theta }(\gamma ))} = \infty $ . Therefore, Theorem 1.4 also applies in this setting.

2. Ergodic properties of Patterson–Sullivan measures

Let G be a connected semisimple real algebraic group. Let $P < G$ be a minimal parabolic subgroup with a fixed Langlands decomposition $P=MAN, $ where A is a maximal real split torus of G, M is a maximal compact subgroup commuting with A, and N is the unipotent radical of P. We fix a positive Weyl chamber $\mathfrak {a}^+\subset \mathfrak {a} = \operatorname {Lie}A$ so that $\log N$ consists of positive root subspaces. Recall that $K< G$ denotes a maximal compact subgroup such that the Cartan decomposition $G=K(\exp \mathfrak a^+) K$ holds and denote by $\mu :G\to \mathfrak {a}^+$ the Cartan projection, that is, $\mu (g) \in \mathfrak {a}^+$ is the unique element such that $g\in K\exp (\mu (g))K$ for $g\in G$ . Let $w_0\in K$ be an element of the normalizer of A such that $\operatorname {Ad}_{w_0}\mathfrak a^+= -\mathfrak a^+$ . The opposition involution $\operatorname {i}:\mathfrak a \to \mathfrak a$ is defined by

(2.1) $$ \begin{align} \operatorname{i} (u)= -\operatorname{Ad}_{w_0} (u) \quad\text{for}\ u\in \mathfrak{a}. \end{align} $$

Note that $\mu (g^{-1})=\operatorname {i} (\mu (g))$ for all $g\in G$ .

Let $\Pi $ denote the set of all simple roots for $(\mathfrak g, \mathfrak a^+)$ . Fix a non-empty subset $\theta \subset \Pi $ . Let $ P_\theta ^-$ and $P_\theta ^+$ be a pair of opposite standard parabolic subgroups of G corresponding to $\theta $ ; here, $P_\theta :=P_\theta ^-$ is chosen to contain P. We set

$$ \begin{align*}\mathcal F_\theta^-=G/P_{\theta}^-\quad \text{and} \quad \mathcal F_\theta^{+}=G/P_{\theta}^{+} .\end{align*} $$

We also write $\mathcal F_\theta =\mathcal F_\theta ^-$ for simplicity. We set $P=P_\Pi $ and $\mathcal F=\mathcal F_\Pi $ . Since $P_\theta ^+$ is conjugate to $P_{\operatorname {i}(\theta )}$ , we have $\mathcal F_{\operatorname {i}(\theta )}=\mathcal F_\theta ^+$ . We say $\xi \in \mathcal F_\theta $ and $\eta \in \mathcal F_{\operatorname {i}(\theta )}$ are in general position if $(\xi , \eta )\in G (P_\theta ^-, P_\theta ^+)$ under the diagonal G-action on $\mathcal F_\theta \times \mathcal F_{\operatorname {i}(\theta )}$ . We write

$$ \begin{align*}\mathcal F_\theta^{(2)}= G (P_\theta^-, P_\theta^+),\end{align*} $$

which is the unique open G-orbit in $\mathcal F_\theta \times \mathcal F_{\operatorname {i}(\theta )}$ .

Let $\mathfrak {a}_\theta =\bigcap _{\alpha \in \Pi - \theta } \ker \alpha $ and denote by $\mathfrak a_{\theta }^*$ the space of all linear forms on $\mathfrak a_{\theta }$ . We set $p_\theta :\mathfrak {a}\to \mathfrak {a}_\theta $ as the unique projection invariant under the subgroup of the Weyl group fixing $\mathfrak a_\theta $ pointwise. Set $\mu _{\theta } := p_{\theta } \circ \mu $ .

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 $g, h\in G$ ,

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

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 =kP_\theta \in \mathcal F_\theta $ for $k\in K$ , we define the $\mathfrak a_{\theta }$ -valued Busemann map $\beta ^{\theta } : \mathcal F_{\theta } \times G \times G \to \mathfrak a_{\theta }$ as

(2.2) $$ \begin{align} \beta_{\xi}^\theta (g, h): = p_\theta ( \beta_{kP} (g, h))\in \mathfrak a_\theta;\end{align} $$

this is well defined [19, §6].

In the rest of this section, let $\Gamma < G$ be a Zariski dense $\theta $ -transverse subgroup as in Definition 1.3. For a $(\Gamma , \theta )$ -proper linear form $\psi \in \mathfrak a_\theta ^*$ , we denote by $\delta _\psi \in (0, \infty ]$ the abscissa of convergence of the series $\mathcal P_{\psi }(s) := \sum _{\gamma \in \Gamma }e^{-s \psi (\mu _{\theta }(\gamma ))}$ ; this is well defined [13, Lemma 4.2]. We set

$$ \begin{align*}\mathcal D_{\Gamma}^{\theta}:=\{\psi\in \mathfrak a_\theta^*: (\Gamma, \theta)\text{-proper, } \delta_\psi=1, \text{ and } \mathcal P_{\psi}(1)= \infty\}.\end{align*} $$

Note that $\psi \circ \operatorname {i}$ can be regarded as a linear form on $\mathfrak a_{\operatorname {i}(\theta )}$ . Using the property that $\operatorname {i} (\mu (g))=\mu (g^{-1})$ for all $g\in G$ , we deduce that $\mathcal P_{\psi } = \mathcal P_{\psi \circ \operatorname {i}}$ and hence $\psi \in \mathcal D_{\Gamma }^{\theta }$ if and only if ${\psi \circ \operatorname {i} \in \mathcal D_{\Gamma }^{\operatorname {i}(\theta )}}$ .

The $\theta $ -limit set $\Lambda _{\theta }$ of $\Gamma $ is the unique $\Gamma $ -minimal subset of $\mathcal F_{\theta }$ [1]. We also write

(2.3) $$ \begin{align} \Lambda_\theta^{(2)} := \{(\xi, \eta)\in \mathcal F_\theta^{(2)}:\xi\in \Lambda_\theta,\eta\in \Lambda_{\operatorname{i}(\theta)}\}.\end{align} $$

The following ergodic property of Patterson–Sullivan measures was obtained by Canary, Zhang, and Zimmer [4] (see also [13, 14]).

Theorem 2.1. [4, Proposition 9.1, Corollary 11.1]

Suppose that $\theta =\operatorname {i}(\theta )$ . Let $\Gamma <G$ be a Zariski dense $\theta $ -transverse subgroup. For any $\psi \in \mathcal D_{\Gamma }^{\theta }$ , there exists a unique $(\Gamma , \psi )$ -Patterson–Sullivan measure $\nu _\psi $ on $\Lambda _{\theta }$ and $\nu _\psi $ is non-atomic. Moreover, the diagonal $\Gamma $ -action on $(\Lambda _{\theta }^{(2)}, (\nu _\psi \times \nu _{\psi \circ \operatorname {i}})|_{\Lambda _{\theta }^{(2)}})$ is ergodic.

3. A property of convergence group actions

In this section, we prove a certain property of convergence group actions which we will need in the proof of our main theorem in the next section. We refer to [2] for basic properties of convergence group actions. Let $\Gamma $ be a countable group acting on a compact metrizable space X (with $\# X\ge 3$ ) by homeomorphisms. This action is called a convergence group action if for any sequence of distinct elements $\gamma _n \in \Gamma $ , there exist a subsequence $\gamma _{n_k}$ and $a, b \in X$ such that as $k \to \infty $ , $\gamma _{n_k}(x) $ converges to $ a $ for all $x\in X-\{b\}$ , uniformly on compact subsets. In this case, we say $\Gamma $ acts on X as a convergence group, which we suppose in the following. Any element $\gamma \in \Gamma $ of infinite order fixes precisely one or two points of X, and $\gamma $ is called parabolic or loxodromic accordingly. In that case, there exist $a_{\gamma }, b_{\gamma } \in X$ , fixed by $\gamma $ , such that $\gamma ^n|_{X- \{b_{\gamma }\}} \to a_\gamma $ uniformly on compact subsets as $n \to \infty $ . We have $\gamma $ loxodromic if and only if $a_{\gamma }\ne b_{\gamma }$ , in which case $a_{\gamma }$ and $b_\gamma $ are called the attracting and repelling fixed points of $\gamma $ , respectively.

We will use the following lemma in the next section.

Lemma 3.1. Let $\Gamma $ be a torsion-free countable group acting on a compact metric space X as a convergence group. For any compact subset W of X with at least two points, the subgroup $\Gamma _W = \{\gamma \in \Gamma : \gamma W = W\}$ acts on $X- W$ properly discontinuously, that is, for any $\eta \in X - W$ , there exists an open neighborhood U of $\eta $ such that $\gamma U \cap U \neq \emptyset $ for $\gamma \in \Gamma _W$ implies $\gamma = e$ .

Proof. Suppose not. Then there exist $\eta \in X - W$ , a decreasing sequence of open neighborhoods $U_n$ of $\eta $ in X with $\bigcap _n U_n=\{\eta \}$ , and a sequence $e\ne \gamma _n\in \Gamma $ such that $\gamma _n W=W$ and $\gamma _n U_n \cap U_n \neq \emptyset $ for each $n\in \mathbb N$ . Hence, there exists a sequence $\eta _n \in U_n\cap \gamma _n^{-1}U_n$ ; so $\eta _n \to \eta $ and $\gamma _n \eta _n \to \eta $ as $n \to \infty $ .

We claim that the elements $\gamma _n$ are all pairwise distinct, possibly after passing to a subsequence. Otherwise, it would mean that, after passing to a subsequence, $\gamma _n$ terms are a constant sequence, say $\gamma _n = \gamma \ne e$ . Since $\gamma \eta = \lim _n \gamma _n \eta _n = \eta $ , $\eta $ must be a fixed point of $\gamma $ . Since $\Gamma $ is torsion-free, $\gamma $ is either parabolic or loxodromic, and in particular it has at most two fixed points in X, including $\eta $ . Since $\eta \not \in W$ and W has at least two points, we can take $w \in W$ which is not fixed by $\gamma $ . Then, as $n\to +\infty $ , $\gamma ^n w\to \eta $ or $\gamma ^{-n}w\to \eta $ . Since W is a compact subset such that $\gamma W=W$ and $\eta \notin W$ , this yields a contradiction.

Therefore, we may assume that $\{\gamma _n\}$ is an infinite sequence of distinct elements. Since the action of $\Gamma $ on X is a convergence group action, there exist a subsequence $\gamma _{n_k}$ and $a, b \in X$ such that as $k\to \infty $ , $\gamma _{n_k}(x)$ converges to $ a$ for all $x\in X - \{b\}$ , uniformly on compact subsets. There are two cases to consider. Suppose that $b=\eta $ . Then $W \subset X - \{b\}$ , and hence $\gamma _{n_k}W \to a$ uniformly as $k \to \infty $ . Since $\gamma _{n_k} W = W$ and W is a compact subset, it follows that $W=\{a\}$ , which contradicts the hypothesis that W consists of at least two elements. Now suppose that $b \neq \eta $ . Since $\eta _{n_k}$ converges to $ \eta $ , we may assume that $\eta _{n_k} \neq b$ for all k. Noting that $\# W \ge 2$ , we can take $w_0 \in W - \{b\}$ . If we now consider the following compact subset:

$$ \begin{align*}W_0 := \{\eta_{n_k} : k \in \mathbb N \} \cup \{\eta, w_0\} \subset X - \{b\},\end{align*} $$

we then have $\gamma _{n_k} W_0 \to a$ uniformly as $k \to \infty $ . Since $\eta _{n_k} \in W_0$ for each k and ${\gamma _{n_k}\eta _{n_k} \to \eta} $ as $k \to \infty $ , we must have

$$ \begin{align*}a = \eta.\end{align*} $$

However, since $w_0 \in W_0\cap W$ , $\gamma _{n_k} w_0 \to \eta $ as $k \to \infty $ . This implies $\eta \in W$ since W is compact and $\gamma _{n_k}w_0\in W$ , yielding a contradiction to the hypothesis $\eta \notin W$ . This completes the proof.

We denote by $\Lambda _X$ the set of all accumulation points of a $\Gamma $ -orbit in X. If $\#\Lambda _X\ge 3$ , the $\Gamma $ -action is called non-elementary and $\Lambda _X$ is the unique $\Gamma $ -minimal subset [2].

A well-known example of a convergence group action is given by a word hyperbolic group $\Gamma $ . Fix a finite symmetric generating subset $S_{\Gamma }$ of $\Gamma $ . A geodesic ray in $\Gamma $ is an infinite sequence $(\gamma _i)_{i=0}^{\infty }$ of elements of $\Gamma $ such that $\gamma _i^{-1}\gamma _{i+1}\in S_{\Gamma }$ for all $i\ge 0$ . The Gromov boundary $\partial \Gamma $ is the set of equivalence classes of geodesic rays, where two rays are equivalent to each other if and only if their Hausdorff distance is finite. The group $\Gamma $ acts on $\partial \Gamma $ by $\gamma \cdot [(\gamma _i)]=[(\gamma \gamma _i)]$ . This action is known to be a convergence group action [2, Lemma 1.11].

Another important example of a convergence group action is the action of a $\theta $ -transverse subgroup $\Gamma $ on $\Lambda _{\theta \cup \operatorname {i}(\theta )}$ .

Proposition 3.2. [10, Theorem 4.21]

For a $\theta $ -transverse subgroup $\Gamma $ , the action of $\Gamma $ on $\Lambda _{\theta \cup \operatorname {i}(\theta )}$ is a convergence group action.

4. Non-concentration property

We fix a non-empty subset $\theta \subset \Pi $ . We first prove the following proposition from which we will deduce Theorem 1.4.

Proposition 4.1. Let $\Gamma < G$ be a torsion-free Zariski dense discrete subgroup admitting a convergence group action on a compact metrizable space X. We assume that this action is $\theta $ -antipodal in the sense that there exist $\Gamma $ -equivariant homeomorphisms $f_{\theta } : \Lambda _X \to \Lambda _{\theta }$ and $f_{\operatorname {i}(\theta )} : \Lambda _X \to \Lambda _{\operatorname {i}(\theta )}$ such that for any $\xi \ne \eta $ in $\Lambda _X$ ,

$$ \begin{align*}(f_{\theta}(\xi), f_{\operatorname{i}(\theta)}(\eta))\in \Lambda_\theta^{(2)}.\end{align*} $$

Let $\nu $ be a $\Gamma $ -quasi-invariant measure on $\Lambda _{\theta }$ such that:

1. (1) $\nu $ is non-atomic;

2. (2) $\Gamma $ acts ergodically on $(\Lambda _{\theta }^{(2)}, (\nu \times \nu _{\operatorname {i}})|_{\Lambda _\theta ^{(2)}})$ for some $\Gamma $ -quasi-invariant measure $\nu _{\operatorname {i}}$ on $\Lambda _{\operatorname {i}(\theta )}$ .

Then, for any proper algebraic subset S of $\mathcal F_{\theta }$ , we have

$$ \begin{align*}\nu(S) = 0.\end{align*} $$

Proof. We first claim that the $\Gamma $ -action on $(\Lambda _\theta \times \Lambda _{\operatorname {i}(\theta )}, \nu \times \nu _{{\operatorname {i}}})$ is ergodic. Set $R:=(\Lambda _\theta \times \Lambda _{\operatorname {i}(\theta )} ) - \Lambda _\theta ^{(2)}$ . Since the $\Gamma $ -action on $(\Lambda _{\theta }^{(2)}, (\nu \times \nu _{\operatorname {i}})|_{\Lambda _\theta ^{(2)}})$ is ergodic, it suffices to show that

$$ \begin{align*}(\nu\times\nu_{{\operatorname{i}}})(R)=0.\end{align*} $$

For $y\in \Lambda _{\operatorname {i}(\theta )}$ , let $R(y):=\{x\in \Lambda _{\theta } : (x,y) \in R\}$ . By the antipodal property of the pair $(f_\theta , f_{\operatorname {i}(\theta )})$ , we have that for each $y\in \Lambda _{\operatorname {i}(\theta )}$ , we have $R(y)=\emptyset $ or $R(y)= \{(f_{\theta } \circ f_{\operatorname {i}(\theta )}^{-1})(y)\}$ and hence $\nu ( R(y))=0$ by the non-atomicity of $\nu $ .

Therefore,

(4.1) $$ \begin{align} (\nu\times\nu_{{\operatorname{i}}})(R)=\int_{y\in \Lambda_{\operatorname{i}(\theta)}}\nu (R(y))\,d\nu_{\operatorname{i}}(y)=0, \end{align} $$

proving the claim.

Now suppose that $\nu (S)> 0$ for some proper algebraic subset $S \subset \mathcal F_{\theta }$ . We may assume that S is irreducible and of minimal dimension among all such algebraic subsets of $\mathcal F_\theta $ . Let $W = f_{\theta }^{-1}(S\cap \Lambda _\theta ) \subset \Lambda _X$ . Since $\nu $ is non-atomic and $\nu (S)> 0$ , we have $\# W = \infty> 2$ . This implies $\#\Lambda _X\ge 3$ . By the property of a non-elementary convergence group action, $\Lambda _X$ is the unique $\Gamma $ -minimal subset of X and there always exists a loxodromic element of $\Gamma $ [2].

Since $\Gamma < G$ is Zariski dense, $\Lambda _\theta $ is Zariski dense in $\mathcal F_\theta $ as well, and hence $\Lambda _{\theta } \not \subset S$ . Therefore, $X - W$ is a non-empty open subset intersecting $\Lambda _X$ . Since $\Gamma $ acts minimally on $\Lambda _X$ and the set of attracting fixed points of loxodromic elements of $\Gamma $ is a non-empty $\Gamma $ -invariant subset, there exists a loxodromic element $\gamma _0 \in \Gamma $ whose attracting fixed point $a_{\gamma _0}$ is contained in $\Lambda _X- W$ . Hence, applying Lemma 3.1 to $\eta =a_{\gamma _0}$ , we have an open neighborhood $U $ of $a_{\gamma _0}$ in $\Lambda _X$ such that

(4.2) $$ \begin{align} \gamma U \cap U = \emptyset \end{align} $$

for all non-trivial $\gamma \in \Gamma $ with $\gamma W = W$ .

Since $\gamma _0^m|_{\Lambda _X - \{b_{\gamma _0} \}} \to a_{\gamma _0}$ uniformly on compact subsets as $m \to +\infty $ and $\#\Lambda _X\ge 3$ , U contains a point $\xi \in \Lambda _X-\{a_{\gamma _0}, b_{\gamma _0}\}$ . By replacing $\gamma _0$ by a large power $\gamma _0^m$ if necessary, we can find an open neighborhood V of $\xi $ contained in $U-\{a_{\gamma _0}\}$ such that $\gamma _0 V \subset U$ and $\gamma _0 V \cap V = \emptyset $ .

We now consider the subset

$$ \begin{align*}S \times f_{\operatorname{i}(\theta)}(V)\end{align*} $$

of $\mathcal F_{\theta } \times \mathcal F_{\operatorname {i}(\theta )}$ . Since $\nu (S)> 0$ and $\nu _{\operatorname {i}}(f_{\operatorname {i}(\theta )}(V))> 0$ , we have that $\Gamma (S \times f_{\operatorname {i}(\theta )}(V))$ has full $\nu \times \nu _{\operatorname {i}}$ -measure by the ergodicity of the $\Gamma $ -action on $(\Lambda _{\theta } \times \Lambda _{\operatorname {i}(\theta )}, \nu \times \nu _{\operatorname {i}})$ . Since $(\nu \times \nu _{\operatorname {i}})(S \times \gamma _0 f_{\operatorname {i}(\theta )}(V))> 0$ , there exists $\gamma \in \Gamma $ such that

$$ \begin{align*}(\nu \times \nu_{\operatorname{i}})( (S \times \gamma_0 f_{\operatorname{i}(\theta)}(V)) \cap (\gamma S \times \gamma f_{\operatorname{i}(\theta)}(V)))> 0.\end{align*} $$

In particular, we have

$$ \begin{align*}\nu(S \cap \gamma S)> 0 \quad \text{and} \quad \nu_{\operatorname{i}}(\gamma_0 f_{\operatorname{i}(\theta)}(V) \cap \gamma f_{\operatorname{i}(\theta)}(V)) > 0.\end{align*} $$

Since S was chosen to be of minimal dimension and irreducible among proper algebraic sets with positive $\nu $ -measure, we must have $S = \gamma S$ . It follows from the $\Gamma $ -invariance of $\Lambda _{\theta }$ that $W = \gamma W$ .

The $\Gamma $ -equivariance of $f_{\operatorname {i}(\theta )}$ implies that

(4.3) $$ \begin{align} \nu_{\operatorname{i}}(f_{\operatorname{i}(\theta)}(\gamma_0 V \cap \gamma V))> 0. \end{align} $$

Since $\gamma _0 V \cap V = \emptyset $ , we have $\gamma \neq e$ . Hence, it follows from $V \subset U$ , $\gamma _0 V \subset U$ , and the choice in equation (4.2) of U that

$$ \begin{align*}\gamma_0 V \cap \gamma V \subset U \cap \gamma U = \emptyset,\end{align*} $$

which gives a contradiction to equation (4.3). This finishes the proof.

4.1. Proof of Theorem 1.4

Let $\Gamma < G$ be a Zariski dense $\theta $ -transverse subgroup and $\nu $ a $(\Gamma , \psi )$ -Patterson–Sullivan measure for a $(\Gamma , \theta )$ -proper linear form $\psi \in \mathfrak a_{\theta }^*$ such that $\sum _{\gamma \in \Gamma } e^{-\psi (\mu _{\theta }(\gamma ))} = \infty $ . We may assume without loss of generality that $\Gamma $ is torsion-free. Indeed, let $\Gamma _0 < \Gamma $ be a torsion-free subgroup of finite index. Then $\Gamma _0$ is also a Zariski dense $\theta $ -transverse subgroup of G. Moreover, $\nu $ is a $(\Gamma _0, \psi )$ -Patterson–Sullivan measure since the limit sets for $\Gamma $ and $\Gamma _0$ are the same. Write $\Gamma = \bigcup _{i = 1}^n \gamma _i \Gamma _0$ for some $\gamma _1, \ldots , \gamma _n \in \Gamma $ . By [1, Lemma 4.6], there exists $C> 0$ such that $\| \mu (\gamma _i \gamma ) - \mu (\gamma )\| \le C$ for all $\gamma \in \Gamma _0$ and $i = 1, \ldots , n$ . Hence, we have that $\psi $ is $(\Gamma _0, \theta )$ -proper as well and

$$ \begin{align*}\infty = \sum_{\gamma \in \Gamma} e^{-\psi(\mu_{\theta}(\gamma))} = \sum_{i = 1}^n \sum_{\gamma \in \Gamma_0} e^{-\psi(\mu_{\theta}(\gamma_i \gamma))} \le n e^{\|\psi\| C} \sum_{\gamma \in \Gamma_0} e^{-\psi(\mu_{\theta}(\gamma))},\end{align*} $$

where $\|\psi \|$ denotes the operator norm of $\psi $ . In particular, $\sum _{\gamma \in \Gamma _0} e^{-\psi (\mu _{\theta }(\gamma ))} = \infty $ . Therefore, replacing $\Gamma $ by $\Gamma _0$ , we assume that $\Gamma $ is torsion-free. By Proposition 3.2, the action of $\Gamma $ on $\Lambda _{\theta \cup \operatorname {i}(\theta )}$ is a convergence group action.

Since there exists a $(\Gamma , \psi )$ -conformal measure, we have $\delta _{\psi } \le 1$ by [13, Lemma 7.3]. Therefore, the hypothesis $\sum _{\gamma \in \Gamma } e^{-\psi (\mu _{\theta }(\gamma ))} = \infty $ implies that $\psi \in \mathcal D_{\Gamma }^{\theta }$ . Moreover, the $\theta $ -antipodality of $\Gamma $ implies that the canonical projections

$$ \begin{align*}f_{\theta} : \Lambda_{\theta \cup \operatorname{i}(\theta)} \to \Lambda_{\theta} \quad \text{and} \quad f_{\operatorname{i}(\theta)} : \Lambda_{\theta \cup \operatorname{i}(\theta)} \to \Lambda_{\operatorname{i}(\theta)}\end{align*} $$

are $\Gamma $ -equivariant $\theta $ -antipodal homeomorphisms [13, Lemma 9.5]. This implies that Theorem 2.1 indeed holds for a general $\theta $ without the hypothesis $\theta =\operatorname {i}(\theta )$ . Hence, $\nu =\nu _\psi $ , $\nu _\psi $ is non-atomic, and the diagonal $\Gamma $ -action on $(\Lambda _{\theta }^{(2)}, (\nu _\psi \times \nu _{\psi \circ \operatorname {i}})|_{\Lambda _{\theta }^{(2)}})$ is ergodic. Since $\nu _{\psi \circ \operatorname {i}}$ is $\Gamma $ -conformal, it is $\Gamma $ -quasi-invariant. Therefore, Theorem 1.4 follows from Proposition 4.1.

We emphasize again that Lemma 3.1 and Proposition 4.1 were introduced to deal with the case when $\operatorname {i}$ is non-trivial. Indeed, when $\operatorname {i}$ is trivial, Theorem 1.4 follows from the following $\theta $ -version of [5, Theorem 9.3].

Theorem 4.2. Let $\Gamma < G$ be a Zariski dense discrete subgroup. Let $\nu $ be a $\Gamma $ -quasi-invariant measure on $\Lambda _{\theta }$ . Suppose that the diagonal $\Gamma $ -action on $(\Lambda _{\theta } \times \Lambda _{\theta }, {\nu \times \nu })$ is ergodic. Then, for any proper algebraic subset S of $\mathcal F_{\theta }$ , we have

$$ \begin{align*}\nu(S) = 0.\end{align*} $$

Proof. The proof is identical to the proof of [5, Theorem 9.3] except that we work with a general $\theta $ . We reproduce it here for the convenience of the readers. Let S be a proper irreducible subvariety of $\mathcal F_\theta $ with $\nu (S)>0$ and of minimal dimension. Since ${(\nu \times \nu ) (S\times S)>0}$ , the $\Gamma $ -ergodicity of $\nu \times \nu $ implies that $(\nu \times \nu )(\Gamma (S\times S))=1$ . It follows that for any $\gamma _0\in \Gamma $ , there exists $\gamma \in \Gamma $ such that $(S \times \gamma _0S)\cap (\gamma S\times \gamma S)$ has positive $\nu \times \nu $ -measure; hence, $\nu (S\cap \gamma S)>0$ and $\nu (\gamma _0 S\cap \gamma S)>0$ . Since S is irreducible and of minimal dimension, it follows that $S=\gamma S=\gamma _0 S$ . Since $\gamma _0\in \Gamma $ was arbitrary, we have $\Gamma S=S$ , which contradicts the Zariski density hypothesis on $\Gamma $ .

We finally mention that the proof of Proposition 4.1 implies the following when the second measure cannot be taken to be the same as the first measure.

Theorem 4.3. Let $\Gamma < G$ be a Zariski dense torsion-free discrete subgroup acting on $\Lambda _\theta $ as a convergence group. Let $\nu $ be a non-atomic $\Gamma $ -quasi-invariant measure on $\Lambda _{\theta }$ . Suppose that the diagonal $\Gamma $ -action on $(\Lambda _{\theta } \times \Lambda _{\theta }, \nu \times \nu ')$ is ergodic for some $\Gamma $ -quasi-invariant measure $\nu '$ on $\Lambda _\theta $ . Then, for any proper algebraic subset S of $\mathcal F_{\theta }$ , we have

$$ \begin{align*}\nu(S) = 0.\end{align*} $$

Proof. Since $\Gamma $ acts ergodically on the entire product space $(\Lambda _{\theta } \times \Lambda _{\theta }, \nu \times \nu ')$ , the first part of the proof of Proposition 4.1 is not relevant. Suppose that S is an irreducible proper subvariety of $\mathcal F_\theta $ and of minimal dimension among all subvarieties with positive $\nu $ -measure. Then, setting $W = S \cap \Lambda _{\theta }$ , as in the proof of Proposition 4.1, we can find non-empty open subsets $V\subset U\subset \Lambda _\theta - W$ such that $ \gamma U \cap U = \emptyset $ for all non-trivial $\gamma \in \Gamma $ with $\gamma W = W$ , and $\gamma _0V\subset U$ and $\gamma _0V\cap V=\emptyset $ for some $\gamma _0\in \Gamma $ . Using $(\nu \times \nu ')(S\times V)>0$ , we then get a contradiction by the same argument as in Proposition 4.1.