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 [Reference Quint19, §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 [Reference Kim, Oh and Wang13, 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 $\mathcal F_{\theta }$ [Reference Benoist1]. 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 [Reference Canary, Zhang and Zimmer4] (see also [Reference Kim, Oh and Wang13, Reference Kim, Oh and Wang14]).

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 [Reference Bowditch2] 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.

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 [Reference Bowditch2].

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 [Reference Bowditch2, Lemma 1.11].

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

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 $ [Reference Bowditch2].

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 [Reference Benoist1, 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 [Reference Kim, Oh and Wang13, 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 [Reference Kim, Oh and Wang13, 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 [Reference Edwards, Lee and Oh5, 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 [Reference Edwards, Lee and Oh5, 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.