Let $G$ be a connected semisimple real algebraic group. The class of transverse subgroups of $G$ includes all discrete subgroups of rank one Lie groups and any subgroups of Anosov or relative Anosov subgroups. Given a transverse subgroup $\Gamma$, we show that the $\Gamma$-action on the Weyl chamber flow space determined by its limit set is properly discontinuous. This allows us to consider the quotient space and define Bowen-Margulis-Sullivan measures. We then establish the ergodic dichotomy for the Weyl chamber flow, in the original spirit of Hopf-Tsuji-Sullivan. We also introduce the notion of growth indicators and discuss their properties and roles in the study of conformal measures, extending the work of Quint. We discuss several applications as well.

We describe multiple correlations of Jordan and Cartan spectra for any finite number of Anosov representations of a finitely generated group. This extends our previous work on correlations of length and displacement spectra for rank one convex cocompact representations. Examples include correlations of the Hilbert length spectra for convex projective structures on a closed surface as well as correlations of eigenvalue gaps and singular value gaps for Hitchin representations. We relate the correlation problem to the counting problem for Jordan and Cartan projections of an Anosov subgroup with respect to a family of carefully chosen truncated hypertubes, rather than in tubes as in our previous work. Hypertubes go to infinity in a linear subspace of directions, while tubes go to infinity in a single direction and this feature presents a novel difficulty in this higher rank correlation problem.

For a geometrically finite Kleinian group Gamma, the Bowen-Margulis-Sullivan measure is finite and is the unique measure of maximal entropy for the geodesic flow, as shown by Sullivan and Otal-Peign & eacute; respectively. Moreover, it is strongly mixing by a result of Babillot. We obtain a higher-rank analogue of this theorem. Given a relatively Anosov subgroup Gamma of a semisimple real algebraic group, there is a family of flow spaces parameterized by linear forms tangent to the growth indicator. We construct a reparameterization of each flow space by the geodesic flow on the Groves-Manning space of Gamma which exhibits exponential expansion along unstable foliations. Using this reparameterization, we prove that the Bowen-Margulis-Sullivan measure of each flow space is finite and is the unique measure of maximal entropy. Moreover, it is strongly mixing.

Let $ G $ be a connected, semisimple real algebraic group and $\Gamma < G$ be a Zariski dense discrete subgroup. Let $N$ denote a maximal horospherical subgroup of $G$, and $P=MAN$ the minimal parabolic subgroup which is the normalizer of $N$. Let $\mathcal{E}$ denote the unique $P$-minimal subset of $\Gamma \backslash G$ and let $\mathcal{E}_0$ be a $P^\circ$-minimal subset. We consider a notion of a horospherical limit point in the Furstenberg boundary $ G/P $ and show that the following are equivalent for any $[g]\in \mathcal{E}_0$: (1) $gP\in G/P$ is a horospherical limit point; (2) $[g]NM$ is dense in $\mathcal{E}$; (3) $[g]N$ is dense in $\mathcal{E}_0$. The equivalence of (1) and (2) is due to Dal'bo in the rank one case. We also observe that unlike convex cocompact groups of rank one Lie groups, the $NM$-minimality of $\mathcal{E}$ does not hold in a general Anosov homogeneous space.

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 $ -Poincar & eacute; series of $\Gamma $ diverges at its abscissa of convergence. In particular, our result also applies to relatively Anosov subgroups.

Let $G$ be a higher rank simple real algebraic group, or more generally, any semisimple real algebraic group with no rank one factors and $X$ the associated Riemannian symmetric space. For any Zariski dense discrete subgroup $\Gamma

Let $\Gamma$ be a Zariski dense discrete subgroup of a connected simple real algebraic group $G_1$. We discuss a rigidity problem for discrete faithful representations $\rho:\Gamma\to G_2$ and a surprising role played by higher rank conformal measures of the associated self-joining group. Our approach recovers rigidity theorems of Sullivan, Tukia and Yue, as well as applies to Anosov representations, including Hitchin representations. More precisely, for a given representation $\rho$ with a boundary map $f$ defined on the limit set $\Lambda$, we ask whether the extendability of $\rho$ to $G$ can be detected by the property that $f$ pushes forward some $\Gamma$-conformal measure class $[\nu_\Gamma]$ to a $\rho(\Gamma)$-conformal measure class $[\nu_{\rho(\Gamma)}]$. When $\Gamma$ is of divergence type in a rank one group or when $\rho$ arises from an Anosov representation, we give an affirmative answer by showing that if the self-joining $\Gamma_\rho=(\text{id} \times \rho)(\Gamma)$ is Zariski dense in $G_1\times G_2$, then the push-forward measures $(\text{id}\times f)_*\nu_\Gamma$ and $(f^{-1}\times \text{id})_*\nu_{\rho(\Gamma)}$, which are higher rank $\Gamma_\rho$-conformal measures, cannot be in the same measure class.

We obtain a bi-Lipschitz rigidity theorem for a Zariski dense discrete subgroup of a connected simple real algebraic group. As an application, we show that any Zariski dense discrete subgroup of a higher rank semisimple algebraic group G cannot have a C^1-smooth antipodal limit set in G/P for any non-maximal parabolic subgroup P.

AbstractFor any integerd≥1{d\geq 1}, we obtain counting and equidistribution results for tori with small volume for a class ofd-dimensional torus packings, invariant under a self-joiningΓρ<∏i=1dPSL2⁡(ℂ){\Gamma_{\rho}<\prod_{i=1}^{d}\operatorname{PSL}_{2}(\mathbb{C})}of a Kleinian group Γ formed by ad-tuple of convex-cocompact representationsρ=(ρ1,…,ρd){\rho=(\rho_{1},\dots,\rho_{d})}. More precisely, if𝒫{\mathcal{P}}is aΓρ{\Gamma_{\rho}}-admissibled-dimensional torus packing, then for any bounded subsetE⊂ℂd{E\subset\mathbb{C}^{d}}with∂⁡E{\partial E}contained in a proper real algebraic subvariety, we havelims→0⁡sδL1⁢(ρ)⋅#⁢{T∈𝒫:Vol⁡(T)>s,T∩E≠∅}=c𝒫⋅ωρ⁢(E∩Λρ).\lim_{s\to 0}{s^{\delta_{L^{1}}({\rho})}}\cdot\#\{T\in\mathcal{P}:% \operatorname{Vol}(T)>s,\,T\cap E\neq\emptyset\}=c_{\mathcal{P}}\cdot\omega_{% \rho}(E\cap\Lambda_{\rho}).HereδL1⁢(ρ){\delta_{L^{1}}(\rho)},0<δL1⁢(ρ)≤2/d{0<\delta_{L^{1}}(\rho)\leq 2/\!{\sqrt{d}}}, denotes the critical exponent of the self-joiningΓρ{\Gamma_{\rho}}with respect to theL1{L^{1}}-metric on the product∏i=1dℍ3{\prod_{i=1}^{d}\mathbb{H}^{3}},Λρ⊂(ℂ∪{∞})d{\Lambda_{\rho}\subset(\mathbb{C}\cup\{\infty\})^{d}}is the limit set ofΓρ{\Gamma_{\rho}}, andωρ{\omega_{\rho}}is a locally finite Borel measure onℂd∩Λρ{\mathbb{C}^{d}\cap\Lambda_{\rho}}which can be explicitly described. The class of admissible torus packings we consider arises naturally from the Teichmüller theory of Kleinian groups. Our work extends previous results of [H. Oh and N. Shah, The asymptotic distribution of circles in the orbits of Kleinian groups, Invent. Math. 187 2012, 1, 1–35] on circle packings (i.e., one-dimensional torus packings) tod-torus packings.

For all Zarski dense Anosov subgroups of a semisimple real algebraic group, we prove that their limit sets are Ahlfors regular for intrinsic conformal premetrics. As a consequence, we obtain that a Patterson-Sullivan measure is Ahlfors regular (and hence equal to the Hausdorff measure) if and only if the associated linear form is symmetric. We also discuss several applications, including analyticity of (p,q)-Hausdorff dimensions on the Teichmüller spaces, new upper bounds on the growth indicator, and L^2-spectral properties of associated locally symmetric manifolds.

Let $G$ be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma <G$, we show that a $\Gamma $-conformal measure is supported on the limit set of $\Gamma $ if and only if its dimension is $\Gamma $-critical. This implies the uniqueness of a $\Gamma $-conformal measure for each critical dimension, answering the question posed in our earlier paper with Edwards []. We obtain this by proving a higher rank analogue of the Hopf-Tsuji-Sullivan dichotomy for the maximal diagonal action. Other applications include an analogue of the Ahlfors measure conjecture for Anosov subgroups.

We establish an analogue of Ratner's orbit closure theorem for any connected closed subgroup generated by unipotent elements in $\operatorname{SO}(d,1)$ acting on the space $\Gamma\backslash \operatorname{SO}(d,1)$, assuming that the associated hyperbolic manifold $M=\Gamma\backslash \mathbb H^d$ is a convex cocompact manifold with Fuchsian ends. For $d=3$, this was proved earlier by McMullen, Mohammadi and Oh. In a higher dimensional case, the possibility of accumulation on closed orbits of intermediate groups causes serious new issues, but in the end, all orbit closures of unipotent flows are relatively homogeneous. Our results imply the following: for any $k\ge 1$, (1) the closure of any $k$-horosphere in $M$ is a properly immersed submanifold; (2) the closure of any geodesic $(k+1)$-plane in $M$ is a properly immersed submanifold; (3) any infinite sequence of maximal properly immersed geodesic $(k+1)$-planes intersecting $\operatorname{core} M$ becomes dense in $M$.

In this paper, we obtain an ergodic dichotomy for directional flows, more generally, subspace flows, for a class of discrete subgroups of a connected semisimple real algebraic group G, called transverse subgroups. The class of transverse subgroups of G includes all discrete subgroups of rank one Lie groups, Anosov subgroups and their relative versions. Let Γ be a Zariski dense θ-transverse subgroup for a subset θ of simple roots. Let L_θ=A_θ S_θ be the Levi subgroup associated with θ where A_θ is the central maximal real split torus and S_θ is the product of a semisimple subgroup and a compact torus. There is a canonical Γ-invariant subspace Ω̃_θ of G/S_θ on which Γ acts properly discontinuously. Setting Ω_θ=Γ\Ω̃_θ, we consider the subspace flow given by A_W=exp W for any linear subspace W< a_θ. Our main theorem is a Hopf-Tsuji-Sullivan type dichotomy for the ergodicity of (Ω_θ, A_W, 𝗆) with respect to a Bowen-Margulis-Sullivan measure 𝗆 satisfying a certain hypothesis. As an application, we obtain the codimension dichotomy for a θ-Anosov subgroup Γ <G: for any subspace W<𝔞_θ containing a vector u in the interior of the θ-limit cone of Γ, we have codim W ≤ 2 if and only if the A_W-action on (Ω_θ, 𝗆_u) is ergodic where 𝗆_u is the Bowen-Margulis-Sullivan measure associated with u.

Let $\Gamma$ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group $G$. For a maximal horospherical subgroup $N$ of $G$, we show that the space of all non-trivial $NM$-invariant ergodic and $A$-quasi-invariant Radon measures on $\Gamma \backslash G$, up to proportionality, is homeomorphic to ${\mathbb R}^{\text{rank}\,G-1}$, where $A$ is a maximal real split torus and $M$ is a maximal compact subgroup which normalizes $N$. One of the main ingredients is to establish the $NM$-ergodicity of all Burger-Roblin measures.

We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic 3-manifold. Our estimates are polynomials in the tight areas and Bowen-Margulis-Sullivan densities of geodesic planes, with degree given by the modified critical exponents.

Let Γ be a Zariski dense discrete subgroup of a connected semisimple real algebraic group G . Let k=rank G . Let ψ _Γ :𝔞→ℝ∪{-∞} be the growth indicator function of Γ , first introduced by Quint. In this paper, we obtain the following pointwise bound of ψ _Γ : for all v∈𝔞 , ψ _Γ (v) ≤min _1≤ i≤ kδ _α _iα _i(v) where Δ ={α _1, ⋯ , α _k} is the set of all simple roots of (𝔤,𝔞) and 0<δ _α _i≤∞ is the critical exponent of Γ associated to α _i . When Γ is Δ -Anosov, there are precisely k -number of directions where the equality is achieved, and the following strict inequality holds for k≥ 2 : for all v∈𝔞-{0} , ψ _Γ (v) <1/k∑ _i=1^k δ _α _iα _i (v). We discuss applications for self-joinings of convex cocompact subgroups in ∏ _i=1^k SO(n_i,1) and Hitchin subgroups of PSL(d, ℝ) . In particular, for a Zariski dense Hitchin subgroup Γ <PSL(d, ℝ) , we obtain that for any v=diag(t_1, ⋯ , t_d)∈𝔞^+ , ψ _Γ (v) ≤min _1≤ i≤ d-1 (t_i -t_i+1).

Let Γ <PSL_2(ℂ)≃Isom^+( ℍ^3) be a finitely generated non-Fuchsian Kleinian group whose ordinary set Ω =𝕊^2-Λ has at least two components. Let ρ : Γ→PSL_2(ℂ) be a faithful discrete non-Fuchsian representation with boundary map f:Λ→𝕊^2 on the limit set. In this paper, we obtain a new rigidity theorem: if f is conformal on Λ , in the sense that f maps every circular slice of Λ into a circle, then f extends to a Möbius transformation g on 𝕊^2 and ρ is the conjugation by g . Moreover, unless ρ is a conjugation, the set of circles C such that f(C∩Λ ) is contained in a circle has empty interior in the space of all circles meeting Λ . This answers a question asked by McMullen on the rigidity of maps Λ→𝕊^2 sending vertices of every tetrahedron of zero-volume to vertices of a tetrahedron of zero-volume. The novelty of our proof is a new viewpoint of relating the rigidity of Γ with the higher rank dynamics of the self-joining (id×ρ )(Γ )<PSL_2( ℂ)×PSL_2(ℂ) .

The classical result of Patterson and Sullivan says that for a non-elementary convex cocompact subgroup $\Gamma<\text{SO}^\circ (n,1)$, $n\ge 2$, the Hausdorff dimension of the limit set of $\Gamma$ is equal to the critical exponent of $\Gamma$. In this paper, we generalize this result for self-joinings of convex cocompact groups in two ways. Let $\Delta$ be a finitely generated group and $\rho_i:\Delta\to \text{SO}^\circ(n_i,1)$ be a convex cocompact faithful representation of $\Delta$ for $1\le i\le k$. Associated to $\rho=(\rho_1, \cdots, \rho_k)$, we consider the following self-joining subgroup of $\prod_{i=1}^k \text{SO}(n_i,1)$: $$\Gamma=\left(\prod_{i=1}^k\rho_i\right)(\Delta)=\{(\rho_1(g), \cdots, \rho_k(g)):g\in \Delta\} .$$ (1). Denoting by $\Lambda\subset \prod_{i=1}^k \mathbb{S}^{n_i-1}$ the limit set of $\Gamma$, we first prove that $$\text{dim}_H \Lambda=\max_{1\le i\le k} \delta_{\rho_i}$$ where $\delta_{\rho_i}$ is the critical exponent of the subgroup $\rho_{i}(\Delta)$. (2). Denoting by $\Lambda_u\subset \Lambda$ the $u$-directional limit set for each $u=(u_1, \cdots, u_k)$ in the interior of the limit cone of $\Gamma$, we obtain that for $k\le 3$, $$ \frac{\psi_\Gamma(u)}{\max_i u_i }\le \text{dim}_H \Lambda_u \le \frac{\psi_\Gamma(u)}{\min_i u_i }$$ where $\psi_\Gamma:\mathbb{R}^k\to \mathbb{R}\cup\{-\infty\}$ is the growth indicator function of $\Gamma$.

For a given d-tuple ρ=(ρ_1,…,ρ_d):Γ→ G of faithful Zariski dense convex cocompact representations of a finitely generated group Γ, we study the correlations of length spectra {ℓ_ρ_i(γ)}_[γ]∈[Γ] and correlations of displacement spectra {𝖽(ρ_i(γ)o,o)}_γ. We prove that for any interior vector 𝗏=(v_1,…,v_d) in the spectrum cone, there exists δ_ρ(𝗏) > 0 such that for any ε_1, …, ε_d>0, there exist c_1,c_2> 0 such that #{[γ]∈ [Γ]: v_iT ≤ℓ_ρ_i(γ)≤ v_i T+ε_i, 1 ≤ i ≤ d }∼ c_1 e^δ_ρ(𝗏)T/ T^(d+1)/2; #{γ∈ Γ: v_iT ≤𝖽(ρ_i(γ)o,o) ≤ v_i T+ε_i, 1 ≤ i ≤ d }∼ c_2 e^δ_ρ(𝗏)T/ T^(d-1)/2. We deduce this result as a special case of our main theorem on the distribution of Jordan projections with holonomies and Cartan projections in tubes of an Anosov subgroup Γ of a semisimple real algebraic group G. We also show that the growth indicator of Γ remains the same when we use Jordan projections instead of Cartan projections and tubes instead of cones, except possibly on the boundary of the limit cone.

Let $G=\operatorname {SO}^\circ (n,1) \times \operatorname {SO}^\circ (n,1)$ and $X=\mathbb {H}^{n}\times \mathbb {H}^{n}$ for $n\ge 2$. For a pair $(\pi _1, \pi _2)$ of non-elementary convex cocompact representations of a finitely generated group $\Sigma$ into $\operatorname {SO}^\circ (n,1)$, let $\Gamma =(\pi _1\times \pi _2)(\Sigma )$. Denoting the bottom of the $L^2$-spectrum of the negative Laplacian on $\Gamma {\setminus } X$ by $\lambda _0$, we show: $L^2(\Gamma {\setminus } G)$ is tempered and $\lambda _0=\frac {1}{2}(n-1)^2$; There exists no positive Laplace eigenfunction in $L^2(\Gamma {\setminus } X)$. In fact, analogues of (1)-(2) hold for any Anosov subgroup $\Gamma$ in the product of at least two simple algebraic groups of rank one as well as for Hitchin subgroups $\Gamma <\operatorname {PSL}_d(\mathbb {R})$, $d\ge 3$. Moreover, if $G$ is a semisimple real algebraic group of rank at least $2$, then (2) holds for any Anosov subgroup $\Gamma$ of $G$.