Searcharxiv⌕ Search

arXiv subjects

Hee Oh

Publications and source records attributed to Hee Oh.

At least 37 records · Page 2Linked to original sources
arXiv2025

Multiple correlations of spectra for higher rank Anosov representations

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 {\it 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.

math.DS↗
arXiv2025

Ergodic dichotomy for subspace flows in higher rank

In this paper, we study the ergodicity of a one-parameter diagonalizable subgroup of a connected semisimple real algebraic group $G$ acting on a homogeneous space or, more generally, a homogeneous-like space, equipped with a Bowen-Margulis-Sullivan type measure. These flow spaces are associated with Anosov subgroups of $G$, or more generally, with transverse subgroups of $G$. We obtain an ergodicity criterion similar to the Hopf-Tsuji-Sullivan dichotomy for the ergodicity of the geodesic flow on hyperbolic manifolds. In addition, we extend this criterion to the action of any connected diagonal subgroup of arbitrary dimension. Our criterion provides a codimension dichotomy on the ergodicity of a connected diagonalizable subgroup for general Anosov subgroups. This generalizes an earlier work by Burger-Landesberg-Lee-Oh for Borel Anosov subgroups.

math.DS↗
arXiv2024

Conformal measure rigidity for representations via self-joinings

Let $Γ$ be a Zariski dense discrete subgroup of a connected simple real algebraic group $G_1$. We discuss a rigidity problem for discrete faithful representations $ρ:Γ\to G_2$ 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 $ρ$ with a boundary map $f$ defined on the limit set $Λ$, we ask whether the extendability of $ρ$ to $G$ can be detected by the property that $f$ pushes forward some $Γ$-conformal measure class $[ν_Γ]$ to a $ρ(Γ)$-conformal measure class $[ν_{ρ(Γ)}]$. When $Γ$ is of divergence type in a rank one group or when $ρ$ arises from an Anosov representation, we give an affirmative answer by showing that if the self-joining $Γ_ρ=(\text{id} \times ρ)(Γ)$ is Zariski dense in $G_1\times G_2$, then the push-forward measures $(\text{id}\times f)_*ν_Γ$ and $(f^{-1}\times \text{id})_*ν_{ρ(Γ)}$, which are higher rank $Γ_ρ$-conformal measures, cannot be in the same measure class.

math.GT↗
arXiv2024

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

Let $G$ be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $Γ<G$ with respect to a parabolic subgroup $P_θ$, we prove that any $Γ$-Patterson-Sullivan measure charges no mass on any proper subvariety of $G/P_θ$. More generally, we prove that for a Zariski dense $θ$-transverse subgroup $Γ<G$, any $(Γ, ψ)$-Patterson-Sullivan measure charges no mass on any proper subvariety of $G/P_θ$, provided the $ψ$-Poincaré series of $Γ$ diverges at $s=1$. In particular, our result also applies to relatively Anosov subgroups.

math.DS↗
arXiv2024

Bi-Lipschitz rigidity of discrete subgroups

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 slim limit set in $G/P$ for any non-maximal parabolic subgroup $P$.

math.GR↗
arXiv2024

On denseness of horospheres in higher rank homogeneous spaces

Let $ G $ be a connected, semisimple real algebraic group and $Γ< G$ 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 $Γ\backslash G$ 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.

math.DS↗
arXiv2023

Torus counting and self-joinings of Kleinian groups

For any $d\geq 1$, we obtain counting and equidistribution results for tori with small volume for a class of $d$-dimensional torus packings, invariant under a self-joining $Γ_ρ<\prod_{i=1}^d\mathrm{PSL}_2(\mathbb{C})$ of a Kleinian group $Γ$ formed by a $d$-tuple of convex cocompact representations $ρ=(ρ_1, \cdots, ρ_d)$. More precisely, if $\mathcal P$ is a $Γ_ρ$-admissible $d$-dimensional torus packing, then for any bounded subset $E\subset \mathbb{C}^d$ with $\partial E$ contained in a proper real algebraic subvariety, we have $$\lim_{s\to 0} { s^{δ_{L^1}(ρ) }} \cdot \#\{T\in \mathcal{P}: \mathrm{Vol} (T)> s,\, T\cap E\neq \emptyset \}= c_{\mathcal P}\cdot ω_ρ (E\cap Λ_ρ).$$ Here $0<δ_{L^1}(ρ)\le 2/\sqrt d$ is the critical exponent of $Γ_ρ$ with respect to the $L^1$-metric on the product $\prod_{i=1}^d \mathbb{H}^3$, $Λ_ρ\subset (\mathbb{C}\cup\{\infty\})^d$ is the limit set of $Γ_ρ$, and $ω_ρ$ is a locally finite Borel measure on $\mathbb{C}^d\cap Λ_ρ$ 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 Oh-Shah on circle packings (i.e. one-dimensional torus packings) to $d$-torus packings.

math.DS↗
arXiv2023

Infinite volume and atoms at the bottom of the spectrum

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 $Γ<G$, we prove that $\operatorname{Vol}(Γ\backslash X)=\infty$ if and only if no positive Laplace eigenfunction belongs to $L^2(Γ\backslash X)$, or equivalently, the bottom of the $L^2$-spectrum is not an atom of the spectral measure of the negative Laplacian. This contrasts with the rank one situation where the square-integrability of the base eigenfunction is determined by the size of the critical exponent relative to the volume entropy of $X$.

math.SP↗
arXiv2023

Ergodic decompositions of geometric measures on Anosov homogeneous spaces

Let $G$ be a connected semisimple real algebraic group and $Γ$ a Zariski dense Anosov subgroup of $G$ with respect to a minimal parabolic subgroup $P$. Let $N$ be the maximal horospherical subgroup of $G$ given by the unipotent radical of $P$. We describe the $N$-ergodic decompositions of all Burger-Roblin measures as well as the $A$-ergodic decompositions of all Bowen-Margulis-Sullivan measures on $Γ\backslash G$. As a consequence, we obtain the following refinement of the main result of [LO]: the space of all {\it non-trivial} $N$-invariant ergodic and $P^\circ$-quasi-invariant Radon measures on $Γ\backslash G$, up to constant multiples, is homeomorphic to ${\mathbb R}^{\text{rank}\,G-1}\times \{1, \cdots, k\}$ where $k$ is the number of $P^\circ$-minimal subsets in $Γ\backslash G$.

math.DS↗
arXiv2023

Tent property of the growth indicator functions and applications

Let $Γ$ be a Zariski dense discrete subgroup of a connected semisimple real algebraic group $G$. Let $k=\operatorname{rank} G$. Let $ψ_Γ:\mathfrak{a} \to \mathbb{R}\cup \{-\infty\}$ 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 $v\in \mathfrak{a}$, $$ ψ_Γ(v) \le \min_{1\le i\le k} δ_{α_i} α_i(v) $$ where $Δ=\{α_1, \cdots, α_k\}$ is the set of all simple roots of $(\mathfrak{g},\mathfrak{a})$ and $0<δ_{α_i}\le \infty$ 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 $k\ge 2$: for all $v\in \mathfrak{a}-\{0\}$, $$ψ_Γ(v) <\frac{1}{k}\sum_{i=1}^k δ_{α_i} α_i (v).$$ We discuss applications for self-joinings of convex cocompact subgroups in $\prod_{i=1}^k \operatorname{SO}(n_i,1)$ and Hitchin subgroups of $\operatorname{PSL}(d, \mathbb{R})$. In particular, for a Zariski dense Hitchin subgroup $Γ<\text{PSL}(d, \mathbb{R})$, we obtain that for any $ v=\operatorname{diag}(t_1, \cdots, t_d)\in \mathfrak{a}^+$, $$ψ_Γ(v) \le \min_{1\le i\le d-1} (t_i -t_{i+1}). $$

math.GT↗
arXiv2023

Temperedness of $L^2(Γ\backslash G)$ and positive eigenfunctions in higher rank

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 $(π_1, π_2)$ of non-elementary convex cocompact representations of a finitely generated group $Σ$ into $\operatorname{SO}^\circ(n,1)$, let $Γ=(π_1\times π_2)(Σ)$. Denoting the bottom of the $L^2$-spectrum of the negative Laplacian on $Γ\backslash X$ by $λ_0$, we show: (1) $L^2(Γ\backslash G)$ is tempered and $λ_0=\frac{1}{2}(n-1)^2$; (2) There exists no positive Laplace eigenfunction in $L^2(Γ\backslash X)$. In fact, analogues of (1)-(2) hold for any Anosov subgroup $Γ$ in the product of at least two simple algebraic groups of rank one as well as for Hitchin subgroups $Γ<\operatorname{PSL}_d(\mathbb R)$, $d\ge 3$. Moreover, if $G$ is a semisimple real algebraic group of rank at least $2$, then (2) holds for any Anosov subgroup $Γ$ of $G$.

math.GT↗
arXiv2023

Rigidity of Kleinian groups via self-joinings

Let $Γ<\mathrm{PSL}_2(\mathbb{C})\simeq \mathrm{Isom}^+(\mathbb{H}^3)$ be a finitely generated non-Fuchsian Kleinian group whose ordinary set $Ω=\mathbb{S}^2-Λ$ has at least two components. Let $ρ: Γ\to \mathrm{PSL}_2(\mathbb{C})$ be a faithful discrete non-Fuchsian representation with boundary map $f:Λ\to \mathbb{S}^2$ on the limit set. In this paper, we obtain a new rigidity theorem: if $f$ is {\it 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 $\mathbb{S}^2$ and $ρ$ is the conjugation by $g$. Moreover, unless $ρ$ is a conjugation, the set of circles $C$ such that $f(C\cap Λ)$ 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 $Λ\to \mathbb{S}^2$ 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 $(\mathrm{id} \times ρ)(Γ)<\mathrm{PSL}_2(\mathbb{C})\times \mathrm{PSL}_2(\mathbb{C})$.

math.GT↗
arXiv2023

Hausdorff dimension of directional limit sets for self-joinings of hyperbolic manifolds

The classical result of Patterson and Sullivan says that for a non-elementary convex cocompact subgroup $Γ<\text{SO}^\circ (n,1)$, $n\ge 2$, the Hausdorff dimension of the limit set of $Γ$ is equal to the critical exponent of $Γ$. In this paper, we generalize this result for self-joinings of convex cocompact groups in two ways. Let $Δ$ be a finitely generated group and $ρ_i:Δ\to \text{SO}^\circ(n_i,1)$ be a convex cocompact faithful representation of $Δ$ for $1\le i\le k$. Associated to $ρ=(ρ_1, \cdots, ρ_k)$, we consider the following self-joining subgroup of $\prod_{i=1}^k \text{SO}(n_i,1)$: $$Γ=\left(\prod_{i=1}^kρ_i\right)(Δ)=\{(ρ_1(g), \cdots, ρ_k(g)):g\in Δ\} .$$ (1). Denoting by $Λ\subset \prod_{i=1}^k \mathbb{S}^{n_i-1}$ the limit set of $Γ$, we first prove that $$\text{dim}_H Λ=\max_{1\le i\le k} δ_{ρ_i}$$ where $δ_{ρ_i}$ is the critical exponent of the subgroup $ρ_{i}(Δ)$. (2). Denoting by $Λ_u\subset Λ$ the $u$-directional limit set for each $u=(u_1, \cdots, u_k)$ in the interior of the limit cone of $Γ$, we obtain that for $k\le 3$, $$ \frac{ψ_Γ(u)}{\max_i u_i }\le \text{dim}_H Λ_u \le \frac{ψ_Γ(u)}{\min_i u_i }$$ where $ψ_Γ:\mathbb{R}^k\to \mathbb{R}\cup\{-\infty\}$ is the growth indicator function of $Γ$.

math.DS↗
arXiv2023

Orbit closures of unipotent flows for hyperbolic manifolds with Fuchsian ends

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 $Γ\backslash \operatorname{SO}(d,1)$, assuming that the associated hyperbolic manifold $M=Γ\backslash \mathbb H^d$ 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 very serious obstacles, and surmounting these via the avoidance theorem (Theorem 7.13) is the heart of this paper. 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 $M$.

math.DS↗
arXiv2023

Dynamics for discrete subgroups of $\operatorname{SL}_2(\mathbb C)$

Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups that "A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: these two theories can be united under the common title of Theory of discrete subgroups of $\operatorname{SL}_2(\mathbb C)$." In this article, we will discuss a few recent advances regarding this missing topic from his book, which were influenced by his earlier works.

math.DS↗
arXiv2023

Anosov groups: local mixing, counting, and equidistribution

Let $G$ be a connected semisimple real algebraic group, and $Γ<G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We describe the asymptotic behavior of matrix coefficients $\langle (\exp tv). f_1, f_2\rangle$ in $L^2(Γ\backslash G)$ as $t\to \infty$ for any $f_1, f_2\in C_c(Γ\backslash G)$ and any vector $v$ in the interior of the limit cone of $Γ$. These asymptotics involve higher rank analogues of Burger-Roblin measures which are introduced in this paper. As an application, for any affine symmetric subgroup $H$ of $G$, we obtain a bisector counting result for $Γ$-orbits with respect to the corresponding generalized Cartan decomposition of $G$. Moreover, we obtain analogues of the results of Duke-Rudnick-Sarnak and Eskin-McMullen for counting discrete $Γ$-orbits in affine symmetric spaces $H\backslash G$.

math.DS↗
arXiv2022

Horospherical invariant measures and a rank dichotomy for Anosov groups

Let $G=\prod_{i=1}^{r} G_i$ be a product of simple real algebraic groups of rank one and $Γ$ an Anosov subgroup of $G$ with respect to a minimal parabolic subgroup. For each $v$ in the interior of a positive Weyl chamber, let $\mathcal R_v\subsetΓ\backslash G$ denote the Borel subset of all points with recurrent $\exp (\mathbb R_+ v)$-orbits. For a maximal horospherical subgroup $N$ of $G$, we show that the $N$-action on ${\mathcal R}_v$ is uniquely ergodic if $r={rank}(G)\le 3$ and $v$ belongs to the interior of the limit cone of $Γ$, and that there exists no $N$-invariant {Radon} measure on $\mathcal R_v$ otherwise.

math.DS↗