arXiv subjects
Publications and source records attributed to Hee Oh.
We give a new proof of the quantitative Oppenheim theorem in signature $(2,2)$, originally proved by Eskin-Margulis-Mozes, by recasting the problem as one about determinant values on lattices in $\operatorname{M}_2(\mathbb R)$.
We study the distribution of determinant values on lattices in $\operatorname{M}_n(\mathbb R)$ for $n\ge 2$. Let $\Lambda<\operatorname{M}_n(\mathbb R)$ be a lattice whose elements have algebraic entries. We prove that if $\det (\Lambda)$ is not contained in a scalar multiple of $\mathbb Z$, then for every $a 0$ depends only on $n$. Under an additional hypothesis on the isotropic subspaces associated with $\Lambda$, which is automatic for $n=2,3$ and is satisfied when $n\geq 4$, by every lattice spanned by scalar multiples of the elementary matrices, we also obtain an asymptotic formula for the singular lattice points. Both conclusions extend to the broader class of Diophantine lattices under the corresponding hypotheses. For $n=2$, our theorem recovers the Eskin-Margulis-Mozes theorem on the quantitative Oppenheim problem for quadratic forms of signature $(2,2)$; more generally, our results may be viewed as higher-degree analogues of that theorem.
Let $\Gamma$ be a discrete subgroup of a connected semisimple real algebraic group of higher rank. The growth indicator function $\psi_\Gamma$ records the directional exponential growth of the Cartan projections of elements of $\Gamma$ in the positive Weyl chamber $\mathfrak a^+$. We prove that if $\Gamma$ is a non-elementary relatively Borel Anosov group, then $\psi_\Gamma$ is strictly concave on non-collinear directions. We prove this by establishing the $\mathcal C^1$-smoothness of the Manhattan hypersurface, defined as the unit level set of the critical-exponent map $\phi\mapsto\delta^\phi(\Gamma)$. More generally, for a non-elementary $\theta$-transverse group, we prove local $\mathcal C^1$-regularity near every point of the $\theta$-Manhattan hypersurface that is positive on the $\theta$-limit cone and has a critical gap at infinity. In particular, the $\theta$-Manhattan hypersurface is globally $\mathcal C^1$ for relatively $\theta$-Anosov groups, and their $\theta$-growth indicator functions are strictly concave on non-collinear directions.
Patterson-Sullivan measures encode the distribution of orbits of discrete group actions near the boundary. In this paper, we prove a global shadow lemma for Patterson-Sullivan measures associated to relatively Morse subgroups of higher-rank semisimple Lie groups. The estimate is uniform for shadows centered at arbitrary points in a Gromov model, including points deep in the cuspidal part. This extends the global shadow lemma of Stratmann-Velani for geometrically finite real hyperbolic groups. As applications, we obtain uniform local estimates for Patterson-Sullivan measures, and we give sufficient conditions under which these measures agree, up to scale, with the Hausdorff measure defined by the associated visual quasi-metric.
We give a dynamical proof of Benoist's non-arithmeticity theorem for Jordan spectra of Zariski dense subgroups of connected semisimple real algebraic groups. After passing to a Zariski dense Schottky subgroup, we use the coding of the limit set to realize Jordan projections as periods of a vector-valued Busemann return map for an expanding map on the Furstenberg boundary. The key step is to prove that a suitable two-branch asymptotic discrepancy is not locally constant on the limit set. We also show that the same criterion applies beyond Lie groups; in particular, it yields a direct density result for multiplier spectra of hyperbolic rational maps whose Julia set is not contained in a circle.
Roblin's theorem asserts that, in rank one, a coamenable normal subgroup has the same critical exponent as its ambient group. Natural higher-rank analogues would predict that coamenability preserves the limit cone and the growth indicator. We show that both assertions fail, even when the quotient is infinite cyclic. For every odd integer $n\geq 3$, we construct a nonempty open family of Zariski-dense Borel--Anosov Schottky subgroups of $\mathrm{SL}_n(\mathbb R)$ admitting cocyclic normal subgroups with strictly smaller limit cones. Moreover, in $\mathrm{SL}_3(\mathbb R)$, we construct cocyclic pairs with the same limit cone but distinct growth indicators at an interior direction. We then identify the precise rigidity that survives. Let $\Gamma$ be a Zariski-dense Borel--Anosov subgroup of a connected semisimple real algebraic group, and let $N\lhd\Gamma$ be coamenable. Then the growth indicators of $N$ and $\Gamma$ agree on the fixed-point set of the opposition involution, and their Riemannian critical exponents are equal. The examples with equal limit cones show that the restriction to opposition-invariant directions is sharp.
We study totally geodesic submanifolds in the convex core of geometrically finite rank-one locally symmetric manifolds. Although the infinite-volume setting can exhibit highly complicated behavior, including geodesic planes with fractal closures, we show that a strong rigidity persists inside the convex core. This rigidity has striking consequences in the infinite volume setting: every maximal totally geodesic submanifold of dimension at least two contained in the convex core is properly immersed and has finite volume, and only finitely many such submanifolds can occur. These results stand in sharp contrast to the behavior in the finite-volume setting. Moreover, combining this finiteness result with the work of Bader-Fisher-Miller-Stover and of Gromov-Schoen, we deduce that any geometrically finite rank-one manifold with infinitely many maximal totally geodesic submanifolds of dimension at least two and of finite volume must be arithmetic.
We report on recent developments in the dynamics and rigidity of infinite-volume homogeneous spaces, viewed through the lens of circles. By addressing four natural questions about circle packings, we highlight the interplay between dynamics, geometry, and rigidity that defines the emerging frontier of homogeneous dynamics.
Ratner's theorem implies topological rigidity of immersed totally geodesic subspaces of noncompact type in finite-volume locally symmetric spaces. In higher rank and infinite volume, however, counter-examples to this rigidity have remained elusive. We construct the first such examples using \emph{floating geodesic planes}. Specifically, we exhibit a Zariski-dense Hitchin surface group $\Gamma < \mathrm{SL}_3(\mathbb{R})$ such that the Hitchin manifold $\Gamma \backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}(3)$ contains immersed floating geodesic planes whose closures are fractal, with non-integer Hausdorff dimensions accumulating at $2$. Moreover, $\Gamma$ can be chosen inside $\mathrm{SL}_3(\mathbb{Z})$.
For a vector $v=(v_1,\dots ,v_n)$ with $v_1>\cdots>v_n$ and $\sum v_i=0$, we study the "directional entropy" of two arithmetic objects: (1) the logarithmic embeddings of degree-$n$ totally real units, and (2) the logarithmic eigenvalue data of $\operatorname{SL}_n(\mathbb Z)$. In each case, the entropy in the direction of $v$ is $\mathsf E_n(v)= \rho_{\operatorname{SL}_n}(v)=\sum_{i=1}^{n-1}(n-i)\,v_i,$ the value of the half-sum of positive roots of $\operatorname{SL}_n(\mathbb R)$ evaluated at $v$. More precisely, the number of objects lying in a thin tube around the ray $\mathbb R_+v$ and of norm at most $T$ grows on the order of $ \exp\!\bigl(\rho_{\operatorname{SL}_n}(v)\,T\bigr)$ as $T\to \infty$. Because each eigenvalue data determines an $\operatorname{SL}_n(\mathbb R)$-conjugacy class, this implies a lower bound of order $\exp\!\bigl(\rho_{\operatorname{SL}_n}(v)T\bigr)$ for the number of $\operatorname{SL}_n(\mathbb Z)$-conjugacy classes with a prescribed eigenvalue data; we also obtain an upper bound of order $\exp\!\bigl(2\rho_{\operatorname{SL}_n}(v)T\bigr)$. A parallel argument for the symplectic lattice $\operatorname{Sp}_{2n}(\mathbb Z)$, taken in the symmetric direction $v=(v_1,\dots ,v_n,-v_n,\dots ,-v_1),\quad v_1>\cdots>v_n>0,$ shows that $\mathsf E_{2n}^{\operatorname{Sp}}(v)=\rho_{\operatorname{Sp}_{2n}}(v)=\sum_{i=1}^n(n+1-i)v_i,$ the half-sum of positive roots of $\operatorname{Sp}_{2n}(\mathbb R)$.
Let $G$ be a connected semisimple real algebraic group. We prove that limit cones vary continuously under deformations of Anosov subgroups of $G$ under a certain convexity assumption, which turns out to be necessary. We apply this result to the notion of sharpness for the action of a discrete subgroup on a non-Riemannian homogeneous space. Finally, we show that, within the space of Anosov representations, the growth indicator, the critical exponents, and the Hausdorff dimension of limit sets (with respect to an appropriate non-Riemannian metric) all vary continuously.
We construct the first example of a Zariski-dense, discrete, non-lattice subgroup $\Gamma_0$ of a higher rank simple Lie group $G$, which is non-tempered in the sense that the quasi-regular representation $L^2(\Gamma_0\backslash G)$ is non-tempered. More precisely, let $n\ge 3$ and let $\Gamma$ be the fundamental group of a closed hyperbolic $n$-manifold that contains a properly embedded totally geodesic hyperplane. We show that there exists a non-empty open subset $\mathcal O$ of $\operatorname{Hom}(\Gamma, \operatorname{SO}(n,2))$ such that for any $\sigma\in \mathcal O$, the subgroup $\sigma(\Gamma)$ is a Zariski-dense and non-tempered Anosov subgroup of $\operatorname{SO}(n,2)$. In addition, the growth indicator of $\sigma(\Gamma)$ is nearly optimal: it almost realizes the supremum of growth indicators among all non-lattice discrete subgroups, a bound imposed by property $(T)$ of $\operatorname{SO}(n,2)$.
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 $P$.
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\'e 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.
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.
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 Teichm\"uller spaces, new upper bounds on the growth indicator, and $L^2$-spectral properties of associated locally symmetric manifolds.
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.
For a given $d$-tuple $\rho=(\rho_1,\dots,\rho_d):\Gamma \to G$ of faithful Zariski dense convex cocompact representations of a finitely generated group $\Gamma$, we study the correlations of length spectra $\{\ell_{\rho_i(\gamma)}\}_{[\gamma]\in[\Gamma]}$ and correlations of displacement spectra $\{\mathsf{d}(\rho_i(\gamma)o,o)\}_{\gamma\in\Gamma}$. We prove that for any interior vector $\mathsf v=(v_1,\dots,v_d)$ in the {\it{spectrum cone}}, there exists $\delta_\rho(\mathsf v) > 0$ such that for any $\varepsilon_1, \dots, \varepsilon_d>0$, there exist $c_1,c_2> 0$ such that \begin{align*} &\#\{[\gamma]\in [\Gamma]: v_iT \le \ell_{\rho_i(\gamma)} \le v_i T+\varepsilon_i, \;1 \le i \le d \} \sim c_1 \frac{e^{\delta_\rho (\mathsf{v})T}}{ T^{{(d+1)}/{2}}};\\ &\#\{\gamma\in \Gamma: v_iT \le \mathsf{d}(\rho_i(\gamma)o,o) \le v_i T+\varepsilon_i, \;1 \le i \le d \} \sim c_2 \frac{e^{\delta_\rho (\mathsf v)T}}{ T^{{(d-1)}/{2}}}. \end{align*} We deduce this result as a special case of our main theorem on the distribution of Jordan projections with holonomies and Cartan projections {\it{in tubes}} of an Anosov subgroup $\Gamma$ of a semisimple real algebraic group $G$. We also show that the growth indicator of $\Gamma$ 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.