About

Hee Oh is the Abraham Robinson Professor of Mathematics at Yale University, specializing in group actions and dynamics. Her research focuses on the dynamics and rigidity of discrete subgroups in higher rank Lie groups, Anosov groups, and hyperbolic manifolds. She has made significant contributions to the understanding of limit cones, critical exponents, and growth indicators of coamenable normal subgroups, as well as the ergodic theory and geometric structures of hyperbolic spaces. Her work often explores the interplay between geometric group theory, ergodic theory, and number theory, with applications to counting problems, spectral theory, and rigidity phenomena. She has collaborated extensively with other mathematicians on topics such as conformal measures, horospherical actions, and the dynamics of Kleinian groups, contributing to the advancement of knowledge in these areas through numerous publications and preprints.

Research topics

• Combinatorics
• Composite material
• Geometry
• Mathematics
• Pure mathematics
• Mathematical analysis
• Materials science
• Statistics
• Chemical engineering
• Chemistry

Selected publications

• Multiple Correlations of Spectra for Higher Rank Anosov Representations

  International Mathematics Research Notices · 2025-03-01 · 1 citations

  articleSenior author

  Abstract 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.

  PublisherDOI

• Ergodic dichotomy for subspace flows in higher rank

  Communications of the American Mathematical Society · 2025-02-19 · 3 citations

  articleOpen access

  In this paper, we study the ergodicity of a one-parameter diagonalizable subgroup of a connected semisimple real algebraic group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , or more generally, with transverse subgroups of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <italic>Borel</italic> Anosov subgroups.

  PublisherOA PDFDOI

• Deformations of Anosov subgroups: Limit cones and growth indicators

  Journal of the London Mathematical Society · 2025-09-01

  articleOpen accessSenior author

  Abstract Let be a connected semisimple real algebraic group. We prove that limit cones vary continuously under deformations of Anosov subgroups of 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.

  PublisherOA PDFDOI

• Rigidity of Kleinian groups via self‐joinings: measure theoretic criterion

  Journal of Topology · 2025-09-01

  articleSenior authorCorresponding

  Abstract Let . Let be a Zariski dense convex cocompact subgroup and be its limit set. Let be a Zariski dense convex cocompact faithful representation and the ‐boundary map. Let When there exists at least one ‐doubly stable circle in (e.g., is disconnected), we prove the following dichotomy: where is the Hausdorff measure of dimension . Moreover, in the former case, we have and is a conjugation by a Möbius transformation on . Our proof uses ergodic theory for directional diagonal flows and conformal measure theory of discrete subgroups of higher rank semisimple Lie groups, applied to the self‐joining subgroup . We also obtain an analogous theorem for any divergence‐type subgroup.

  PublisherDOI

• Fractal closures of geodesic planes in Hitchin manifolds

  ArXiv.org · 2025-09-22

  preprintOpen accessSenior author

  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 $Γ&lt; \mathrm{SL}_3(\mathbb{R})$ such that the Hitchin manifold $Γ\backslash \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}(3)$ contains immersed floating geodesic planes whose closures are fractal, with non-integer Hausdorff dimensions accumulating at $2$. Moreover, $Γ$ can be chosen inside $\mathrm{SL}_3(\mathbb{Z})$.

  PublisherOA PDFDOI

• Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 2025-07-02

  articleSenior author

  Abstract For a geometrically finite Kleinian group Γ, 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–Peigné 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 Γ 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 Γ 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.

  PublisherDOI

• Counting totally real units and eigenvalue patterns in $\rm{SL}_n(\mathbb Z)$ and $\rm{Sp}_{2n}(\mathbb Z)$ in thin tubes

  ArXiv.org · 2025-05-19

  preprintOpen access1st authorCorresponding

  For a vector $v=(v_1,\dots ,v_n)$ with $v_1&gt;\cdots&gt;v_n$ 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 $\mathsf E_n(v)= ρ_{\operatorname{SL}_n}(v)=\sum_{i=1}^{n-1}(n-i)\,v_i,$ 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(ρ_{\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 $\exp\!\bigl(ρ_{\operatorname{SL}_n}(v)T\bigr)$ for the number of $\operatorname{SL}_n(\mathbb Z)$-conjugacy classes with a prescribed eigenvalue data; we also obtain an upper bound of order $\exp\!\bigl(2ρ_{\operatorname{SL}_n}(v)T\bigr)$. A parallel argument for the symplectic lattice $\operatorname{Sp}_{2n}(\mathbb Z)$, taken in the symmetric direction $v=(v_1,\dots ,v_n,-v_n,\dots ,-v_1),\quad v_1&gt;\cdots&gt;v_n&gt;0,$ shows that $\mathsf E_{2n}^{\operatorname{Sp}}(v)=ρ_{\operatorname{Sp}_{2n}}(v)=\sum_{i=1}^n(n+1-i)v_i,$ the half-sum of positive roots of $\operatorname{Sp}_{2n}(\mathbb R)$.

  PublisherOA PDFDOI

• Properness and finiteness of totally geodesic submanifolds in the convex core

  ArXiv.org · 2025-11-12

  preprintOpen accessSenior author

  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.

  PublisherOA PDFDOI

• Properly discontinuous actions, growth indicators, and conformal measures for transverse subgroups

  Mathematische Annalen · 2025-09-23 · 1 citations

  article
  PublisherDOI

• Dynamics and Rigidity through the Lens of Circles

  ArXiv.org · 2025-10-12

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

Recent grants

• Thin Groups and Dynamics

  NSF · $600k · 2014–2019

• Dynamics and Kleinian Groups

  NSF · $531k · 2019–2025

• Equidistribution on homogeneous spaces for orbits of discrete groups beyond lattices

  NSF · $204k · 2011–2013

• Counting and Equidistribution on homogeneous spaces

  NSF · $309k · 2006–2012

• Equidistribution problems and semisimple Lie groups

  NSF · $143k · 2003–2007

Frequent coauthors

• Minju Lee

  34 shared

• Amir Mohammadi

  30 shared

• Nimish A. Shah

  25 shared

• Alex Kontorovich

  18 shared

• Dongryul M. Kim

  16 shared

• Yves Benoist

  13 shared

• S. F. Edwards

  13 shared

• Dale Winter

  Winchester Hospital

  10 shared

Labs

• Hee Oh LabPI

Education

• Ph. D, Mathematics

  Yale

  1997