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Dongryul M. Kim

Publications and source records attributed to Dongryul M. Kim.

At least 19 recordsLinked to original sources
arXiv2026

Strict concavity of the growth indicator function for relatively Anosov groups

Let $Γ$ be a discrete subgroup of a connected semisimple real algebraic group of higher rank. The growth indicator function $ψ_Γ$ records the directional exponential growth of the Cartan projections of elements of $Γ$ in the positive Weyl chamber $\mathfrak a^+$. We prove that if $Γ$ is a non-elementary relatively Borel Anosov group, then $ψ_Γ$ 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 $ϕ\mapstoδ^ϕ(Γ)$. More generally, for a non-elementary $θ$-transverse group, we prove local $\mathcal C^1$-regularity near every point of the $θ$-Manhattan hypersurface that is positive on the $θ$-limit cone and has a critical gap at infinity. In particular, the $θ$-Manhattan hypersurface is globally $\mathcal C^1$ for relatively $θ$-Anosov groups, and their $θ$-growth indicator functions are strictly concave on non-collinear directions.

math.DG↗
arXiv2026

Determining subgroups via stationary measures

In this paper, we consider random walks on the isometry groups of general metric spaces. Under some mild conditions, we show that if two non-elementary random walks on a discrete subgroup of the isometry group have non-singular stationary measures, then subgroups generated by the random walks are commensurable. This result in particular applies to Gromov hyperbolic spaces and Teichmüller spaces. As a specific application, we prove singularity between stationary measures associated to random walks on different fiber subgroups of the fundamental group of a hyperbolic 3-manifold fibering over the circle.

math.GT↗
arXiv2026

A global shadow lemma for relatively Morse groups in higher rank

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.

math.GT↗
arXiv2026

Classification of horospherical invariant measures in higher rank: The Full Story

In this paper, we classify horospherical invariant Radon measures for Anosov subgroups of arbitrary semisimple real algebraic groups. This generalizes the works of Burger and Roblin in rank one to higher ranks. At the same time, this extends the works of Furstenberg, Veech, and Dani, and a special case of Ratner's theorem for finite-volume homogeneous spaces to infinite-volume Anosov homogeneous spaces. Especially, this resolves the open problems proposed by Landesberg--Lee--Lindenstrauss--Oh and by Oh. Our measure classification is in fact for a more general class of discrete subgroups, including relatively Anosov subgroups with respect to any parabolic subgroups, not necessarily minimal. We also obtain results for their normal subgroups. Our method is rather geometric, not relying on continuous flows or ergodic theorems.

math.DS↗
arXiv2026

Non-arithmeticity of length spectra of subgroups of mapping class groups

In this paper, we prove that every non-elementary subgroup of the mapping class group of a surface has non-arithmetic Teichmüller length spectrum. Namely, Teichmüller translation lengths of its pseudo-Anosov elements generate a dense additive subgroup of $\mathbb{R}$. We prove this by introducing the notion of cross-ratios on $\mathcal{MF}$ and $\mathcal{PMF}$, and studying its geometric and dynamical properties, despite the lack of negatively curved features of the Teichmüller space nor the conformal geometry on $\mathcal{PMF}$.

math.GT↗
arXiv2026

Normal generators for mapping class groups

In this chapter, we discuss normal generators for mapping class groups of surfaces. Especially, we focus on the relation between normal generation of a mapping class with its asymptotic translation lengths on the Teichmüller space and the curve graph of the underlying surface. We also discuss several open questions.

math.GT↗
arXiv2026

Rigidity for Patterson--Sullivan systems with applications to random walks and entropy rigidity

In this paper we introduce Patterson--Sullivan systems, which consist of a group action on a compact metrizable space and a quasi-invariant measure which behaves like a classical Patterson--Sullivan measure. For such systems we prove a generalization of Tukia's measurable boundary rigidity theorem. We then apply this generalization to (1) study the singularity conjecture for Patterson--Sullivan measures (or, conformal densities) and stationary measures of random walks on isometry groups of Gromov hyperbolic spaces, mapping class groups, and discrete subgroups of semisimple Lie groups; (2) prove versions of Tukia's theorem for word hyperbolic groups, Teichmüller spaces, and higher rank symmetric spaces; and (3) in a companion paper prove an entropy rigidity result for Anosov groups with Lipschitz limit sets.

math.GT↗
arXiv2026

Vector-valued horofunction boundaries and Patterson--Sullivan measures

In higher rank, there is a well-studied theory of Patterson--Sullivan measures supported on partial flag manifolds. However, establishing the existence and uniqueness of such measures is a difficult question. In this paper, we develop a theory for Patterson--Sullivan measures supported on certain vector-valued horofunction boundaries of the associated symmetric space, where existence is straightforward. We also introduce a notion of shadows for this compactification and establish a shadow lemma. For transverse groups, we prove uniqueness and ergodicity results.

math.GT↗
arXiv2026

Measurable boundary maps and Patterson--Sullivan measures for non-Borel Anosov groups on the Furstenberg boundary

In this paper we develop a theory for Patterson--Sullivan measures for non-Borel Anosov groups on the Furstenberg boundary. Previously, such a theory has been successfully developed for measures supported on the partial flag manifold associated to the Anosov condition, which coincides with the Furstenberg boundary only under the strongest Anosov condition, Borel Anosov. We establish existence, uniqueness, and ergodicity results for the measures on the Furstenberg boundary under arbitrary Anosov conditions; we show ergodicity of Bowen--Margulis--Sullivan measures on the homogeneous space; and we establish strict convexity results for the critical exponent associated to functionals on the entire Cartan subspace. Using this strict convexity, we establish an entropy rigidity result for Anosov groups with Lipschitz limit set. A key tool we develop is a new sufficient condition for the existence of a measurable boundary map associated to a Zariski dense representation. This result not only applies to Anosov groups, but also to transverse groups, mapping class groups, and discrete subgroups of the isometry groups of Gromov hyperbolic spaces.

math.GT↗
arXiv2025

Ahlfors regularity of Patterson-Sullivan measures of Anosov groups and applications

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.

math.GR↗
arXiv2025

Invariant measures on the space of measured laminations for subgroups of mapping class group

For a non-elementary subgroup of the mapping class group of a surface, we study its invariant Radon measures on the space of measured laminations, by classifying them on the recurrent measured laminations. In particular, given a divergence-type subgroup, we show the uniquely ergodic by explicitly constructing the ergodic measure. This generalizes Lindenstrauss--Mirzakhani's result and Hamenstädt's result for the full mapping class group, in which case the ergodic measure is the Thurston measure. As a special case, we deduce that for a convex cocompact subgroup, every invariant ergodic Radon measure on the space of all measured laminations is either the unique measure on recurrent measured laminations, or a counting measure on the orbit of a non-recurrent measured lamination. Our method is geometric and does not rely on continuous or homogeneous flows on the ambient space or a dynamical system associated with a finite measure space. This leads to a unifying approach for various metric spaces, including Teichmüller spaces and partially $\operatorname{CAT}(-1)$ spaces.

math.DS↗
arXiv2025

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

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 $Γ$, we show that the $Γ$-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.

math.DS↗
arXiv2025

Rigidity of Kleinian groups via self-joinings: measure theoretic criterion

Let $n, m\ge 2$. Let $Γ<\text{SO}^\circ(n+1,1)$ be a Zariski dense convex cocompact subgroup and $Λ\subset\mathbb{S}^n$ be its limit set. Let $ρ: Γ\to \text{SO}^\circ(m+1,1)$ be a Zariski dense convex cocompact faithful representation and $f:Λ\to \mathbb{S}^{m}$ the $ρ$-boundary map. Let $$Λ_f:= \bigcup \left\{ C \cap Λ: \begin{matrix} C \subset \mathbb{S}^n \text{ is a circle such that} \\ f(C \cap Λ) \text{ is contained in a proper sphere } \text{in } \mathbb{S}^m \end{matrix} \right\}.$$ When there exists at least one $Λ$-doubly stable circle in $\mathbb{S}^n$ (e.g., $Ω=\mathbb{S}^n-Λ$ is disconnected), we prove the following dichotomy: $$\text{either}\quad Λ_f= Λ\quad \text{ or } \quad \mathcal{H}^δ(Λ_f) =0,$$ where $\mathcal{H}^δ$ is the Hausdorff measure of dimension $δ=\dim_H Λ$. Moreover, in the former case, we have $n=m$ and $ρ$ is a conjugation by a Möbius transformation on $\mathbb{S}^n$. 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 $Γ_ρ=(\operatorname{id} \times ρ)(Γ) < \text{SO}^\circ(n+1,1)\times \text{SO}^\circ(m+1,1)$. We also obtain an analogous theorem for any divergence-type subgroup.

math.GT↗
arXiv2025

Conformal measure rigidity and ergodicity of horospherical foliations

In this paper, we prove two main theorems: conformal measure rigidity and ergodicity of horospherical foliations, especially in higher rank. Both theorems are new even for relatively Anosov groups. First, we establish a higher rank extension of rigidity theorems of Sullivan, Tukia, Yue, and Kim-Oh for representations of rank one discrete subgroups of divergence type, in terms of the push-forwards of conformal measures via boundary maps. We consider a certain class of higher rank discrete subgroups, which we call hypertransverse subgroups. It includes all rank one discrete subgroups, Anosov subgroups, relatively Anosov subgroups, and their subgroups. Our proof of the rigidity theorem generalizes the idea of Kim-Oh to self-joinings of higher rank hypertransverse subgroups, overcoming the lack of $\mathrm{CAT}(-1)$ geometry on symmetric spaces. In contrast to the work of Sullivan, Tukia, and Yue, our argument is closely related to the study of horospherical foliations. We also show the ergodicity of horospherical foliations with respect to Burger-Roblin measures. This generalizes the classical work of Hedlund, Burger, and Roblin in rank one and of Lee-Oh for Borel Ansov subgroups in higher rank. Moreover, we describe the ergodic decomposition of Burger-Roblin measures and Bowen-Margulis-Sullivan measures when a given parabolic subgroup is minimal.

math.GT↗
arXiv2025

A note on the singularity conjecture for infinite covolume discrete subgroups

We consider random walks on semisimple Lie groups where the support of the step distribution generates (as a group) a Zariski dense discrete subgroup of infinite covolume. When the semisimple Lie group has property (T), we show that the stationary measure on the Furstenberg boundary is singular to the Lebesgue measure class. This result does not require any condition on the moment or symmetry of the step distribution. When the semisimple Lie group has rank one and the step distribution has a finite first moment, we again show that the stationary measure on the Furstenberg boundary is singular to the Lebesgue measure class. For general semisimple Lie groups, we also obtain a sufficient condition for the singularity of the stationary measure and a general Patterson-Sullivan measure.

math.GT↗
arXiv2025

Horocycles in hyperbolic 3-manifolds with round Sierpiński limit sets

Let M be a geometrically finite hyperbolic 3-manifold whose limit set is a round Sierpiński gasket, i.e. M is geometrically finite and acylindrical with a compact, totally geodesic convex core boundary. In this paper, we classify orbit closures of the 1-dimensional horocycle flow on the frame bundle of M. As a result, the closure of a horocycle in M is a properly immersed submanifold. This extends the work of McMullen-Mohammadi-Oh where M is further assumed to be convex cocompact.

math.DS↗
arXiv2025

Simple length spectra as moduli for hyperbolic surfaces and rigidity of length identities

In this article, we revisit classical length identities enjoyed by simple closed curves on hyperbolic surfaces. We state and prove the rigidity of such identities over Teichmüller spaces. Due to this rigidity, certain collections of simple closed curves which minimally intersect are characterized on generic hyperbolic surfaces by their lengths. As an application, we construct a meagre set $V$ in the Teichmüller space of a topological orientable surface $S$, possibly of infinite type. Then the isometry class of a (Nielsen-convex) hyperbolic structure on $S$ outside $V$ is characterized by its unmarked simple length spectrum. Namely, we show that the simple length spectra can be used as moduli for generic hyperbolic surfaces. In the case of compact surfaces, an analogous result using length spectra was obtained by Wolpert.

math.GT↗