arXiv subjects

Dongryul M. Kim

Publications and source records attributed to Dongryul M. Kim.

32 records · Page 2Linked to original sources

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.

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arXiv2025

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

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.

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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.

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arXiv2024

On the asymptotic translation lengths on the sphere complexes and the generalized fibered cone

In this paper, we study the asymptotic translation lengths on the sphere complexes of monodromies of a manifold fibered over the circle. Given a compact mapping torus, we define a cone in the first cohomology which we call the generalized fibered cone, and show that every primitive integral element gives a fibration over the circle. Moreover, we prove that the generalized fibered cone is a rational slice of Fried's cone, which is defined as the dual of homological directions, an analogue of Thurston's fibered cone. As a consequence of our description of the generalized fibered cone, we provide each proper subcone of the generalized fibered cone with a uniform upper bound for asymptotic translation lengths of monodromies on sphere complexes of fibers in the proper subcone. Our upper bound is purely in terms of the dimension of the proper subcone. We also deduce similar estimates for asymptotic translation lengths of some mapping classes on finite graphs constructed in the works of Dowdall--Kapovich--Leininger, measured on associated free-splitting complexes and free-factor complexes. Moreover, as an application of our result, we prove that the asymptote for the minimal asymptotic translation length of the genus $g$ handlebody group on the disk complex is $1/g^2$, the same as the one on the curve complex.

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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.

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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.

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arXiv2023

Minimal asymptotic translation lengths on curve complexes and homology of mapping tori

Let $S_g$ be a closed orientable surface of genus $g > 1$. Consider the minimal asymptotic translation length $L_{\mathcal{T}}(k, g)$ on the Teichmüller space of $S_g$, among pseudo-Anosov mapping classes of $S_g$ acting trivially on a $k$-dimensional subspace of $H_1(S_g)$, $0 \le k \le 2g$. The asymptotics of $L_{\mathcal{T}}(k, g)$ for extreme cases $k = 0, 2g$ have been shown by several authors. Jordan Ellenberg asked whether there is a lower bound for $L_{\mathcal{T}}(k, g)$ interpolating the known results on $L_{\mathcal{T}}(0, g)$ and $L_{\mathcal{T}}(2g, g)$, which was affirmatively answered by Agol, Leininger, and Margalit. In this paper, we study an analogue of Ellenberg's question, replacing Teichmüller spaces with curve complexes. We provide lower and upper bound on the minimal asymptotic translation length $L_{\mathcal{C}}(k, g)$ on the curve complex, whose lower bound interpolates the known results on $L_{\mathcal{C}}(0, g)$ and $L_{\mathcal{C}}(2g, g)$. Finally, for each $g$, we construct a non-Torelli pseudo-Anosov $f_g \in \operatorname{Mod}(S_g)$ which does not normally generates $\operatorname{Mod}(S_g)$ and so that the asymptotic translation length of $f_g$ on curve complexes decays more quickly than a constant multiple of $1/g$ as $g \to \infty$. From this, we provide a restriction on how small the asymptotic translation lengths on curve complexes should be if the similar phenomenon as in the work of Lanier and Margalit on Teichmüller spaces holds for curve complexes.

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arXiv2023

Linear growth of translation lengths of random isometries on Gromov hyperbolic spaces and Teichmüller spaces

We investigate the translation lengths of group elements that arise in random walks on the isometry groups of Gromov hyperbolic spaces. In particular, without any moment condition, we prove that non-elementary random walks exhibit at least linear growth of translation lengths. As a corollary, almost every random walk on mapping class groups eventually becomes pseudo-Anosov and almost every random walk on $\mathrm{Out}(F_n)$ eventually becomes fully irreducible. If the underlying measure further has finite first moment, then the growth rate of translation lengths is equal to the drift, the escape rate of the random walk. We then apply our technique to investigate the random walks induced by the action of mapping class groups on Teichm{ü}ller spaces. In particular, we prove the spectral theorem under finite first moment condition, generalizing a result of Dahmani and Horbez.

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arXiv2023

Reducible normal generators for mapping class groups are abundant

In this article, we study the normal generation of the mapping class group. We first show that a mapping class is a normal generator if its restriction on the invariant subsurface normally generates the (pure) mapping class group of the subsurface. As an application, we provided a criterion for reducible mapping classes to normally generate the mapping class groups in terms of its asymptotic translation lengths on Teichmüller spaces. This is an analogue to the work of Lanier-Margalit dealing with pseudo-Anosov normal generators.

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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}^+$, $$ψ_Γ(v) \le \min_{1\le i\le d-1} (t_i -t_{i+1}). $$

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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 $(\mathrm{id} \times ρ)(Γ)<\mathrm{PSL}_2(\mathbb{C})\times \mathrm{PSL}_2(\mathbb{C})$.

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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 $Γ$.

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arXiv2021

Topological entropy of pseudo-Anosov maps from a typical Thurston's construction

In this paper, we develop a way to extract information about a random walk associated with a typical Thurston's construction. We first observe that a typical Thurston's construction entails a free group of rank 2. We also present a proof of the spectral theorem for random walks associated with Thurston's construction that have finite second moment with respect to the Teichmüller metric. Its general case was remarked by Dahmani and Horbez. Finally, under a hypothesis not involving moment conditions, we prove that random walks eventually become pseudo-Anosov. As an application, we first discuss a random analogy of Kojima and McShane's estimation of the hyperbolic volume of a mapping torus with pseudo-Anosov monodromy. As another application, we discuss non-probabilistic estimations of stretch factors from Thurston's construction and the powers for Salem numbers to become the stretch factors of pseudo-Anosovs from Thurston's construction.

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arXiv2020

On the surjectivity of the Symplectic representation of the mapping class group

In this note, we study the symplectic representation of the mapping class group. In particular, we discuss the surjecivity of the representation restricted to certain mapping classes. It is well-known that the representation itself is surjective. In fact the representation is still surjective after restricting on pseudo-Anosov mapping classes. However, we show that the surjectivity does not hold when the representation is restricted on orientable pseudo-Anosovs, even after reducing its codomain to integer symplectic matrices with a bi-Perron leading eigenvalue. In order to prove the non-surjectivity, we explicitly construct an infinite family of symplectic matrices with a bi-Perron leading eigenvalue which cannot be obtained as the symplectic representation of an orientable pseudo-Anosov mapping class.

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