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Sungwoon Kim

Publications and source records attributed to Sungwoon Kim.

At least 19 recordsLinked to original sources
arXiv2026

Local $C^0$-semi-rigidity of meandering-hyperbolic actions

Meandering hyperbolicity, introduced by Kapovich, Kim, and Lee, extends classical hyperbolicity beyond the setting of word-hyperbolic groups. In this paper, we prove that every meandering-hyperbolic action is locally semi-rigid in the $C^0$ topology. This extends the previously established stability theory in the Lipschitz $C^0$ topology to the full $C^0$ topology. Consequently, we recover the $C^0$-local semi-rigidity of both boundary actions of word-hyperbolic groups and cocompact lattices in semisimple Lie groups from a single dynamical principle.

math.GT↗
arXiv2024

Mitigating Adversarial Perturbations for Deep Reinforcement Learning via Vector Quantization

Recent studies reveal that well-performing reinforcement learning (RL) agents in training often lack resilience against adversarial perturbations during deployment. This highlights the importance of building a robust agent before deploying it in the real world. Most prior works focus on developing robust training-based procedures to tackle this problem, including enhancing the robustness of the deep neural network component itself or adversarially training the agent on strong attacks. In this work, we instead study an input transformation-based defense for RL. Specifically, we propose using a variant of vector quantization (VQ) as a transformation for input observations, which is then used to reduce the space of adversarial attacks during testing, resulting in the transformed observations being less affected by attacks. Our method is computationally efficient and seamlessly integrates with adversarial training, further enhancing the robustness of RL agents against adversarial attacks. Through extensive experiments in multiple environments, we demonstrate that using VQ as the input transformation effectively defends against adversarial attacks on the agent's observations.

cs.LG↗
arXiv2022

Structural stability of meandering-hyperbolic group actions

In his 1985 paper Sullivan sketched a proof of his structural stability theorem for differentiable group actions satisfying certain expansion-hyperbolicity axioms. In this paper we relax Sullivan's axioms and introduce a notion of "meandering hyperbolicity" for group actions on geodesic metric spaces. This generalization is substantial enough to encompass actions of certain non-hyperbolic groups, such as actions of "uniform lattices" in semisimple Lie groups on flag manifolds. At the same time, our notion is sufficiently robust and we prove that meandering-hyperbolic actions are still structurally stable. We also prove some basic results on meandering-hyperbolic actions and give other examples of such actions.

math.GR↗
arXiv2022

A note on the integrality of volumes of representations

Let $Γ$ be a torsion-free, non-uniform lattice in $\mathrm{SO}(2n,1)$. We present an elementary, combinatorial-geometrical proof of a theorem of Bucher, Burger, and Iozzi which states that the volume of a representation $ρ:Γ\to\mathrm{SO}(2n,1)$, properly normalized, is an integer if $n$ is greater than or equal to $2$.

math.GT↗
arXiv2022

Weakly positive and directed Anosov representations

Given a finitely generated group $Γ$, a directed graph $Λ$, and a map $R:Λ\toΓ$, we introduce the notion of an $(R,Λ)$-directed Anosov representation. This is a weakening of the notion of Anosov representations. Our main theorem gives a procedure to construct $(R,Λ)$-directed Anosov representations using Fock-Goncharov positivity. As an application of our main theorem, we construct large families of primitive stable representations from $F_2$ to $\mathrm{PGL}(V)$, including non-discrete and non-faithful examples.

math.GT↗
arXiv2020

Primitive stable representations in higher rank semisimple Lie groups

We study primitive stable representations of free groups into higher rank semisimple Lie groups and their properties. Let $Σ$ be a compact, connected, orientable surface (possibly with boundary) of negative Euler characteristic. We first verify the $σ_{mod}$-regularity for convex projective structures and positive representations. Then we show that the holonomies of convex projective structures and positive representations on $Σ$ are all primitive stable if $Σ$ has one boundary component.

math.GT↗
arXiv2016

On boundary maps of Anosov representations of a surface group to SL(3,R)

We prove that Anosov representations from a surface group to SL(3,R) are uniquely determined by their boundary maps if and only if they do not factor over a completely reducible representation. Furthermore we discuss representations not distinguished by their boundary maps, and we give a lower bound for the number of mapping class orbits of components of the space of Anosov representations.

math.GT↗
arXiv2015

Simplicial volume, Barycenter method, and Bounded cohomology

We show that codimension one dimensional Jacobian of the barycentric straightening map is uniformly bounded for most of the higher rank symmetric spaces. As a consequence, we prove that the locally finite simplicial volume of most $\mathbb Q$-rank $1$ locally symmetric spaces is positive, which has been open for many years. Finally we improve the degree theorem for $\mathbb Q$-rank $1$ locally symmetric spaces of Connell and Farb. We also address the issue of surjectivity of the comparison map in real rank $2$ case.

math.GT↗
arXiv2015

Complex and Quaternionic hyperbolic Kleinian groups with real trace fields

Let $Γ$ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of $Γ$ is contained in $\mathbb R$, $Γ$ preserves a totally geodesic submanifold of constant negative sectional curvature. Furthermore if $Γ$ is irreducible, $Γ$ is a Zariski dense irreducible discrete subgroup of SO(n,1) up to conjugation. This is an analog of a theorem of Maskit for general semisimple Lie groups of rank $1$.

math.GT↗
arXiv2014

On the equivalence of the definitions of volume of representations

Let G be a rank 1 simple Lie group and M be a connected orientable aspherical tame manifold. Assume that each end of M has amenable fundamental group. There are several definitions of volume of representations of the fundamental group of M into G. We give a new definition of volume of representations and furthermore, show that all definitions so far are equivalent.

math.GT↗
arXiv2013

On deformation spaces of nonuniform hyperbolic lattices

Let $Γ$ be a nonuniform lattice acting on real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of $Γ$ in SO(n,1). Furthermore, in dimensions 2 and 3, there is a semialgebraic subset of the representation variety such that the volume of a representation is constant on connected components of the semialgebraic subset. Our approach gives a new proof of the local rigidity theorem for nonuniform hyperbolic lattices and the analogue of Soma's theorem, which shows that the number of orientable hyperbolic manifolds dominated by a closed, connected, orientable 3-manifold is finite, for noncompact 3-manifolds.

math.GT↗
arXiv2013

Simplicial volume of compact manifolds with amenable boundary

Let $M$ be the interior of a connected, oriented, compact manifold $V$ of dimension at least 2. If each path component of $\partial V$ has amenable fundamental group, then we prove that the simplicial volume of $M$ is equal to the relative simplicial volume of $V$ and also to the geometric (Lipschitz) simplicial volume of any Riemannian metric on $M$ whenever the latter is finite. As an application we establish the proportionality principle for the simplicial volume of complete, pinched negatively curved manifolds of finite volume.

math.GT↗
arXiv2012

On the limit set of Anosov representations

We study the limit set of discrete subgroups arising from Anosov representations. Specially we study the limit set of discrete groups arising from strictly convex real projective structures and Anosov representations from a finitely generated word hyperbolic group into a semisimple Lie group.

math.GT↗
arXiv2012

Homological and Bloch invariants for Q-rank one spaces and flag structures

We use group homology to define invariants in algebraic K-theory and in an analogue of the Bloch group for Q-rank one lattices and for some other geometric structures. We also show that the Bloch invariants of CR structures and of flag structures can be recovered by a fundamental class construction.

math.GT↗