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Suhyoung Choi

Publications and source records attributed to Suhyoung Choi.

At least 19 recordsLinked to original sources
arXiv2026

The stability of Margulis space-times with parabolic holonomy elements

Let $E$ be a flat Lorentzian space of signature $(2,1)$. A Margulis space-time is a noncompact complete flat Lorentzian $3$-manifold $E/\Gamma$, where the holonomy group $\Gamma$ is a free group of rank $g\geq 2$ acting freely and properly discontinuously by isometries. We consider the case where $\Gamma$ contains a parabolic element. We show that sufficiently small deformations of $\Gamma$ still act properly discontinuously on $E$ provided their linear parts are Fuchsian; moreover, the number of conjugacy classes of parabolic elements may increase or decrease under deformation. Our proof combines our previous compactification of $E/\Gamma$ relative to parabolic holonomy elements with a partial generalization of the work of Carri\`ere. However, this result depends only on the parts on parabolic actions of our earlier work. We believe that the shortness of the proof of this openness result is of independent interest.

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arXiv2025

Partially hyperbolic flows on flat vector bundles with an application to complete affine manifolds

Let $N$ be a manifold of dimension $m$ with a flat vector bundle given by a representation $\rho:\pi_1(N) \rightarrow \mathrm{GL}(n, \mathbf{R})$ where $\pi_1(N)$ is finitely generated. The holonomy group $\rho$ is a $k$-partially hyperbolic holonomy representation if the flat bundle pulled back over the unit tangent bundle of a sufficiently large compact submanifold of $N$ splits into expanding, neutral, and contracting subbundles along the geodesic flow, where the expanding and contracting subbundles are $k$-dimensional with $k < n/2$. Suppose that each element of $\rho(\pi_1(N))$ has an eigenvalue of norm $1$, or, alternatively, $\rho$ has some singular values of subexponential growth in terms of word length. We show that $\rho$ is a $P$-Anosov representation for a parabolic subgroup $P$ of $\mathrm{GL}(n, \mathbf{R})$ if and only if $\rho$ is a partially hyperbolic representation. We are going to primarily employ representation theory techniques. As an application, we will show that the equivalence holds when $N$ is a complete affine $n$-manifold, and $\rho$ is a linear part of the holonomy representation. This had never been done over the full general linear group.

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arXiv2025

The smoothness of the real projective deformation spaces of orderable Coxeter 3-polytopes

A Coxeter polytope is a convex polytope in a real projective space equipped with linear reflections in its facets, such that the orbits of the polytope under the action of the group generated by the linear reflections tessellate a convex subset in the real projective space. Vinberg proved that the group generated by these reflections acts properly discontinuously on the interior of this convex subset, thus inducing a natural orbifold structure on the polytope. In this paper, we consider labeled combinatorial polytopes $\mathcal{G}$ associated to such orbifolds, and study the deformation space $\mathcal{C} (\mathcal{G})$ of Coxeter polytopes realizing $\mathcal{G}$. We prove that if $\mathcal G$ is orderable and of normal type, and its underlying combinatorial polytope is not a cone over a polygon, then the deformation space $\mathcal C(\mathcal G)$ is a smooth manifold. This result is obtained by analyzing a natural map from $\mathcal C(\mathcal G)$ to a smooth manifold called the realization space.

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arXiv2024

Complete affine manifolds with Anosov holonomy groups

Let $N$ be a complete affine manifold $\mathbb{A}^n/Γ$ of dimension $n$, where $Γ$ is an affine transformation group acting on the complete affine space $\mathbb{A}^n$, and $K(Γ, 1)$ is realized as a finite CW-complex. $N$ has a $k$-partially hyperbolic holonomy group if the tangent bundle pulled back over the unit tangent bundle of a sufficiently large compact subset splits into expanding, neutral, and contracting subbundles along the geodesic flow, where the expanding and contracting subbundles are $k$-dimensional with $k < n/2$. In part 1, we will demonstrate that the complete affine $n$-manifold has a $P$-Anosov linear holonomy group for a parabolic subgroup $P$ of $\mathrm{GL}(n, \mathbb{R})$ if and only if it has a partially hyperbolic linear holonomy group. This had never been done over the full general linear group before this paper. Part 1 will primarily employ representation theory techniques. In part 2, we demonstrate that if the holonomy group is partially hyperbolic of index $k$, where $k < n/2$, then $\mathrm{cd}(Γ) \leq n-k$. Moreover, if a finitely-presented affine group $Γ$ acts properly discontinuously and freely on $\mathbb{A}^n$ with a $k$-Anosov linear subgroup for $k \leq n/2$, then $\mathrm{cd}(Γ) \leq n-k$. Also, there exists a compact collection of $n-k$-dimensional affine subspaces where $Γ$ acts. The techniques employed here mostly stem from the coarse geometry theory. Canary and Tsouvalis previously proved the same result using the powerful method of Bestvina and Mess for word hyperbolic groups; however, our approach differs in that our method projects the holonomy cover to a stable affine subspace, and we plan to generalize to relative Anosov groups.

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arXiv2024

Deformations of Margulis space-times with parabolics

Let $E$ be a flat Lorentzian space of signature $(2, 1)$. A Margulis space-time is a noncompact complete Lorentz flat $3$-manifold $E/Γ$ with a free isometry group $Γ$ of rank $g \geq 2$. We consider the case when $Γ$ contains a parabolic element. We show that sufficiently small deformations of $Γ$ still act properly on $E$. We use our previous work showing that $E/Γ$ can be compactified relative to a union of solid tori and some old idea of Carrière in his famous work. We will show that the there is also a decomposition of $E/Γ$ by crooked planes that are disjoint and embedded in a generalized sense. These can be perturbed so that $E/Γ$ decomposes into cells. This partially affirms the conjecture of Charette-Drumm-Goldman.

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arXiv2023

Complete affine manifolds with Anosov holonomy groups II: partially hyperbolic holonomy and cohomological dimensions

Let $N$ be a complete affine manifold $A^n/Γ$ of dimension $n$ where $Γ$ is an affine transformation group and $K(Γ, 1)$ is realized as a finite CW-complex. $N$ has a partially hyperbolic holonomy group if the tangent bundle pulled over the unit tangent bundle over a sufficiently large compact part splits into expanding, neutral, and contracting subbundles along the geodesic flow. We show that if the holonomy group is partially hyperbolic of index $k$, $k < n/2$, then $\mathrm{cd}(Γ) \leq n-k$. Moreover, if a finitely-presented affine group $Γ$ acts on $A^n$ properly discontinuously and freely with the $k$-Anosov linear group for $k \leq n/2$, then $\mathrm{cd}(Γ) \leq n-k$. Also, there exists a compact collection of $n-k$-dimensional affine subspaces where $Γ$ acts on. The techniques here are mostly from coarse geometry.

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arXiv2023

The volumes of the Hitchin-Riemann moduli spaces are infinite

In this study, we prove that the actions of the mapping class groups on a large range of higher Teichm\"uller spaces with a rank of at least two possess infinite Atiyah-Bott-Goldman covolume. This result encompasses $\mathsf{G}$-Hitchin components of a higher rank split real form $\mathsf{G}$ and each component of the space of $\mathsf{Sp}_{2n}(\mathbb{R})$-maximal representations where $n \geq 2$. To achieve this outcome, we employ Goldman flows to find an infinite series of subsets of identical volume, the images of which in the quotient space are all mutually disjoint.

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arXiv2022

$\mathbf{RP}^n \# \mathbf{RP}^n$ and some others admit no real projective structure

A manifold $M$ possesses a real projective structure if it has an atlas consisting of charts mapping to $\mathbf{S}^n$, where the transition maps lie in $\mathrm{SL}_\pm(n+1, \mathbf{R})$. In this context, we present a concise proof demonstrating that $\mathbf{RP}^n\#\mathbf{RP}^n$ and a few other manifolds do not possess a real projective structure when $n\geq3$. Notably, our proof is shorter than those provided by Cooper-Goldman for $n=3$ and \c{C}oban for $n\geq 4$. To do this, we reprove the classification of closed real projective manifolds with infinite-cyclic holonomy groups by Benoist due to a small error. We will leverage the concept of the octantizability of real projective manifolds with nilpotent holonomy groups, as introduced by Benoist and Smillie, which serves as a powerful tool.

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arXiv2022

Tameness of Margulis space-times with parabolics

Let $\mathbf{E}$ be a flat Lorentzian space of signature $(2, 1)$. A Margulis space-time is a noncompact complete flat Lorentzian $3$-manifold $\mathbf{E}/Γ$ with a free holonomy group $Γ$ of rank $\mathbf{g}, \mathbf{g} \geq 2$. We consider the case when $Γ$ contains a parabolic element. We obtain a characterization of proper $Γ$-actions in terms of Margulis and Drumm-Charette invariants. We show that $\mathbf{E}/Γ$ is homeomorphic to the interior of a compact handlebody of genus $\mathbf{g}$ generalizing our earlier result. Also, we obtain a bordification of the Margulis space-time with parabolics by adding a real projective surface at infinity giving us a compactification as a manifold relative to parabolic end neighborhoods. Our method is to estimate the translational parts of the affine transformation group and use some $3$-manifold topology.

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arXiv2022

Deformation spaces of Coxeter truncation polytopes

A convex polytope $P$ in the real projective space with reflections in the facets of $P$ is a Coxeter polytope if the reflections generate a subgroup $Γ$ of the group of projective transformations so that the $Γ$-translates of the interior of $P$ are mutually disjoint. It follows from work of Vinberg that if $P$ is a Coxeter polytope, then the interior $Ω$ of the $Γ$-orbit of $P$ is convex and $Γ$ acts properly discontinuously on $Ω$. A Coxeter polytope $P$ is $2$-perfect if $P \smallsetminus Ω$ consists of only some vertices of $P$. In this paper, we describe the deformation spaces of $2$-perfect Coxeter polytopes $P$ of dimension $d \geqslant 4$ with the same dihedral angles when the underlying polytope of $P$ is a truncation polytope, i.e. a polytope obtained from a simplex by successively truncating vertices. The deformation spaces of Coxeter truncation polytopes of dimension $d = 2$ and $d = 3$ were studied respectively by Goldman and the third author.

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arXiv2022

Symplectic coordinates on the deformation spaces of convex projective structures on 2-orbifolds

Let $\mathcal{O}$ be a closed orientable 2-orbifold of negative Euler characteristic. Huebschmann constructed the Atiyah-Bott-Goldman type symplectic form $ω$ on the deformation space $\mathcal{C}(\mathcal{O})$ of convex projective structures on $\mathcal{O}$. We show that the deformation space $\mathcal{C}(\mathcal{O})$ of convex projective structures on $\mathcal{O}$ admits a global Darboux coordinates system with respect to $ω$. To this end, we show that $\mathcal{C}(\mathcal{O})$ can be decomposed into smaller symplectic spaces. In the course of the proof, we also study the deformation space $\mathcal{C}(\mathcal{O})$ for an orbifold $\mathcal{O}$ with boundary and construct the symplectic form on the deformation space of convex projective structures on $\mathcal{O}$ with fixed boundary holonomy.

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arXiv2019

Symplectic coordinates on $\mathrm{PSL}_3(\mathbb{R})$-Hitchin components

Goldman parametrizes the $\mathrm{PSL}_3(\mathbb{R})$-Hitchin component of a closed oriented hyperbolic surface of genus $g$ by $16g-16$ parameters. Among them, $10g-10$ coordinates are canonical. We prove that the $\mathrm{PSL}_3(\mathbb{R})$-Hitchin component equipped with the Atiyah-Bott-Goldman symplectic form admits a global Darboux coordinate system such that the half of its coordinates are canonical Goldman coordinates. To this end, we show a version of the action-angle principle and the Zocca-type decomposition formula for the symplectic form of H. Kim and Guruprasad-Huebschmann-Jeffrey-Weinstein given to symplectic leaves of the Hitchin component.

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arXiv2018

Convex and concave decompositions of affine $3$-manifolds

A (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an affine space $\mathbb{R}^3$ with transition maps in the affine transformation group $\mathrm{Aff}(\mathbb{R}^3)$. We will show that a connected closed affine $3$-manifold is either an affine Hopf $3$-manifold or decomposes canonically to concave affine submanifolds with incompressible boundary, toral $π$-submanifolds and $2$-convex affine manifolds, each of which is an irreducible $3$-manifold. It follows that if there is no toral $π$-submanifold, then $M$ is prime. Finally, we prove that if a closed affine manifold is covered by a domain in $\mathbb{R}^{n}$, then $M$ is irreducible or is an affine Hopf manifold.

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arXiv2018

Convex projective generalized Dehn filling

For $d=4, 5, 6$, we exhibit the first examples of complete finite volume hyperbolic $d$-manifolds $M$ with cusps such that infinitely many $d$-orbifolds $M_{m}$ obtained from $M$ by generalized Dehn filling admit properly convex real projective structures. The orbifold fundamental groups of $M_m$ are Gromov-hyperbolic relative to a collection of subgroups virtually isomorphic to $\mathbb{Z}^{d-2}$, hence the images of the developing maps of the projective structures on $M_m$ are new examples of divisible properly convex domains of the projective $d$-space which are not strictly convex, in contrast to the previous examples of Benoist.

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arXiv2017

Topological tameness of Margulis spacetimes

We show that Margulis spacetimes without parabolic holonomy are topologically tame. A Margulis spacetime is the quotient of the $3$-dimensional Minkowski space by a free proper isometric action of the free group of rank $\geq 2$. We will use our particular point of view that the Margulis spacetime is a manifold-with-boundary with an $\mathbb{R} P^3$-structure in an essential way. The basic tools are a bordification by a closed $\mathbb{R} P^2$-manifold with free holonomy group, and the work of Goldman, Labourie, and Margulis on geodesics in the Margulis spacetimes and $3$-manifold topology.

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arXiv2017

The convex real projective orbifolds with radial or totally geodesic ends: a survey of some partial results

A real projective orbifold has a radial end if a neighborhood of the end is foliated by projective geodesics that develop into geodesics ending at a common point. It has a totally geodesic end if the end can be completed to have the totally geodesic boundary. The purpose of this paper is to announce some partial results. A real projective structure sometimes admits deformations to parameters of real projective structures. We will prove a homeomorphism between the deformation space of convex real projective structures on an orbifold $\mathcal{O}$ with radial or totally geodesic ends with various conditions with the union of open subspaces of strata of the corresponding subset of \[ Hom(π_{1}(\mathcal{O}), PGL(n+1, \mathbb{R}))/PGL(n+1, \mathbb{R}).\] Lastly, we will talk about the openness and closedness of the properly (resp. strictly) convex real projective structures on a class of orbifold with generalized admissible ends.

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arXiv2017

A classification of radial or totally geodesic ends of real projective orbifolds I: a survey of results

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the original ones. The purpose of this paper is to understand the structures of ends of real projective $n$-dimensional orbifolds. In particular, these have the radial or totally geodesic ends. Hyperbolic manifolds with cusps and hyper-ideal ends are examples. For this, we will study the natural conditions on eigenvalues of holonomy representations of ends when these ends are manageably understandable. We will show that only the radial or totally geodesic ends of lens type or horospherical ends exist for strongly irreducible properly convex real projective orbifolds under the suitable conditions. The purpose of this article is to announce these results.

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