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Gye-Seon Lee

Publications and source records attributed to Gye-Seon Lee.

18 recordsLinked to original sources
arXiv2026

Convex cocompact groups with three-dimensional limit sets

We provide a general construction of convex cocompact hyperbolic reflection groups with three-dimensional limit sets. More precisely, our construction takes as input an arbitrary simplicial complex L of dimension 3 on n vertices, and outputs a convex cocompact right-angled reflection group acting on real hyperbolic n-space whose nerve is precisely the Przytycki-Świątkowski subdivision of L. Moreover, the output reflection group is a thin subgroup of an n-dimensional cocompact arithmetic hyperbolic lattice. This answers affirmatively a question of M. Kapovich concerning the existence of a convex cocompact group acting on some real hyperbolic space with limit set a Čech cohomology sphere other than the standard sphere.

math.GR↗
arXiv2026

Convex cocompact groups in real hyperbolic spaces with limit set a Pontryagin sphere

We exhibit two examples of convex cocompact subgroups of the isometry groups of real hyperbolic spaces with limit set a Pontryagin sphere: one generated by $50$ reflections of $\mathbb{H}^4$, and the other by a rotation of order $21$ and a reflection of $\mathbb{H}^6$. For each of them, we also locate convex cocompact subgroups with limit set a Menger curve.

math.GT↗
arXiv2026

Anosov triangle reflection groups in SL(3,R)

We identify all Anosov representations of compact hyperbolic triangle reflection groups into the higher rank Lie group $\mathrm{SL}(3,\mathbb R)$. Specifically, we prove that such a representation is Anosov if and only if either it lies in the Hitchin component of the representation space, or it lies in the "Barbot component" and the product of the three generators of the triangle group has distinct real eigenvalues. Unlike representations in the Hitchin component, Anosov representations in the Barbot component have non-convex boundary maps.

math.GT↗
arXiv2025

Zariski-Closures of Linear Reflection Groups

We give necessary and sufficient conditions for a linear reflection group in the sense of Vinberg to be Zariski-dense in the ambient projective general linear group. As an application, we show that every irreducible right-angled Coxeter group of rank $N \geq 3$ virtually embeds Zariski-densely in $\mathrm{SL}_n(\mathbb{Z})$ for all $n \geq N$. This allows us to settle the existence of Zariski-dense surface subgroups of $\mathrm{SL}_n(\mathbb{Z})$ for all $n \geq 3$. Among the other applications are examples of Zariski-dense one-ended finitely generated subgroups of $\mathrm{SL}_n(\mathbb{Z})$ that are not finitely presented for all $n \geq 6$.

math.GT↗
arXiv2024

Convex cocompactness for Coxeter groups

We investigate representations of Coxeter groups into $\mathrm{GL}(n,\mathbb{R})$ as geometric reflection groups which are convex cocompact in the projective space $\mathbb{P}(\mathbb{R}^n)$. We characterize which Coxeter groups admit such representations, and we fully describe the corresponding spaces of convex cocompact representations as reflection groups, in terms of the associated Cartan matrices. The Coxeter groups that appear include all infinite, word hyperbolic Coxeter groups; for such groups the representations as reflection groups that we describe are exactly the projective Anosov ones. We also obtain a large class of nonhyperbolic Coxeter groups, thus providing many examples for the theory of nonhyperbolic convex cocompact subgroups in $\mathbb{P}(\mathbb{R}^n)$ developed in arXiv:1704.08711.

math.GR↗
arXiv2023

Cusped Borel Anosov representations with positivity

We show that if a cusped Borel Anosov representation from a lattice $Γ\subset \mathsf{PGL}_2(\mathbb{R})$ to $\mathsf{PGL}_d(\mathbb{R})$ contains a unipotent element with a single Jordan block in its image, then it is necessarily a (cusped) Hitchin representation. We also show that the amalgamation of a Hitchin representation with a cusped Borel Anosov representation that is not Hitchin is never cusped Borel Anosov.

math.DG↗
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.

math.GT↗
arXiv2022

Discrete Coxeter groups

Coxeter groups are a special class of groups generated by involutions. They play important roles in the various areas of mathematics. This survey particularly focuses on how one uses Coxeter groups to construct interesting examples of discrete subgroups of Lie groups.

math.GT↗
arXiv2021

A small closed convex projective 4-manifold via Dehn filling

In order to obtain a closed orientable convex projective four-manifold with small positive Euler characteristic, we build an explicit example of convex projective Dehn filling of a cusped hyperbolic four-manifold through a continuous path of projective cone-manifolds.

math.GT↗
arXiv2020

Hitchin components for orbifolds

We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of higher Teichmüller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball and we compute its dimension explicitly. We then give applications to the study of the pressure metric, cyclic Higgs bundles, and the deformation theory of real projective structures on $3$-manifolds.

math.GT↗
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.

math.GT↗
arXiv2018

Anti-de Sitter strictly GHC-regular groups which are not lattices

For $d=4, 5, 6, 7, 8$, we exhibit examples of $\mathrm{AdS}^{d,1}$ strictly GHC-regular groups which are not quasi-isometric to the hyperbolic space $\mathbb{H}^d$, nor to any symmetric space. This provides a negative answer to Question 5.2 in [9A12] and disproves Conjecture 8.11 of Barbot-Mérigot [BM12]. We construct those examples using the Tits representation of well-chosen Coxeter groups. On the way, we give an alternative proof of Moussong's hyperbolicity criterion [Mou88] for Coxeter groups built on Danciger-Guéritaud-Kassel [DGK17] and find examples of Coxeter groups $W$ such that the space of strictly GHC-regular representations of $W$ into $\mathrm{PO}_{d,2}(\mathbb{R})$ up to conjugation is disconnected.

math.GT↗
arXiv2018

Pappus Theorem, Schwartz Representations and Anosov Representations

In the paper "Pappus's theorem and the modular group", R. Schwartz constructed a 2-dimensional family of faithful representations $ρ_Θ$ of the modular group $\mathrm{PSL}(2,\mathbb{Z})$ into the group $\mathscr{G}$ of projective symmetries of the projective plane via Pappus Theorem. The image of the unique index 2 subgroup $\mathrm{PSL}(2,\mathbb{Z})_o$ of $\mathrm{PSL}(2,\mathbb{Z})$ under each representation $ρ_Θ$ is in the subgroup $\mathrm{PGL}(3,\mathbb{R})$ of $\mathscr{G}$ and preserves a topological circle in the flag variety, but $ρ_Θ$ is not Anosov. In her PhD Thesis, V. P. Valério elucidated the Anosov-like feature of Schwartz representations: For every $ρ_Θ$, there exists a 1-dimensional family of Anosov representations $ρ^\varepsilon_Θ$ of $\mathrm{PSL}(2,\mathbb{Z})_o$ into $\mathrm{PGL}(3,\mathbb{R})$ whose limit is the restriction of $ρ_Θ$ to $\mathrm{PSL}(2,\mathbb{Z})_o$. In this paper, we improve her work: For each $ρ_Θ$, we build a 2-dimensional family of Anosov representations of $\mathrm{PSL}(2,\mathbb{Z})_o$ into $\mathrm{PGL}(3,\mathbb{R})$ containing $ρ^\varepsilon_Θ$ and a 1-dimensional subfamily of which can extend to representations of $\mathrm{PSL}(2,\mathbb{Z})$ into $\mathscr{G}$. Schwartz representations are therefore, in a sense, the limits of Anosov representations of $\mathrm{PSL}(2,\mathbb{Z})$ into $\mathscr{G}$.

math.DS↗
arXiv2017

Convex projective structures on non-hyperbolic three-manifolds

Y. Benoist proved that if a closed three-manifold M admits an indecomposable convex real projective structure, then M is topologically the union along tori and Klein bottles of finitely many sub-manifolds each of which admits a complete finite volume hyperbolic structure on its interior. We describe some initial results in the direction of a potential converse to Benoist's theorem. We show that a cusped hyperbolic three-manifold may, under certain assumptions, be deformed to convex projective structures with totally geodesic torus boundary. Such structures may be convexly glued together whenever the geometry at the boundary matches up. In particular, we prove that many doubles of cusped hyperbolic three-manifolds admit convex projective structures.

math.GT↗
arXiv2016

Deformations of convex real projective manifolds and orbifolds

In this survey, we study representations of finitely generated groups into Lie groups, focusing on the deformation spaces of convex real projective structures on closed manifolds and orbifolds, with an excursion on projective structures on surfaces. We survey the basics of the theory of character varieties, geometric structures on orbifolds, and Hilbert geometry. The main examples of finitely generated groups for us will be Fuchsian groups, 3-manifold groups and Coxeter groups.

math.GT↗
arXiv2014

Projective deformations of weakly orderable hyperbolic Coxeter orbifolds

A Coxeter $n$-orbifold is an $n$-dimensional orbifold based on a polytope with silvered boundary facets. Each pair of adjacent facets meet on a ridge of some order $m$, whose neighborhood is locally modeled on ${\mathbb R}^n$ modulo the dihedral group of order $2m$ generated by two reflections. For $n \geq 3$, we study the deformation space of real projective structures on a compact Coxeter $n$-orbifold $Q$ admitting a hyperbolic structure. Let $e_+(Q)$ be the number of ridges of order $\geq 3$. A neighborhood of the hyperbolic structure in the deformation space is a cell of dimension $e_+(Q) - n$ if $n=3$ and $Q$ is weakly orderable, i.e., the faces of $Q$ can be ordered so that each face contains at most $3$ edges of order $2$ in faces of higher indices, or $Q$ is based on a truncation polytope.

math.GT↗
arXiv2010

Projective Deformations of Hyperbolic Coxeter 3-Orbifolds

By using Klein's model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev's theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real projective structure on some such 3-orbifolds deforms into a family of real projective structures that are not induced from hyperbolic structures. In this paper, we find new classes of compact and complete hyperbolic reflection 3-orbifolds with such deformations. We also explain numerical and exact results on projective deformations of some compact hyperbolic cubes and dodecahedra.

math.GT↗