arXiv subjects

Suhyoung Choi

Publications and source records attributed to Suhyoung Choi.

36 records · Page 2Linked to original sources

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↗

arXiv2015

A classification of radial and totally geodesic ends of properly convex real projective orbifolds III: the convex but nonproperly convex and non-complete-affine radial ends

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. In previous papers, we classified properly convex or complete radial ends under suitable conditions. In this paper, we will study radial ends that are convex but not properly convex nor complete affine. The main techniques are the theory of Fried and Goldman on affine manifolds, and a generalization of the work on Riemannian foliations by Molino, Carrière, and so on. We will show that these are quasi-joins of horospheres and totally geodesic radial ends. These are deformations of joins of horospheres and totally geodesic radial ends.

math.GT↗

arXiv2015

The classification of ends of properly convex real projective orbifolds II: Properly convex radial ends and totally geodesic ends

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 properly convex ends of real projective $n$-dimensional orbifolds. In particular, these have the radial or totally geodesic ends. For this, we will study the natural conditions on eigenvalues of holonomy representations of ends when these ends are manageably understandable. In this paper, we only study the properly convex ends. The main techniques are the Vinberg duality and a generalization of the work of Goldman, Labourie, and Margulis on flat Lorentzian $3$-manifolds. Finally, we show that a noncompact strongly tame properly convex real projective orbifold with generalized lens-type or horospherical ends satisfying some topological conditions always has a strongly irreducible holonomy group.

math.GT↗

arXiv2014

Closed flat affine 3-manifolds are prime

An (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an affine space ${\mathbf R}^3$ with transition maps in the affine transformation group $Aff({\mathbf R}^3)$. Equivalently an affine $3$-manifold is a $3$-manifold with a flat torsion-free affine connection. We show that a closed affine $3$-manifold is either irreducible or is finitely covered by an affine Hopf manifold. A real projective $3$-manifold is a manifold with an atlas of charts to a real projective space ${\mathbf R} P^3$ with transition maps in the projective transformation group $PGL(4, {\mathbf R})$. Using the convex concave decomposition of real projective manifolds, we will show that a closed real projective $3$-manifold decomposes into concave affine submanifolds, toral $π$-submanifolds and $2$-convex real projective manifolds.

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↗

arXiv2014

A classification of radial and totally geodesic ends of properly convex real projective orbifolds

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $SL(n+1, \mathbb{R})$ or $PGL(n+1, \mathbb{R})$. 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. The main techniques are the theory of Fried and Goldman on affine manifolds, a generalization of the work of Goldman, Labourie, and Margulis on flat Lorentzian $3$-manifolds and the work on Riemannian foliations by Molino, Carrière, and so on. 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.

math.GT↗

arXiv2014

The convex real projective orbifolds with radial or totally geodesic ends: The closedness and openness of deformations

A real projective orbifold is an $n$-dimensional orbifold modeled on $\mathbb{RP}^n$ with the group $PGL(n+1, \mathbb{R})$. We concentrate on an orbifold that contains a compact codimension $0$ submanifold whose complement is a union of neighborhoods of ends, diffeomorphic to closed $(n-1)$-dimensional orbifolds times intervals. A real projective orbifold has a {\it 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 {\it totally geodesic end} if the end can be completed to have the totally geodesic boundary. The orbifold is said to be {\it convex} if any path can be homotopied to a projective geodesic with endpoints fixed. A real projective structure sometimes admits deformations to parameters of real projective structures. We will prove the local homeomorphism between the deformation space of convex real projective structures on such an orbifold with radial or totally geodesic ends with various conditions with the $PGL(n+1, \mathbb{R})$-character space of the fundamental group with corresponding conditions. We will use a Hessian argument to show that under a small deformation, a properly (resp. strictly) convex real projective orbifold with generalized admissible ends will remain properly and properly (resp. strictly) convex with generalized admissible ends. Lastly, we will prove the openness and closedness of the properly (resp. strictly) convex real projective structures on a class of orbifold with generalized admissible ends, where we need the theory of Crampon-Marquis and Cooper, Long and Tillmann on the Margulis lemma for convex real projective manifolds. The theory here partly generalizes that of Benoist on closed real projective orbifolds.

math.GT↗

arXiv2013

The definability criterions for convex projective polyhedral reflection groups

Following Vinberg, we find the criterions for a subgroup generated by reflections $Γ\subset \SL^{\pm}(n+1,\mathbb{R})$ and its finite-index subgroups to be definable over $\mathbb{A}$ where $\mathbb{A}$ is an integrally closed Noetherian ring in the field $\mathbb{R}$. We apply the criterions for groups generated by reflections that act cocompactly on irreducible properly convex open subdomains of the $n$-dimensional projective sphere. This gives a method for constructing injective group homomorphisms from such Coxeter groups to $\SL^{\pm}(n+1,\mathbb{Z})$. Finally we provide some examples of $\SL^{\pm}(n+1,\mathbb{Z})$-representations of such Coxeter groups. In particular, we consider simplicial reflection groups that are isomorphic to hyperbolic simplicial groups and classify all the conjugacy classes of the reflection subgroups in $\SL^{\pm}(n+1,\mathbb{R})$ that are definable over $\mathbb{Z}$. These were known by Goldman, Benoist, and so on previously.

math.GT↗

arXiv2010

The two components of the SO(3)-character space of the fundamental group of a closed surface of genus 2

We use geometric techniques to explicitly find the topological structure of the space of SO(3)-representations of the fundamental group of a closed surface of genus 2 quotient by the conjugation action by SO(3). There are two components of the space. We will describe the topology of both components and describe the corresponding SU(2)-character spaces by parametrizing them by spherical triangles. There is the sixteen to one branch-covering for each component, and the branch locus is a union of 2-spheres or 2-tori. Along the way, we also describe the topology of both spaces. We will later relate this result to future work into higher-genus cases and the SL(3,R)-representations.

math.GN↗

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↗

arXiv2006

The PL-methods for hyperbolic 3-manifolds to prove tameness

Using PL-methods, we prove the Marden's conjecture that a hyperbolic 3-manifold $M$ with finitely generated fundamental group and with no parabolics are topologically tame. Our approach is to form an exhaustion $M_i$ of $M$ and modify the boundary to make them 2-convex. We use the induced path-metric, which makes the submanifold $M_i$ negatively curved and with Margulis constant independent of $i$. By taking the convex hull in the cover of $M_i$ corresponding to the core, we show that there exists an exiting sequence of surfaces $Σ_i$. Some of the ideas follow those of Agol. We drill out the covers of $M_i$ by a core $\core$ again to make it negatively curved. Then the boundary of the convex hull of $Σ_i$ is shown to meet the core. By the compactness argument of Souto, we show that infinitely many of $Σ_i$ are homotopic in $M - \core^o$. Our method should generalize to a more wider class of piecewise hyperbolic manifolds.

math.GT↗

arXiv2004

Drilling cores of hyperbolic 3-manifolds to prove tameness

We supply a proof of the fact that a hyperbolic 3-manifold $M$ with finitely generated fundamental group and with no parabolics is topologically tame. This proves the Marden's conjecture. Our approach is to form an exhaustion $M_i$ of $M$ and modify the boundary to make them 2-convex. We use the induced path-metric, which makes the submanifold $M_i$ $δ$-hyperbolic and with Margulis constants independent of $i$. By taking the convex hull in the cover of $M_i$ corresponding the core, we show that there exists an exiting sequence of surfaces $Σ_i$. We drill out the covers of $M_i$ by a core $C$ again to make it $δ$-hyperbolic. Then the boundary of the convex hull of $Σ_i$ is shown to meet the core. By the compactness argument of Souto, we show that infinitely many of $Σ_i$ are homotopic in $M - C^o$.

math.GT↗

arXiv2004

The maximal tubes under the deformations of a class of 3-dimensional hyperbolic cone-manifolds

Recently, Hodgson and Kerckhoff found a small bound on Dehn surgered 3-manifolds from hyperbolic knots not admitting hyperbolic structures using deformations of hyperbolic cone-manifolds. They asked whether the area normalized meridian length squared of maximal tubular neighborhoods of the singular locus of the cone-manifold is decreasing and that summed with the cone angle squared is increasing as we deform the cone-angles. We confirm this near 0 cone-angles for an infinite family of hyperbolic cone-manifolds obtained by Dehn surgeries along the Whitehead link complements. The basic method is based on explicit holonomy computations using the A-polynomials and finding the maximal tubes. One of the key tool is the Taylor expression of a geometric component of the zero set of the A-polynomial in terms of the cone-angles. We also show a sequence of Taylor expressions for Dehn surgered manifolds converges to one for the limit hyperbolic manifold.

math.GT↗

arXiv2003

Geometric structures on orbifolds and holonomy representations

An orbifold is a topological space modeled on quotient spaces of a finite group actions. We can define the universal cover of an orbifold and the fundamental group as the deck transformation group. Let $G$ be a Lie group acting on a space $X$. We show that the space of isotopy-equivalence classes of $(G,X)$-structures on a compact orbifold $Σ$ is locally homeomorphic to the space of representations of the orbifold fundamental group of $Σ$ to $G$ following the work of Thurston, Morgan, and Lok. This implies that the deformation space of $(G, X)$-structures on $Σ$ is locally homeomorphic to the space of representations of the orbifold fundamental group to $G$ when restricted to the region of proper conjugation action by $G$.

math.GT↗

arXiv2003

The deformation spaces of convex RP^2-structures on 2-orbifolds

We determine that the deformation space of convex real projective structures, that is, projectively flat torsion-free connections with the geodesic convexity property on a compact 2-orbifold of negative Euler characteristic is homeomorphic to a cell of certain dimension. The basic techniques are from Thurston's lecture notes on hyperbolic 2-orbifolds, and the previous work of Goldman on convex real projective structures on surfaces.

math.GT↗

arXiv2000

The decomposition and classification of radiant affine 3-manifolds

An affine manifold is a manifold with torsion-free flat affine connection. A geometric topologist's definition of an affine manifold is a manifold with an atlas of charts to the affine space with affine transition functions; a radiant affine manifold is an affine manifold with holonomy consisting of affine transformations fixing a common fixed point. We decompose an orientable closed radiant affine 3-manifold into radiant 2-convex affine manifolds and radiant concave affine 3-manifolds along mutually disjoint totally geodesic tori or Klein bottles using the convex and concave decomposition of real projective $n$-manifolds developed earlier. Then we decompose a 2-convex radiant affine manifold into convex radiant affine manifolds and concave-cone affine manifolds. To do this, we will obtain certain nice geometric objects in the Kuiper completion of holonomy cover. The equivariance and local finiteness property of the collection of such objects will show that their union covers a compact submanifold of codimension zero, the complement of which is convex. Finally, using the results of Barbot, we will show that a closed radiant affine 3-manifold admits a total cross section, confirming a conjecture of Carrière, and hence every radiant affine 3-manifold is homeomorphic to a Seifert fibered space with trivial Euler number, or a virtual bundle over a circle with fiber homeomorphic to a torus.

dg-ga↗

arXiv1999

The universal cover of an affine three-manifold with holonomy of shrinkable dimension $\leq 2$

An affine manifold is a manifold with an affine structure, i.e. a torsion-free flat affine connection. We show that the universal cover of a closed affine 3-manifold $M$ with holonomy group of shrinkable dimension (or discompacité in French) less than or equal to two is diffeomorphic to $\bR^3$. Hence, $M$ is irreducible. This follows from two results: (i) a simply connected affine 3-manifold which is 2-convex is diffeomorphic to $\bR^3$, whose proof using the Morse theory takes most of this paper; and (ii) a closed affine manifold of holonomy of shrinkable dimension less or equal to $d$ is $d$-convex. To prove (i), we show that 2-convexity is a geometric form of topological incompressibility of level sets. As a consequence, we show that the universal cover of a closed affine three-manifold with parallel volume form is diffeomorphic to $\bR^3$, a part of the weak Markus conjecture. As applications, we show that the universal cover of a hyperbolic 3-manifold with cone-type singularity of arbitrarily assigned cone-angles along a link removed with the singular locus is diffeomorphic to $\bR^3$. A fake cell has an affine structure as shown by Gromov. Such a cell must have a concave point at the boundary.

dg-ga↗

arXiv1998

Canonical decomposition of manifolds with flat real projective structure into (n-1)-convex manifolds and concave affine manifolds

We try to understand the geometric properties of $n$-manifolds ($n\geq 2$) with geometric structures modeled on $(\bR P^n, \PGL(n+1, \bR))$, i.e., $n$-manifolds with projectively flat torsion free affine connections. We define the notion of $i$-convexity of such manifolds due to Carriére for integers $i$, $1 \leq i \leq n-1$, which are generalization of convexity. Given a real projective $n$-manifold $M$, we show that the failure of an $(n-1)$-convexity of $M$ implies an existence of a certain geometric object, $n$-crescent, in the completion $\che M$ of the universal cover $\tilde M$ of $M$. We show that this further implies the existence of a particular type of affine submanifold in $M$ and give a natural decomposition of $M$ into simpler real projective manifolds, some of which are $(n-1)$-convex and others are affine, more specifically concave affine. We feel that it is useful to have such decomposition particularly in dimension three. Our result will later aid us to study the geometric and topological properties of radiant affine 3-manifolds leading to their classification. We get a consequence for affine Lie groups.

math.GT↗