The variety of characters in PSL(2,C)

Dominating surface-group representations into $\mathrm{PSL}_2 (\mathbb{C})$ in the relative representation variety

Let $ρ$ be a representation of the fundamental group of a punctured surface into $\mathrm{PSL}_2 (\mathbb{C})$ that is not Fuchsian. We prove that there exists a Fuchsian representation that strictly dominates $ρ$ in the simple length spectrum, and preserves the boundary lengths. This extends a result of Gueritaud-Kassel-Wolff to the case of $\mathrm{PSL}_2 (\mathbb{C})$-representations. Our proof involves straightening the pleated plane in $\mathbb{H}^3$ determined by the Fock-Goncharov coordinates of a framed representation, and applying strip-deformations.

$PSL_2(\mathbb{C})$-character varieties and Seifert fibered cosmetic surgeries

We study small Seifert possibly chiral cosmetic surgeries on not necessarily null-homologous knot in rational homology spheres. Using $PSL_2(\mathbb{C})$-character variety theory we give a sharp bound on the number of slopes producing the same small Seifert manifold if the ambient manifold satisfies some representation theoretic conditions.

Weyl chamber length compactification of the ${\rm PSL}(2,\mathbb R)\times{\rm PSL}(2,\mathbb R)$ maximal character variety

We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group $Γ$ of a closed hyperbolic surface $Σ$ in ${\rm PSL}(2,\mathbb R)^n$. We identify the boundary with the sphere $\mathbb P((\mathcal{ML})^n)$, where $\mathcal{ML}$ is the space of measured geodesic laminations on $Σ$. In the case $n=2$, we give a geometric interpretation of the boundary as the space of homothety classes of $\mathbb R^2$-mixed structures on $Σ$. We associate to such a structure a dual tree-graded space endowed with an $\mathbb R_+^2$-valued metric,…

从纽结图出发的纽结群的几何结构与$PSL_2(\mathbb{C})$表示

Geometric structures and $PSL_2(\mathbb{C})$ representations of knot groups from knot diagrams

该研究提出一种仅用满足弱限制的纽结图推导纽结群到$PSL_2(\mathbb{C})$表示簇典范分支方程组的新算法，无需纽结补的多面体分解，可手工计算、处理无穷族纽结，还能显式描述双曲纽结的几何表示对应的双曲结构，并给出闭合交替辫的示例公式。

On Culler-Shalen seminorms and Dehn filling

Consider the exterior M of a hyperbolic knot lying in a closed, connected, orientable 3-manifold. Culler and Shalen defined norm on H_1(dM;R) using the SL(2,C) character variety of pi_1(M). The Culler-Shalen norm encodes many topological properties of M; in particular it provides information about Dehn fillings of M. Their construction may be applied to arbitrary curves in the SL_2(C)-character variety of a connected, compact, orientable, irreducible 3-manifold whose boundary is a torus, though in this generality one can only guarantee that it will define a seminorm. The first half of this pap…

$PSL(2,C)$-特征簇中$CP^1$-结构的颈部收缩

Neck-Pinching of $CP^1$-structures in the $PSL(2,C)$-character variety

本文研究了亏格至少为2的闭定向曲面上$CP^1$-结构的颈部收缩退化，在复结构沿单一环路收缩的假设下，利用发展映射、全纯二次微分和褶曲曲面描述了路径极限，并给出了极限和乐为平凡表示的实例。

Shearing deformations of Hitchin representations and the Atiyah-Bott-Goldman symplectic form

The Hitchin component Hit_n(S) of a closed surface S is a preferred component of the character variety X_PSL_n(R)(S) consisting of homomorphisms from the fundamental group pi_1(S) to the Lie group PSL_n(R)(S), whose elements enjoy remarkable geometric and dynamical properties. We consider a certain type of deformations of the elements of Hit_n(S), called shearing deformations, and compute their pairing for the Atiyah-Bott-Goldman symplectic form of the character variety.

Character varieties, A-polynomials, and the AJ Conjecture

We establish some facts about the behavior of the rational-geometric subvariety of the $SL_2(\c)$ or $PSL_2(\c)$ character variety of a hyperbolic knot manifold under the restriction map to the $SL_2(\c)$ or $PSL_2(\c)$ character variety of the boundary torus, and use the results to get some properties about the A-polynomials and to prove the AJ conjecture for certain class of knots in $S^3$ including in particular any $2$-bridge knot over which the double branched cover of $S^3$ is a lens space of prime order.

The modular action on PSL(2,R)-characters in genus 2

We explore the dynamics of the action of the mapping class group in genus 2 on the PSL(2,R)-character variety. We prove that this action is ergodic on the connected components of Euler class 1 and -1, as it was conjectured by Goldman. In the connected component of Euler class 0 there are two invariant open subsets, on one of them the action is ergodic. In this process we give a partial answer to a question of Bowditch.

Separable-stable representations of a compression body

Let M be a hyperbolizable, nontrivial compression body without toroidal boundary components. In this paper, we characterize which discrete and faithful representations of the fundamental group of M into PSL(2,C) are separable-stable. The set of separable-stable representations forms a domain of discontinuity for the action of the outer automorphism group of the fundamental group of M on the PSL(2,C)-character variety of M.

A PSL(2,C) Casson Invariant

We use intersection theory techniques to define an invariant of closed 3-manifolds counting the characters of irreducible representations of the fundamental group in PSL(2,C). We note several properties of the invariant and compute the invariant for certain Seifert fibered spaces and for some Dehn surgeries on twist knots. We discuss the relationship between this invariant and the SL(2,C)-invariant.

A counterexample to the simple loop conjecture for PSL(2,R)

In this note, we give an explicit counterexample to the simple loop conjecture for representations of surface groups into PSL(2,R). Specifically, we show that for any surface with negative Euler characteristic and genus at least 1, there are uncountably many non-conjugate, non-injective homomorphisms of its fundamental group into PSL(2,R) that kill no simple closed curve (nor any power of a simple closed curve). This result is not new -- work of Louder and Calegari for representations of surface groups into SL(2, C) applies to the PSL(2,R) case, but our approach here is explicit and elementary…

The asymptotic behavior of the Reidemeister torsion for Seifert manifolds and PSL(2;R)-representations of Fuchsian groups

We show that a PSL(2;R)-representation of a Fuchsian group induces the asymptotics of the Reidemeister torsion for the Seifert manifold corresponding to the euler class of the PSL(2;R)-representation. We also show that the limit of leading coefficient of the Reidemeister torsion is determined by the euler class of a PSL(2;R)-representation of a Fuchsian group. In particular, the leading coefficient of the Reidemeister torsion for the unit tangent bundle over a two-orbifold converges to $-χ\log2$ where $χ$ is the Euler characteristic of the two-orbifold. We also give a relation between $\mathbb…

Deformations of reducible representations of 3-manifold groups into PSL_2(C)

Let M a 3-manifold with torus boundary which is a rational homology circle. We study deformations of reducible representations of p_1(M) into PSL_2(C) associated to a simple zero of the twisted Alexander polynomial. We also describe the local structure of the representation and character varieties.

On dynamics of Out(F_n) on PSL(2,C) characters

This note introduces and studies an open set of PSL(2,C) characters of a nonabelian free group, on which the action of the outer automorphism group is properly discontinuous, and which is strictly larger than the set of discrete, faithful convex-cocompact (i.e. Schottky) characters. This implies, in particular, that the outer automorphism group does not act ergodically on the set of characters with dense image. Hence there is a difference between the geometric (discrete vs. dense) decomposition of the characters, and a natural dynamical decomposition.

Quasi-isometric rigidity for PSL(2,Z[1/p])

We prove that PSL(2,Z[1/p]) gives the first example of groups which are not quasi-isometric to each other but have the same quasi-isometry group. Namely, PSL(2,Z[1/p]) and PSL(2,Z[1/q]) are not quasi-isometric unless p=q, and, independent of p, the quasi-isometry group of PSL(2,Z[1/p]) is PSL(2,Q). In addition, we characterize PSL(2,Z[1/p]) uniquely among all finitely generated groups by its quasi-isometry type.

Symmetries and Detection of Surfaces by the Character Variety

We extend Culler and Shalen's construction of detecting essential surfaces in 3-manifolds to 3-orbifolds. We do so in the setting of the $\mathrm{SL}_2(\mathbb{C})$ character variety, and following Boyer and Zhang in the $\mathrm{PSL}_2(\mathbb{C})$ character variety as well. We show that any slope detected on a canonical component of the $\mathrm{(P)SL}_2(\mathbb{C})$ character variety of a one cusped hyperbolic 3-manifold with symmetries must be the slope of a symmetric surface. As an application, we show that for each symmetric double twist knot there are slopes which are strongly detected …

关于映射类群作用于相对$\text{PSL}(2,\text{ℝ})$-特征簇的动力学研究

On dynamics of the Mapping class group action on relative $\text{PSL}(2,\mathbb{R})$-Character Varieties

本文研究带孔曲面相对$\text{PSL}(2,\text{ℝ})$-特征簇上映射类群作用，定义简单稳定性并证明其构成不连续域，还证明特定双曲锥面完整群的本原稳定性并给出无限族非离散例子。

A birationality result for character varieties

Let M be an orientable, cusped hyperbolic 3-manifold of finite volume. We show that the restriction map from a Dehn surgery component in the PSL(2,C)-character variety of M to the character variety of the boundary of M is a birational isomorphism onto its image. This generalises a result by Nathan Dunfield. A key step in our proof is the exactness of Craig Hodgson's volume differential on the eigenvalue variety.

QHI Theory, II: Dilogarithmic and Quantum Hyperbolic Invariants of 3-Manifolds with PSL(2,C)-Characters

We give parallel constructions of an invariant R(W,f), based on the classical Rogers dilogarithm, and of quantum hyperbolic invariants (QHI), based on the Faddeev-Kashaev quantum dilogarithms, for flat PSL(2,C)-bundles f over closed oriented 3-manifolds W. All these invariants are explicitely computed as a sum or state sums over the same hyperbolic ideal tetrahedra of the idealization of any fixed simplicial 1-cocycle description of (W,f) of a special kind, called a D-triangulation. R(W,f) recovers the volume and the Chern-Simons invariant of f, and we conjecture that it determines the semi-cl…

3D Quantum Hyperbolic Field Theory

We construct a new family of exact quantum field theories modeled on hyperbolic geometry, called {\it quantum hyperbolic field theories} (QHFTs). The QHFTs are defined for a $(2+1)$-bordism category based on the set of compact oriented 3-manifolds $Y$, equipped with properly embedded framed links $L_\Ff$ and with flat connections $ρ$ of principal $PSL(2,\C)$-bundles over $Y \setminus L_\Ff$, with arbitrary holonomy at the link meridians. A main point is the introduction of new parameters for the space of all $PSL(2,\C)$-characters of a punctured surface. Each QHFT associates to a triple $(Y,L_…

Reidemeister torsion, hyperbolic three-manifolds, and character varieties

This is a survey on Reidemeister torsion for hyperbolic three-manifolds of finite volume. Torsions are viewed as topological invariants and also as functions on the variety of representations in $\operatorname{ SL}_2(\mathbb C)$. In both cases, the torsions may also be computed after composing with finite dimensional representations of $\operatorname{ SL}_2(\mathbb C)$. In addition the paper deals with the torsion of the adjoint representation as a function on the variety of $\operatorname{ PSL}_{n+1}(\mathbb C)$-characters, using that the first cohomology group with coefficients twisted by th…

On character varieties, sets of discrete characters, and non-zero degree maps

In this paper we use character variety methods to study homomorphisms between the fundamental groups of 3-manifolds, in particular those induced by non-zero degree maps. A {\it knot manifold} is a compact, connected, irreducible, orientable 3-manifold whose boundary is an incompressible torus. A {\it virtual epimorphism} is a homomorphism whose image is of finite index in its range. We show that the existence of such homomorphisms places constraints on the algebraic decomposition of a knot manifold's $PSL_2(\mathbb C)$-character variety and consequently determine a priori bounds on the number …

Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds

This paper proves a theorem about Dehn surgery using a new theorem about PSL(2, C) character varieties. Confirming a conjecture of Boyer and Zhang, this paper shows that a small hyperbolic knot in a homotopy sphere having a non-trivial cyclic slope r has an incompressible surface with non-integer boundary slope strictly between r-1 and r+1. A corollary is that any small knot which has only integer boundary slopes has Property P. The proof uses connections between the topology of the complement of the knot, M, and the PSL(2, C) character variety of M that were discovered by Culler and Shalen. T…

Quantum Hyperbolic Invariants Of 3-Manifolds With PSL(2,C)-Characters

We construct {\it quantum hyperbolic invariants} (QHI) for triples $(W,L,ρ)$, where $W$ is a compact closed oriented 3-manifold, $ρ$ is a flat principal bundle over $W$ with structural group $PSL(2,\mc)$, and $L$ is a non-empty link in $W$. These invariants are based on the Faddeev-Kashaev's {\it quantum dilogarithms}, interpreted as matrix valued functions of suitably decorated hyperbolic ideal tetrahedra. They are explicitely computed as state sums over the decorated hyperbolic ideal tetrahedra of the {\it idealization} of any fixed {\it $\Dd$-triangulation}; the $\Dd$-triangulations are sim…

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 e…

Super-maximal representations from fundamental groups of punctured surfaces to $\mathrm{PSL}(2,\mathbb{R})$

We study a particular class of representations from the fundamental groups of punctured spheres $Σ_{0,n}$ to the group $\text{PSL} (2,\mathbb R)$ (and their moduli spaces), that we call \emph{super-maximal}. Super-maximal representations are shown to be \emph{totally non hyperbolic}, in the sense that every simple closed curve is mapped to a non hyperbolic element. They are also shown to be \emph{geometrizable} (appart from the reducible super-maximal ones) in the following very strong sense : for any element of the Teichmüller space $\mathcal T_{0,n}$, there is a unique holomorphic equivarian…

Six-point configurations in the hyperbolic plane and ergodicity of the mapping class group

Let $X$ be the space of isometry classes of ordered sextuples of points in the hyperbolic plane such that the product of the six corresponding rotations of angle $π$ is the identity. This space $X$ is closely related to the PSL$_2(\mathbb{R})$-character variety of the genus 2 surface $Σ$. In this article we study the topology and the natural symplectic structure on $X$, and we describe the action of the mapping class group of $Σ$ on $X$. This completes the classification of the ergodic components of the character variety in genus 2 initiated in our previous work.

Hasse-Weil zeta functions of ${\rm SL}_2$-character varieties of closed orientable hyperbolic $3$-manifolds

It is proved that the Hasse-Weil zeta functions of the canonical components of the ${\rm SL}_2$ (${\rm PSL}_2$)-character varieties of closed orientable complete hyperbolic $3$-manifolds of finite volume are equal to the Dedekind zeta functions of their trace fields (invariant trace fields). When the closed $3$-manifold is arithmetic, the special value at $s=2$ of the Hasse-Weil zeta function of the canonical component of the ${\rm PSL}_2$-character variety is expressed in terms of the hyperbolic volume of the manifold up to rational numbers.

四元数莫比乌斯群中可逆性的代数刻画

Algebraic characterization of reversibility in the quaternionic Möbius group

本文通过代数方法研究四元数莫比乌斯群中元素的可逆性，提出基于矩阵代表元的判据，并刻画可逆元素的反演对称性。

关于一次 punctured torus 系列的特征品种拓扑

On the topology of character varieties of once-punctured torus bundles

本文研究了一次 punctured torus 系列的特征品种拓扑，通过分析纤维上的特征并构建分支覆盖映射，扩展了 Boyer 等人的成果，并展示了无限家族的超几何 bundles 的 canonical curves 特征具有无界 genus。

一个通过树的核实现的极大分支的闭球紧化

A closed ball compactification of a maximal component via cores of trees

本文证明了曲面群表示进入PSL(2,R)×PSL(2,R)的特征簇中极大分支的紧化是闭球，映射类群作用于其上，并研究了该作用的动力学及边界点的几何描述。

PSL(2, K)中的共轭类

Conjugacy classes in PSL(2, K)

本文研究PSL(2, K)在二次形式上的共轭类划分及其几何解释，结合Hilbert符号和模算术轨道的几何特性。

通过 Fock-Goncharov 坐标控制曲面群表示

Dominating surface-group representations via Fock-Goncharov coordinates

本文利用 Fock-Goncharov 坐标和加权平面网络的权重矩阵等线性代数方法，证明了对穿孔曲面上的一般复表示，存在一个在长度谱上控制它的正实表示，且保持外围曲线长度不变。

Hitchin characters and geodesic laminations

For a closed surface S, the Hitchin component Hit_n(S) is a preferred component of the character variety consisting of group homomorphisms from the fundamental group pi_1(S) to the Lie group PSL_n(R). We construct a parametrization of the Hitchin component that is well-adapted to a maximal geodesic lamination on the surface. This is a natural extension of Thurston's parametrization of the Teichmueller space of S by shear coordinates associated to a maximal geodesic lamination, corresponding to the case n=2. However, significantly new ideas are needed in this higher dimensional case. The articl…

Ergodicity of the mapping class group action on Deroin-Tholozan representations

This note investigates the dynamics of the mapping class group action on the character variety of super-maximal representations of the fundamental group of a punctured sphere into $\text{PSL}(2,\mathbb{R})$, discovered by Deroin and Tholozan. We apply symplectic methods developed by Goldman and Xia to prove that the action is ergodic.

Monodromy groups of $\mathbb{C}\mathrm{P}^1$-structures on punctured surfaces

For a punctured surface $S$, we characterize the representations of its fundamental group into $\mathrm{PSL}_2 (\mathbb{C})$ that arise as the monodromy of a meromorphic projective structure on $S$ with poles of order at most two and no apparent singularities. This proves the analogue of a theorem of Gallo-Kapovich-Marden concerning $\mathbb{C}\mathrm{P}^1$-structures on closed surfaces, and settles a long-standing question about characterizing monodromy groups for the Schwarzian equation on punctured spheres. The proof involves a geometric interpretation of the Fock-Goncharov coordinates of t…

Fundamental classes of 3-manifold group representations in SL(4,R)

We compute the fundamental class (in the extended Bloch group) for representations of fundamental groups of 3-manifolds to SL(4,R) that factor over SL(2,C), in particular for those factoring over the isomorphism PSL(2,C) = S0(3,1). We also discuss consequences for the number of connected components of SL(4,R)-character varieties, and we show that there are knots with arbitrarily many components of vanishing Chern-Simons invariant in their SL(n,C)-character varieties

Dynamics on character varieties: a survey

We survey recent work on the dynamics of the outer automorphism group of a word hyperbolic group on spaces of (conjugacy classes of) representations ofthe group into a semi-simple Lie group G. All these results are motivated by the fact that the mapping class group of a closed surface acts properly discontinuously on the Teichmuller space of the surface. We explore two settings. 1) The work of Guichard, Labourie and Wienhard establishing the proper discontinuity of the action of outer automorphism groups on spaces of Anosov representations. 2) The work of Canary, Gelander, Lee, Magid, Minsky a…

全椭圆曲面群表示

Totally elliptic surface group representations

本文完全刻画了到 PSL₂ℝ 和 PSL₂ℂ 的全椭圆曲面群表示，证明其仅为紧子群表示或 Deroin–Tholozan 表示。

Higher Teichmüller Spaces: from SL(2,R) to other Lie groups

The first part of this paper surveys several characterizations of Teichmüller space as a subset of the space of representation of the fundamental group of a surface into PSL(2,R). Special emphasis is put on (bounded) cohomological invariants which generalize when PSL(2,R) is replaced by a Lie group of Hermitian type. The second part discusses underlying structures of the two families of higher Teichmüller spaces, namely the space of maximal representations for Lie groups of Hermitian type and the space of Hitchin representations or positive representations for split real simple Lie groups.

On Character varieties of two-bridge knot groups

We find explicit models for the PSL(2,C)- and SL(2,C)-character varieties of the fundamental groups of complements in S^3 of an infinite family of two-bridge knots that contains the twist knots. We compute the genus of the components of these character varieties, and deduce upper bounds on the degree of the associated trace fields. We also show that these knot complements are fibered if and only if they are commensurable to a fibered knot complement in a Z/2Z-homology sphere, resolving a conjecture of Hoste and Shanahan.

模 knots 的链接数

Linking numbers of modular knots

研究模 knots 与 trefoil 的链接数，推导出具有多种数学风味的公式，并证明链接配对的非退化性。

模群到SL(3,R)/SO(3)等距群的表示

Representations of the modular group into the isometries of SL(3,R)/SO(3)

本文研究了模群到SL(3,R)/SO(3)等距群的表示，证明了该空间中某些远离Schwartz表示的表示为Anosov表示，并指出主要结果此前由Schwartz通过不同方法证明。

Quantum coadjoint action and the $6j$-symbols of $U_qsl_2$

We review the representation theory of the quantum group $U_εsl_2\mathbb{C}$ at a root of unity $ε$ of odd order, focusing on geometric aspects related to the 3-dimensional quantum hyperbolic field theories (QHFT). Our analysis relies on the quantum coadjoint action of De Concini-Kac-Procesi, and the theory of Heisenberg doubles of Poisson-Lie groups and Hopf algebras. We identify the 6j-symbols of generic representations of $U_εsl2\mathbb{C}$, the main ingredients of QHFT, with a bundle morphism defined over a finite cover of the algebraic quotient $PSL_2\mathbb{C}/!/PSL_2\mathbb{C}$, of degr…

四元数射影变换的分类与分解

Classification and Decomposition of Quaternionic Projective Transformations

本文研究四元数射影线性群$\text{PSL}(3,\text{H})$的可逆性问题，借此给出其动力学类型的代数刻画，还将$\text{SL}(3,\text{H})$元素分解为简单元乘积，并重新给出$\text{SL}(3,\text{R})$元素的完整分类。

Bowditch's Q-conditions and Minsky's primitive stability

For the action of the outer automorphism group of the rank two free group on the corresponding variety of PSL(2,C) characters, two domains of discontinuity have been known to exist that are strictly larger than the set of Schottky characters. One is introduced by Bowditch in 1998 (followed by Tan, Wong and Zhang in 2008) and the other by Minsky in 2013. We prove that these two domains are equal. We then show that they are contained in the set of characters having what we call the bounded intersection property.