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.
Let $M_ϕ$ be a surface bundle over a circle with monodromy $ϕ:S \rightarrow S$. We study deformations of certain reducible representations of $π_1(M_ϕ)$ into $\text{SL}(n,\mathbb{C})$, obtained by composing a reducible representation into $\text{SL}(2,\mathbb{C})$ with the irreducible representation $\text{SL}(2,\mathbb{C}) \rightarrow \text{SL}(n,\mathbb{C})$. In particular, we show that under certain conditions on the eigenvalues of $ϕ^*$, the reducible representation is contained in a $(n+1+k)(n-1)$ dimensional component of the representation variety, where $k$ is the number of components o…
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.
The Bianchi groups ${\rm Bi}(d)={\rm PSL}(2,\mathcal{O}_d) < {\rm PSL}(2,\C)$ (where $\mathcal{O}_d$ denotes the ring of integers of $\Q (i\sqrt{d})$, with $d \geqslant 1$ squarefree) can be viewed as subgroups of ${\rm SO}(3,1)$ under the isomorphism ${\rm PSL}(2,\C) \simeq {\rm SO}^0(3,1)$. We study the deformations of these groups into the larger Lie groups ${\rm SU}(3,1)$ and ${\rm SL}(4,\R)$ for small values of $d$. In particular we show that ${\rm Bi}(3)$, which is rigid in ${\rm SO}(3,1)$, admits a 1-dimensional deformation space into ${\rm SU}(3,1)$ and ${\rm SL}(4,\R)$, whereas any de…
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…
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.
In this paper, we construct cataclysm deformations for $θ$-Anosov representations into a semisimple non-compact connected real Lie group $G$ with finite center, where $θ\subset Δ$ is a subset of the simple roots that is invariant under the opposition involution. These generalize Thurston's cataclysms on Teichmüller space and Dreyer's cataclysms for Borel-Anosov representations into $\mathrm{PSL}(n, \mathbb{R})$. We express the deformation also in terms of the boundary map. Furthermore, we show that cataclysm deformations are additive and behave well with respect to composing a representation w…
We introduce higher strip deformations, which give a way of constructing affine deformations of discrete free groups in the image of the irreducible representation $\operatorname{PSL}_2\mathbb{R}\to \operatorname{SO}(2n,2n-1)$. We use the Margulis invariant to find a properness condition for these deformations, showing that each convex cocompact free group admits affine deformations acting properly on $\mathbb{R}^{4n-1}$. Using our method we can also construct proper affine actions of virtually free groups, which we demonstrate by finding proper actions of $C_2 \star C_3$ on $\mathbb{R}^7.$
The germ of the universal isomonodromic deformation of a logarithmic connection on a stable n-pointed genus g curve always exists in the analytic category. The first part of this paper investigates under which conditions it is the analytic germification of an algebraic isomonodromic deformation. Up to some minor technical conditions, this turns out to be the case if and only if the monodromy of the connection has finite orbit under the action of the mapping class group. The second part of this paper studies the dynamics of this action in the particular case of reducible rank 2 representations …
Let $Γ$ be a subgroup of $PSL(2,R)$ generated by three parabolic transformations. The main goal of this paper is to present an algorithm to determine whether or not $Γ$ is discrete. Historically discreteness algorithms have been considered within several broader mathematical paradigms: the discreteness problem, the construction and deformation of hyperbolic structures on surfaces and notions of automata for groups. Each of these approaches yield equivalent results. The second goal of this paper is to give an exposition of the basic ideas needed to interpret these equivalences, emphasizing rela…
We study the topology and geometry of compact complex manifolds associated to Anosov representations of surface groups and other hyperbolic groups in a complex semisimple Lie group $G$. These manifolds are obtained as quotients of the domains of discontinuity in generalized flag varieties $G/P$ constructed by Kapovich-Leeb-Porti (arXiv:1306.3837), and in some cases by Guichard-Wienhard (arXiv:1108.0733). For $G$-Fuchsian representations and their Anosov deformations, where $G$ is simple, we compute the homology of the domains of discontinuity and of the quotient manifolds. For $G$-Fuchsian and…
We describe the geometry of the character variety of representations of the fundamental group of the complement of a Hopf link with $n$ twists, namely $Γ_{n}=\langle x,y \,| \, [x^n,y]=1 \rangle$ into the group $\mathrm{SU}(r)$. For arbitrary rank, we provide geometric descriptions of the loci of irreducible and totally reducible representations. In the case $r = 2$, we provide a complete geometric description of the character variety, proving that this $\mathrm{SU}(2)$-character variety is a deformation retract of the larger $\mathrm{SL}(2,\mathbb{C})$-character variety, as conjectured by Flo…
Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进学会数学研究所）；Institute for Advanced Study, Massey University（梅西大学高等研究院）；Department of Mathematics, The University of Auckland（奥克兰大学数学系）
