Related papers: Positive surface group representations in PO(p,q)

Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact representations into rank-one Lie groups.…

We prove that any hyperbolic group acting properly discontinuously and cocompactly on a $\mathrm{CAT}(0)$ cube complex admits a projective Anosov representation into $\mathrm{SL}(d, \mathbb{R})$ for some $d$. More specifically, we show that…

This article considers $G$-Anosov representations of a fixed closed oriented Riemann surface $\Sigma$ of genus at least $2$. Here, $G$ is the Lie group $\text{PSp}(2n,\mathbb{R}$), $\text{PSO}(n,n)$ or $\text{PSO}(n,n+1)$. It proves that…

Let Gamma be a cocompact lattice in SO(1,n). A representation rho: Gamma \to SO(2,n) is quasi-Fuchsian if it is faithfull, discrete, and preserves an acausal subset in the boundary of anti-de Sitter space - a particular case is the case of…

We obtain restrictions on which groups can admit relatively Anosov representations into specified target Lie groups, by examining the topology of possible Bowditch boundaries and how they interact with the Anosov limit maps. For instance,…

We study the character variety of representations of the fundamental group of a closed surface of genus $g\geq2$ into the Lie group SO(n,n+1) using Higgs bundles. For each integer $0<d\leq n(2g-2),$ we show there is a smooth connected…

Let $\Sigma_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We study actions of the mapping class group of $\Sigma_{g,n}$ via Hodge-theoretic and arithmetic techniques. We show that if $$\rho: \pi_1(\Sigma_{g,n})\to…

Let $\Gamma$ be a finite group acting faithfully and linearly on a vector space $V$. Let $T(V)$ ($S(V)$) be the tensor (symmetric) algebra associated to $V$ which has a natural $\Gamma$ action. We study generalized quadratic relations on…

We construct for each conformal structure on a closed orientable surface of genus at least 2 a proper slice in the character variety of representations of the associated surface group into SL(3,R) that belongs to the Barbot component and…

We compute the twisted cohomology of the mapping class group with level structures, with coefficients in the $r$-tensor powers of the Prym representations for any positive integer $r$. When $r\ge 2$, we show that the cohomology exhibits…

Let $\Gamma$ be a one-ended, torsion-free hyperbolic group and let $G$ be a semisimple Lie group with finite center. Using the canonical JSJ splitting due to Sela, we define amalgam Anosov representations of $\Gamma$ into $G$ and prove that…

We show the set of faithful representations of a closed orientable hyperbolic surface group is dense in both irreducible components of the PSL(2,K) representation variety, where K is the field of real or complex numbers, answering a…

A result due to R. Greenberg gives a relation between the cardinality of Selmer groups of elliptic curves over number fields and the characteristic power series of Pontryagin duals of Selmer groups over cyclotomic $\mathbb Z_p$-extensions…

In the paper "Pappus's theorem and the modular group", R. Schwartz constructed a 2-dimensional family of faithful representations $\rho_\Theta$ of the modular group $\mathrm{PSL}(2,\mathbb{Z})$ into the group $\mathscr{G}$ of projective…

We consider polynomials on the intersection of the closed positive orthant with the height-$1$ level hypersurface of certain polynomials with positive coefficients. We show that any polynomial strictly positive on such a semi-algebraic set…

We define a new family of discrete representations of relatively hyperbolic groups which unifies many existing definitions and examples of geometrically finite behavior in higher rank. The definition includes the relative Anosov…

Let $\Gamma$ be a finitely generated group, and let $\op{Rep}(\Gamma, \SO(2,n))$ be the moduli space of representations of $\Gamma$ into $\SO(2,n)$ ($n \geq 2$). An element $\rho: \Gamma \to \SO(2,n)$ of $\op{Rep}(\Gamma, \SO(2,n))$ is…

We define the notion of affine Anosov representations of word hyperbolic groups into the affine group $\mathsf{SO}^0(n+1,n)\ltimes\mathbb{R}^{2n+1}$. We then show that a representation $\rho$ of a word hyperbolic group is affine Anosov if…

We prove that a word hyperbolic group which admits a $P_{2q+1}$-Anosov representation into $\mathsf{PGL}(4q+2, \mathbb{R})$ contains a finite-index subgroup which is either free or a surface group. As a consequence, we give an affirmative…

The notion of Anosov representations has been introduced by Labourie in his study of the Hitchin component for SL(n,R). Subsequently, Anosov representations have been studied mainly for surface groups, in particular in the context of higher…