Related papers: Strongly Dense Representations of Hyperbolic 3-Man…

In this note, we study deformations of discrete and Zariski dense subgroups of SU(2, 1) in quaternionic hyperbolic space. Specifi- cally we consider two examples coming from representations of 3-manifold groups (the figure eight knot and…

We prove that generic Hitchin representations are strongly dense: every pair of non commuting elements in their image generate a Zariski-dense subgroup of SL_n(R). The proof uses a theorem of Rapinchuk, Benyash-Krivetz and Chernousov, to…

For each integer $n \geq 3$, we exhibit a nonuniform arithmetic lattice in $\mathrm{SO}(n,1)$ containing Zariski-dense surface subgroups.

We construct Zariski-dense surface subgroups in infinitely many commensurability classes of uniform lattices of the split real Lie groups $\operatorname{SL}(n,\mathbb{R})$, $\operatorname{Sp}(2n,\mathbb{R})$, $\operatorname{SO}(k+1,k)$, and…

We prove that any nonabelian, non-Fuchsian representation of a surface group into PSL(2,R) is the holonomy of a folded hyperbolic structure on the surface. Using similar ideas, we establish that any non-Fuchsian representation rho of a…

Let $\Gamma$ denote a lattice in $SU(1,p)$, with $p$ greater than 1. We show that there exists no Zariski dense maximal representation with target $SU(m,n)$ if $n>m>1$. The proof is geometric and is based on the study of the rigidity…

Let $\rho$ be a maximal representation of a uniform lattice $\Gamma\subset{\rm SU}(n,1)$, $n\geq 2$, in a classical Lie group of Hermitian type $H$. We prove that necessarily $H={\rm SU}(p,q)$ with $p\geq qn$ and there exists a holomorphic…

We show that for a representation of the fundamental group of a triangulated closed 3-manifold (not necessarily hyperbolic) into $\PSL$ so that any edge loop has non-trivial image under the representation, there exist uncountably many…

For $\textrm{SL}(n,\mathbb{R})$ ($n\geq3$), $\textrm{SO}(n+1,n)$ ($n\geq2$), $\textrm{Sp}(2n,\mathbb{R})$ ($n\geq2$) and for the adjoint real split form of the exceptional group $\textrm{G}_2$, we exhibit non-uniform lattices in which we…

We classify $SU(2)$-cyclic and $SU(2)$-abelian 3-manifolds, for which every representation of the fundamental group into $SU(2)$ has cyclic or abelian image respectively, among geometric 3-manifolds which are not hyperbolic. As an…

For any maximal surface group representation into $\mathrm{SO}_0(2,n+1)$, we introduce a non-degenerate scalar product on the the first cohomology group of the surface with values in the associated flat bundle. In particular, it gives rise…

We study deformations of non-cocompact lattices of ${\rm SO}(n,1)$ into ${\rm SU}(n,1)$ and ${\rm SO}(n+1,1)$. A necessary condition for these deformations to remain discrete and faithful (when $n \geqslant 3$) is for the parabolic…

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…

We prove that all lattices of Sp(2n,R), except those commensurable with Sp(4k+2,Z) when n=2k+1, contain the image of infinitely many mapping class group orbits of Zariski-dense maximal representation that are continuous deformations of…

Given a number field $K$, we show that certain $K$-integral representations of closed surface groups can be deformed to being Zariski dense while preserving many useful properties of the original representation. This generalizes a method…

In this notes, we characterize discrete subgroups of PU(2,1), holomorphic isometric group of complex hyperbolic space, which have an invariant totally geodesic submanifold.

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…

For a complete hyperbolic three manifold M, we consider the representations of its fundamental group obtained by composing a lift of the holonomy with complex finite dimensional representations of SL(2,C). We prove a vanishing result for…

Given a finite volume hyperbolic 3-manifold, we compose a lift of the holonomy in SL(2,C) with the n-dimensional irreducible representation of SL(2,C) in SL(n,C). In this paper we give local coordinates of the SL(n,C)-character variety…

In this note, we study deformations of a non-uniform real hyperbolic lattice in quaternionic hyperbolic spaces. Specially we show that the representations of the fundamental group of the figure eight knot complement into PU(2,1) cannot be…