63 research outputs found

### Duality and invariants of representations of fundamental groups of 3-manifolds into PGL(3,C)

We determine the explicit transformation under duality of generic
configurations of four flags in \PGL(3,\bC) in cross-ratio coordinates. As an
application we prove invariance under duality of an invariant in the Bloch
group obtained from decorated triangulations of 3-manifolds.Comment: Revised version, 29 pages,4 figure

### Dimension of character varieties for $3$-manifolds

Let $M$ be a $3$-manifold, compact with boundary and $\Gamma$ its fundamental
group. Consider a complex reductive algebraic group G. The character variety
$X(\Gamma,G)$ is the GIT quotient $\mathrm{Hom}(\Gamma,G)//G$ of the space of
morphisms $\Gamma\to G$ by the natural action by conjugation of $G$. In the
case $G=\mathrm{SL}(2,\mathbb C)$ this space has been thoroughly studied.
Following work of Thurston, as presented by Culler-Shalen, we give a lower
bound for the dimension of irreducible components of $X(\Gamma,G)$ in terms of
the Euler characteristic $\chi(M)$ of $M$, the number $t$ of torus boundary
components of $M$, the dimension $d$ and the rank $r$ of $G$. Indeed, under
mild assumptions on an irreducible component $X_0$ of $X(\Gamma,G)$, we prove
the inequality $\mathrm{dim}(X_0)\geq t \cdot r - d\chi(M).$Comment: 12 pages, 1 figur

### Branched Spherical CR structures on the complement of the figure eight knot

We obtain a branched spherical CR structure on the complement of the figure
eight knot with a given holonomy representation (called rho_2). There are
essentially two boundary unipotent representations from the complement of the
figure eight knot into PU(2,1), we call them rho_1 and rho_2. We make explicit
some fundamental differences between these two representations. For instance,
seeing the figure eight knot complement as a surface bundle over the circle,
the behaviour of of the fundamental group of the fiber under the representation
is a key difference between rho_1 and rho_2

### Eigenvalues of Products of Unitary Matrices and Lagrangian Involutions

This paper introduces a submanifold of the moduli space of unitary
representations of the fundamental group of a punctured sphere with fixed local
monodromy. The submanifold is defined via products of involutions through
Lagrangian subspaces. We show that the moduli space of Lagrangian
representations is a Lagrangian submanifold of the moduli of unitary
representations.Comment: 35 pages, 2 figures, to appear in Topolog

### The geometry of the Eisenstein-Picard modular group

The Eisenstein-Picard modular group ${\rm PU}(2,1;\mathbb {Z}[\omega])$ is defined to be the subgroup of ${\rm PU}(2,1)$ whose entries lie in the ring $\mathbb {Z}[\omega]$, where $\omega$ is a cube root of unity. This group acts isometrically and properly discontinuously on ${\bf H}^2_\mathbb{C}$, that is, on the unit ball in $\mathbb {C}^2$ with the Bergman metric. We construct a fundamental domain for the action of ${\rm PU}(2,1;\mathbb {Z}[\omega])$ on ${\bf H}^2_\mathbb {C}$, which is a 4-simplex with one ideal vertex. As a consequence, we elicit a presentation of the group (see Theorem 5.9). This seems to be the simplest fundamental domain for a finite covolume subgroup of ${\rm PU}(2,1)

### A Flag structure on a cusped hyperbolic 3-manifold with unipotent holonomy

International audienceA Flag structure on a 3-manifold is an (X;G) structure where G = SL(3,R) and X is the space of flags on the 2-dimensional projective space. We construct a flag structure on a cusped hyperbolic manifold with unipotent boundary holonomy. The holonomy representation can be obtained from a punctured torus group representation into SL(3,R) which is equivariant under a pseudo-Anosov

### Configurations of flags in orbits of real forms

In this paper we start the study of configurations of flags in closed orbits of real forms using mainly tools of GIT. As an application, using cross ratio coordinates for generic configurations, we identify boundary unipotent representations of the fundamental group of the figure eight knot complement into real forms of PGL(4, C)

- …