Computing and
Understanding
Representation
Varieties
Efficiently

Duality and invariants of representations of fundamental groups of 3manifolds into PGL(3,C)
E. Falbel, Qingxue Wang
We determine the explicit transformation under duality of generic configurations of four flags in PGL(3,C) in crossratio coordinates. As an application we prove the invariance under duality of an invariant in the Bloch group obtained from decorated triangulations of 3manifolds.
FockGoncharov coordinates for rank two Lie groups
Christian K. Zickert
This paper defines Ptolemy coordinates for all simple Lie groups of rank 2.
Triangulation independent Ptolemy varieties
Matthias Goerner, Christian K. Zickert
This paper defines triangulation independent refinements of all variants of the Ptolemy variety. Changing the triangulation changes the variety by a biregular isomorphism.
Ptolemy coordinates, Dehn invariant and the Apolynomial
Christian K. Zickert
This paper defines an enhanced Ptolemy variety for representations that are not necessarily boundaryunipotent, and gives a formula for the Dehn invariant. The enhanced Ptolemy variety projects to the Apolynomial curve.
The Ptolemy field of 3manifold representations
Stavros Garoufalidis, Matthias Goerner, Christian K. Zickert
This paper defines the Ptolemy field of a representation, and proves that it is equal to the trace field.
The symplectic properties of the PGL(n,C)gluing equations
Stavros Garoufalidis, Christian K. Zickert
This paper proves that the PGL(n,C)gluing equations satisfy symplectic properties analogous to the properties satisfied by Thurston's gluing equations. Some results were independently proved by Antonin Guilloux.
Local rigidity for PGL(3,C)representations of 3manifold groups
Nicolas Bergeron, Elisha Falbel, Antonin Guilloux, PierreVincent Koseleff, Fabrice Rouillier
Let M be a noncompact hyperbolic 3manifold that has a triangulation by positively oriented
ideal tetraedra. We explain how to produce local coordinates for the variety defined
by the gluing equations for SL(3;C)representations. In particular we prove local rigidity of the "geometric"
representation in SL(3;C), recovering a recent result of MenalFerrer and Porti.
More generally we give a criterion for local rigidty of SL(3;C)representations
and provide detailed analysis of the figure eight knot sister manifold exhibiting
the different possibilities that can occur.
Branched Spherical CR structures on the complement of the figure eight knot
Elisha Falbel, Jieyan Wang
To be written.
Gluing equations for PGL(n,C)representations of 3manifolds
Stavros Garoufalidis, Matthias Goerner, Christian K. Zickert
This paper defines shape coordinates for representations in PGL(n,C). The shape coordinates are 3dimensional analogues of the FockGoncharov Xcoordinates, and satisfy relations generalizing Thurston's gluing equations for representations in PGL(2,C).
The complex volume of SL(n,C)representations of 3manifolds
Stavros Garoufalidis, Dylan P. Thurston, Christian K. Zickert
This paper defines Ptolemy coordinates for boundaryunipotent representations in SL(n,C). These coordinates are 3dimensional analogues of the FockGoncharov Acoordinates. The main result is that a Ptolemy assignment determines an element in the extended Bloch group, and that Neumann's extension of Rogers dilogarithm computes the complex volume. Explicit formulas for a representation are given in terms of the coordinates, and the behavior of the coordinates under the canonical irreducible representation of SL(2,C) in SL(n,C) are given. The paper also proves that the Ptolemy variety detects the geometric representation whenever all edges are essential.
Tetrahedra of flags, volume and homology of SL(3)
Nicolas Bergeron, Elisha Falbel, Antonin Guilloux
In the paper we define a "volume" for simplicial complexes of flag tetrahedra. This generalizes and unifies the classical
volume of hyperbolic manifolds and the volume of CR tetrahedra complexes considered by Falbel and FalbelWang. We describe when this volume
belongs to the Bloch group and more generally describe a variation formula in terms of boundary data. In doing so, we recover
and generalize results of NeumannZagier, Neumann, and
Kabaya. Our approach is very related to the work of Fock and Goncharov.
The extended Bloch group and algebraic Ktheory
Christian K. Zickert
The main result of this paper is that the extended Bloch group of a number field is isomorphic to indecomposable algebraic Ktheory. The proof uses rudimentary versions of the shape and Ptolemy coordinates developed in later papers.
The volume and ChernSimons invariant of a representation
Christian K. Zickert
This paper gives an explicit algorithm for computing the volume and ChernSimons invariant of a hyperbolic 3manifold, and more generally, a parabolic representation in PSL(2,C). The formula uses shape and Ptolemy coordinates for n=2.
