145 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
Gluing equations for PGL(n,C)-representations of 3-manifolds
In a previous paper, we parametrized boundary-unipotent representations of a
3-manifold group into SL(n,C) using Ptolemy coordinates, which were inspired by
A-coordinates on higher Teichm\"uller space due to Fock and Goncharov. In this
paper, we parametrize representations into PGL(n,C) using shape coordinates
which are a 3-dimensional analogue of Fock and Goncharov's X-coordinates. These
coordinates satisfy equations generalizing Thurston's gluing equations. These
equations are of Neumann-Zagier type and satisfy symplectic relations with
applications in quantum topology. We also explore a duality between the Ptolemy
coordinates and the shape coordinates.Comment: 47 pages, 28 figure
On SL(3,)-representations of the Whitehead link group
We describe a family of representations in SL(3,) of the
fundamental group of the Whitehead link complement. These representations
are obtained by considering pairs of regular order three elements in
SL(3,) and can be seen as factorising through a quotient of
defined by a certain exceptional Dehn surgery on the Whitehead link. Our main
result is that these representations form an algebraic component of the
SL(3,)-character variety of .Comment: 20 pages, 3 figures, 4 tables, and a companion Sage notebook (see the
references) v2: A few corrections and improvement
A Spectral Perspective on Neumann-Zagier
We provide a new topological interpretation of the symplectic properties of
gluing equations for triangulations of hyperbolic 3-manifolds, first discovered
by Neumann and Zagier. We also extend the symplectic properties to more general
gluings of PGL(2,C) flat connections on the boundaries of 3-manifolds with
topological ideal triangulations, proving that gluing is a K_2 symplectic
reduction of PGL(2,C) moduli spaces. Recently, such symplectic properties have
been central in constructing quantum PGL(2,C) invariants of 3-manifolds. Our
methods adapt the spectral network construction of Gaiotto-Moore-Neitzke to
relate framed flat PGL(2,C) connections on the boundary C of a 3-manifold to
flat GL(1,C) connections on a double branched cover S -> C of the boundary.
Then moduli spaces of both PGL(2,C) connections on C and GL(1,C) connections on
S gain coordinates labelled by the first homology of S, and inherit symplectic
properties from the intersection form on homology.Comment: 53 + 12 page
Absolute profinite rigidity and hyperbolic geometry
We construct arithmetic Kleinian groups that are profinitely rigid in the
absolute sense: each is distinguished from all other finitely generated,
residually finite groups by its set of finite quotients. The Bianchi group
with is rigid in
this sense. Other examples include the non-uniform lattice of minimal co-volume
in and the fundamental group of the Weeks manifold
(the closed hyperbolic -manifold of minimal volume).Comment: v2: 35 pages. Final version. To appear in the Annals of Mathematics,
Vol. 192, no. 3, November 202
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