145 research outputs found

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

    Full text link
    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

    Full text link
    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,C\mathbb C)-representations of the Whitehead link group

    Get PDF
    We describe a family of representations in SL(3,C\mathbb C) of the fundamental group π\pi of the Whitehead link complement. These representations are obtained by considering pairs of regular order three elements in SL(3,C\mathbb C) and can be seen as factorising through a quotient of π\pi 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,C\mathbb C)-character variety of π\pi.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

    Get PDF
    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

    Full text link
    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 PSL(2,Z[ω])\mathrm{PSL}(2,\mathbb{Z}[\omega]) with ω2+ω+1=0\omega^2+\omega+1=0 is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in PSL(2,C)\mathrm{PSL}(2,\mathbb{C}) and the fundamental group of the Weeks manifold (the closed hyperbolic 33-manifold of minimal volume).Comment: v2: 35 pages. Final version. To appear in the Annals of Mathematics, Vol. 192, no. 3, November 202
    corecore