We introduce a family of matrix dilogarithms, which are automorphisms of C^N tensor C^N, N being any odd positive integer, associated to hyperbolic ideal tetrahedra equipped with an additional decoration. The matrix dilogarithms satisfy fundamental five-term identities that correspond to decorated versions of the 2 --> 3 move on 3-dimensional triangulations. Together with the decoration, they arise from the solution we give of a symmetrization problem for a specific family of basic matrix dilogarithms, the classical (N=1) one being the Rogers dilogarithm, which only satisfy one special instance of five-term identity. We use the matrix dilogarithms to construct invariant state sums for closed oriented 3-manifolds $W$ endowed with a flat principal PSL(2,C)-bundle rho, and a fixed non empty link L if N>1, and for (possibly "marked") cusped hyperbolic 3-manifolds M. When N=1 the state sums recover known simplicial formulas for the volume and the Chern-Simons invariant. When N>$, the invariants for M are new; those for triples (W,L,rho) coincide with the quantum hyperbolic invariants defined in [Topology 43 (2004) 1373-1423], though our present approach clarifies substantially their nature. We analyse the structural coincidences versus discrepancies between the cases N=1 and N>1, and we formulate "Volume Conjectures", having geometric motivations, about the asymptotic behaviour of the invariants when N tends to infinity.
We give parallel constructions of an invariant R(W,f), based on the classical Rogers dilogarithm, and of quantum hyperbolic invariants (QHI), based on the Faddeev-Kashaev quantum dilogarithms, for flat PSL(2,C)-bundles f over closed oriented 3-manifolds W. All these invariants are explicitely computed as a sum or state sums over the same hyperbolic ideal tetrahedra of the idealization of any fixed simplicial 1-cocycle description of (W,f) of a special kind, called a D-triangulation. R(W,f) recovers the volume and the Chern-Simons invariant of f, and we conjecture that it determines the semi-cl…
Let M be a twisted interval bundle over a nonorientable hyperbolizable surface. Let X(M) be the PSL(2,C)-character variety of π_1(M). We examine the dynamics of the action of Out(π_1(M)) on X(M), and in particular, we find an open set on which the action is properly discontinuous that is strictly larger than the interior of the deformation space of marked hyperbolic manifolds homotopy equivalent to M. Further, we identify which discrete and faithful representations can lie in a domain of discontinuity for the action of Out(π_1(M)) on X(M).
We define the notion of spectral network on manifolds of dimension $\le 3$. For a manifold $X$ equipped with a spectral network, we construct equivalences between Chern-Simons invariants of flat ${\mathrm {SL}}(2,{\mathbb C})$-bundles over $X$ and Chern-Simons invariants of flat ${\mathbb C}^\times$-bundles over ramified double covers $\widetilde X$. Applications include a new viewpoint on dilogarithmic formulas for Chern-Simons invariants of flat ${\mathrm {SL}}(2,{\mathbb C})$-bundles over triangulated 3-manifolds, and an explicit description of Chern-Simons lines of flat ${\mathrm {SL}}(2,{…
Let M be a compact, orientable, hyperbolizable 3-manifold with incompressible boundary which is not an interval bundle. We study the dynamics of the action of the outer automorphism group of the fundamental group of M on the relative PSL(2,C)-character variety.
Karen Uhlenbeck's compactness theorem for sequences of connections with L2 bounds on curvature applies only to connections on principal bundles with compact structure group. This article states and proves an extension of Uhlenbecks theorem that describes sequences of connections on principal PSL(2;C) bundles over compact three dimensional manifolds.
We construct {\it quantum hyperbolic invariants} (QHI) for triples $(W,L,ρ)$, where $W$ is a compact closed oriented 3-manifold, $ρ$ is a flat principal bundle over $W$ with structural group $PSL(2,\mc)$, and $L$ is a non-empty link in $W$. These invariants are based on the Faddeev-Kashaev's {\it quantum dilogarithms}, interpreted as matrix valued functions of suitably decorated hyperbolic ideal tetrahedra. They are explicitely computed as state sums over the decorated hyperbolic ideal tetrahedra of the {\it idealization} of any fixed {\it $\Dd$-triangulation}; the $\Dd$-triangulations are sim…
We show that a PSL(2;R)-representation of a Fuchsian group induces the asymptotics of the Reidemeister torsion for the Seifert manifold corresponding to the euler class of the PSL(2;R)-representation. We also show that the limit of leading coefficient of the Reidemeister torsion is determined by the euler class of a PSL(2;R)-representation of a Fuchsian group. In particular, the leading coefficient of the Reidemeister torsion for the unit tangent bundle over a two-orbifold converges to $-χ\log2$ where $χ$ is the Euler characteristic of the two-orbifold. We also give a relation between $\mathbb…
In this paper we complete the topological description of the space of representations of the fundamental group of a punctured surface in SL(2,R) with prescribed behavior at the punctures and nonzero Euler number, following the strategy employed by Hitchin in the unpunctured case and exploiting Hitchin-Simpson correspondence between flat bundles and Higgs bundles in the parabolic case. This extends previous results by Boden-Yokogawa and Nasatyr-Steer. A relevant portion of the paper is intended to give an overview of the subject.
Let M be the moduli space of irreducible flat PSL(2,R) connections on a punctured surface of finite type with parabolic holonomies around punctures. By using a notion of admissibility of an ideal arc, M is covered by dense open subsets associated to ideal triangulations of the surface. A principal bundle over M is constructed which, when restricted to the Teichmuller component of M, is isomorphic to the decorated Teichmuller space of Penner. The construction gives a generalization to M of Penner's coordinates for the Teichmuller space.
We introduce systems of objects and operators in linear monoidal categories called $\hat Ψ$-systems. A $\hat Ψ$-system satisfying several additional assumptions gives rise to a topological invariant of triples (a closed oriented 3-manifold $M$, a principal bundle over $M$, a link in $M$). This construction generalizes the quantum dilogarithmic invariant of links appearing in the original formulation of the volume conjecture. We conjecture that all quantum groups at odd roots of unity give rise to $\hat Ψ$-systems and we verify this conjecture in the case of the Borel subalgebra of quantum $sl_…
We construct a new family of exact quantum field theories modeled on hyperbolic geometry, called {\it quantum hyperbolic field theories} (QHFTs). The QHFTs are defined for a $(2+1)$-bordism category based on the set of compact oriented 3-manifolds $Y$, equipped with properly embedded framed links $L_\Ff$ and with flat connections $ρ$ of principal $PSL(2,\C)$-bundles over $Y \setminus L_\Ff$, with arbitrary holonomy at the link meridians. A main point is the introduction of new parameters for the space of all $PSL(2,\C)$-characters of a punctured surface. Each QHFT associates to a triple $(Y,L_…
Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathrm{PSL}_2(\mathbb{R})$. The complement of any finite number of orbits is a hyperbolic $3$-manifold, which thus has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics
We review the representation theory of the quantum group $U_εsl_2\mathbb{C}$ at a root of unity $ε$ of odd order, focusing on geometric aspects related to the 3-dimensional quantum hyperbolic field theories (QHFT). Our analysis relies on the quantum coadjoint action of De Concini-Kac-Procesi, and the theory of Heisenberg doubles of Poisson-Lie groups and Hopf algebras. We identify the 6j-symbols of generic representations of $U_εsl2\mathbb{C}$, the main ingredients of QHFT, with a bundle morphism defined over a finite cover of the algebraic quotient $PSL_2\mathbb{C}/!/PSL_2\mathbb{C}$, of degr…
Cauchy-compact flat spacetimes with extreme BTZ are Lorentzian analogue of complete hyperbolic surfaces of finite volume. Indeed, the latter are 2-manifolds locally modeled on the hyperbolic plane, with group of isometries $\mathrm{PSL}_2(\R)$, admitting finitely many cuspidal ends while the regular part of the former are 3-manifolds locally models on 3 dimensionnal Minkowski space, with group of isometries $\mathrm{PSL}_2(\R)\ltimes \RR^3$, admitting finitely many ends whose neighborhoods are foliated by cusps. We prove a Theorem akin to the classical parametrization result for finite volume …
In this paper, it is shown that every closed hyperbolic 3-manifold contains an immersed quasi-Fuchsian closed subsurface of odd Euler characteristic. The construction adopts the good pants method, and the primary new ingredient is an enhanced version of the connection principle, which allows one to connect any two frames with a path of frames in a prescribed relative homology class of the frame bundle. The existence result is applied to show that every uniform lattice of $\mathrm{PSL}(2,\mathbb{C})$ admits an exhausting nested sequence of sublattices with exponential homological torsion growth…
We introduce coordinates for a principal bundle $S\tilde T(F)$ over the super Teichmueller space $ST(F)$ of a surface $F$ with $s\geq 1$ punctures that extend the lambda length coordinates on the decorated bundle $\tilde T(F)=T(F)\times {\mathbb R}_+^s$ over the usual Teichmueller space $T(F)$. In effect, the action of a Fuchsian subgroup of $PSL(2,{\mathbb R})$ on Minkowski space ${\mathbb R}^{2,1}$ is replaced by the action of a super Fuchsian subgroup of $OSp(1|2)$ on the super Minkowski space ${\mathbb R}^{2,1|2}$, where $OSp(1|2)$ denotes the orthosymplectic Lie supergroup, and the lambda…
For any triple $(W,L,ρ)$, where W is a closed connected and oriented 3-manifold, L is a link in W and $ρ$ is a flat principal B-bundle over W (B is the Borel subgroup of $SL(2,\mc)$), one constructs a $\Dd$-scissors congruence class $\cG_{\Dd}(W,L,ρ)$ which belongs to a (pre)-Bloch group $\Pp (\Dd)$. The class $\cG_{\Dd}(W,L,ρ)$ may be represented by $\Dd$-triangulations $\Tt=(T,H,\Dd)$ of $(W,L,ρ)$. For any $\Tt$ and any odd integer $N>1$, one defines a ``quantization'' $\Tt_N$ of $\Tt$ based on the representation theory of the quantum Borel subalgebra $\Ww_N$ of $U_q(sl(2,\mc))$ specialized …
