Taut ideal triangulations of 3-manifolds

A taut ideal triangulation of a 3-manifold is a topological ideal triangulation with extra combinatorial structure: a choice of transverse orientation on each ideal 2-simplex, satisfying two simple conditions. The aim of this paper is to demonstrate that taut ideal triangulations are very common, and that their behaviour is very similar to that of a taut foliation. For example, by studying normal surfaces in taut ideal triangulations, we give a new proof of Gabai's result that the singular genus of a knot in the 3-sphere is equal to its genus.Comment: Published in Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol4/paper12.abs.htm

Non ambiguous structures on 3-manifolds and quantum symmetry defects

The state sums defining the quantum hyperbolic invariants (QHI) of hyperbolic oriented cusped $3$-manifolds can be split in a "symmetrization" factor and a "reduced" state sum. We show that these factors are invariants on their own, that we call "symmetry defects" and "reduced QHI", provided the manifolds are endowed with an additional "non ambiguous structure", a new type of combinatorial structure that we introduce in this paper. A suitably normalized version of the symmetry defects applies to compact $3$-manifolds endowed with $PSL_2(\mathbb{C})$-characters, beyond the case of cusped manifolds. Given a manifold $M$ with non empty boundary, we provide a partial "holographic" description of the non-ambiguous structures in terms of the intrinsic geometric topology of $\partial M$. Special instances of non ambiguous structures can be defined by means of taut triangulations, and the symmetry defects have a particularly nice behaviour on such "taut structures". Natural examples of taut structures are carried by any mapping torus with punctured fibre of negative Euler characteristic, or by sutured manifold hierarchies. For a cusped hyperbolic $3$-manifold $M$ which fibres over $S^1$, we address the question of determining whether the fibrations over a same fibered face of the Thurston ball define the same taut structure. We describe a few examples in detail. In particular, they show that the symmetry defects or the reduced QHI can distinguish taut structures associated to different fibrations of $M$. To support the guess that all this is an instance of a general behaviour of state sum invariants of 3-manifolds based on some theory of 6j-symbols, finally we describe similar results about reduced Turaev-Viro invariants.Comment: 58 pages, 32 figures; exposition improved, ready for publicatio

Ratio coordinates for higher Teichm\"uller spaces

We define new coordinates for Fock-Goncharov's higher Teichm\"uller spaces for a surface with holes, which are the moduli spaces of representations of the fundamental group into a reductive Lie group $G$. Some additional data on the boundary leads to two closely related moduli spaces, the $\mathscr{X}$-space and the $\mathscr{A}$-space, forming a cluster ensemble. Fock and Goncharov gave nice descriptions of the coordinates of these spaces in the cases of $G = PGL_m$ and $G=SL_m$, together with Poisson structures. We consider new coordinates for higher Teichm\"uller spaces given as ratios of the coordinates of the $\mathscr{A}$-space for $G=SL_m$, which are generalizations of Kashaev's ratio coordinates in the case $m=2$. Using Kashaev's quantization for $m=2$, we suggest a quantization of the system of these new ratio coordinates, which may lead to a new family of projective representations of mapping class groups. These ratio coordinates depend on the choice of an ideal triangulation decorated with a distinguished corner at each triangle, and the key point of the quantization is to guarantee certain consistency under a change of such choices. We prove this consistency for $m=3$, and for completeness we also give a full proof of the presentation of Kashaev's groupoid of decorated ideal triangulations.Comment: 42 pages, 6 figure