Bruno Dular

We show that non-Fuchsian geometrically finite hyperbolic structures on a given hyperbolizable 3-manifold M are uniquely determined, up to isotopy, by their bending laminations. Consequently, the bending map from the space of non-Fuchsian geometrically finite structures on M, endowed with the strong topology, to the space of bending laminations, endowed with Lecuire's tubular topology, is a homeomorphism.

This extends the convex co-compact case established with Schlenker. The proof combines hyperbolic Dehn filling, continuity and properness of the bending map (Lecuire) and real-analyticity of its fibres (Bonahon). We also establish a criterion for contractibility of the boundary fibres of a continuous extension of a homeomorphism, extending Finney's theorem to a boundary setting. This topological result may be of independent interest.

Source: arXiv

Convex co-compact 3-dimensional hyperbolic manifolds are uniquely determined by the pleating measured lamination on the boundary of their convex core.

Source: arXiv

Goussarov-Habiro's theory of clasper surgeries defines a filtration of the monoid of string links L(m) on m strands, in a way that geometrically realizes the Feynman diagrams appearing in low-dimensional and quantum topology. Concretely, L(m) is filtered by C_n-equivalence, for n ≥ 1, which is defined via local moves that can be seen as higher crossing changes.

The graded object associated to the Goussarov-Habiro filtration is the Goussarov-Habiro Lie algebra of string links ℒL(m). We give a concrete presentation, in terms of primitive Feynman (tree) diagrams and relations (1T, AS, IHX, STU²), of the rational Goussarov-Habiro Lie algebra ℒL(m)_ℚ. To that end, we investigate cycles in graphs of forests: flip graphs associated to forest diagrams and their STU relations.

As an application, we give an alternative diagrammatic proof of Massuyeau's rational version of the Goussarov-Habiro conjecture for string links, which relates indistinguishability under finite type invariants of degree <n and C_n-equivalence.

Source: arXiv

It was recently proved that quasi-Fuchsian manifolds are uniquely determined by their bending laminations. This paper concerns a similar result for certain non-quasi-Fuchsian manifolds : those obtained by degenerating one end but not the other, i.e. those appearing in boundaries of Bers slices. More precisely, we show that such hyperbolic manifolds are uniquely determined by the end structure of the degenerated end and the bending lamination of the other.

The end structure consists of the parabolic locus, which is a multicurve, together with ending laminations or conformal structures on each components of its complement.

Source: arXiv

We study the period of the linear map T: ℤ_mⁿ → ℤ_mⁿ: (a₀, …, a_n−1) ↦ (a₀+a₁, …, a_n−1+a₀) as a function of m and n, where ℤ_m stands for the ring of integers modulo m. Since this map is a variant of the Ducci sequence, several known results are adapted in the context of T. The main theorem of this paper states that the period modulo m can be deduced from the prime factorization of m and the periods of its prime factors.

We also characterize the tuples that belong to a cycle when m is prime.

Source: arXiv