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Bruno Dular

Publications and source records attributed to Bruno Dular.

4 recordsLinked to original sources
arXiv2025

Bending parameterization of one-sided degenerated Kleinian surface groups

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.

math.GT↗
arXiv2024

Primitive Feynman diagrams and the rational Goussarov--Habiro Lie algebra of string links

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 $n\geq 1$, 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 $\mathcal{L} L(m)$. We give a concrete presentation, in terms of primitive Feynman (tree) diagrams and relations ($\text{1T}$, $\text{AS}$, $\text{IHX}$, $\text{STU}^2$), of the rational Goussarov-Habiro Lie algebra $\mathcal{L} L(m)_{\mathbb{Q}}$. To that end, we investigate cycles in graphs of forests: flip graphs associated to forest diagrams and their $\text{STU}$ 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.

math.GT↗
arXiv2019

Cycles of Sums of Integers

We study the period of the linear map $T:\mathbb{Z}_m^n\rightarrow \mathbb{Z}_m^n:(a_0,\dots,a_{n-1})\mapsto(a_0+a_1,\dots,a_{n-1}+a_0)$ as a function of $m$ and $n$, where $\mathbb{Z}_m$ 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.

math.NT↗