arXiv subjects

Antonin Guilloux

Publications and source records attributed to Antonin Guilloux.

At least 19 recordsLinked to original sources

arXiv2025

FAMED by computer: proving the Andersen-Kashaev volume conjecture for 42,000 knots

The FAMED condition is a combinatorial property for ideal triangulations of $3$-manifolds, which was introduced in 2024 by the first and last authors in order to study the Andersen--Kashaev volume conjecture. They notably proved that this conjecture is true for all FAMED geometric triangulations of one-cusped hyperbolic $3$-manifolds with trivial second homology. In this paper, using a straightforward computer implementation in Regina and Snappy, we find FAMED geometric triangulations for more than 42.000 complements of knots in $S^3$, including all knots with $12$ crossings or fewer and all knots whose complement can be triangulated with $23$ tetrahedra or fewer. As a consequence, the Andersen-Kashaev conjecture is now proven to be true for as many new examples. Along the way, we find several new insights about the FAMED property, which have great value in the quest of a general proof of the Andersen-Kashaev volume conjecture for every knot complement.

math.GT↗

arXiv2024

Hausdorff dimension, diverging Schottky representations and the infinite dimensional hyperbolic space

One of our main goals in this paper is to understand the behavior of limit sets of a diverging sequence of Schottky groups in the group of isometries of the N-dimensional hyperbolic space. This leads us to a generalization of a classical theorem of Bowen on variations of Hausdorff dimension of limit sets; and to a method of transforming a diverging sequence of Schottky groups into an almost converging sequence in the group of isometries of the infinite dimensional hyperbolic space. Our results apply in particular to an example of McMullen and generalize a previous work by Mehmeti and Dang.

math.DS↗

arXiv2024

Limits of limit sets in rank-one symmetric spaces

We consider the question of continuity of limit sets for sequences of geometrically finite subgroups of isometry groups of rank-one symmetric spaces, and prove analogues of classical (Kleinian) theorems in this context. In particular we show that, assuming strong convergence of the sequence of subgroups, the limit sets vary continuously with respect to Hausdorff distance, and if the sequence is weakly type-preserving, the sequence of Cannon-Thurston maps also converges uniformly to a limiting Cannon-Thurston map. Our approach uses the theory of extended geometrically finite representations, developed recently by the second author.

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arXiv2024

A Hilbert metric for bounded symmetric domains

Bounded symmetric domains carry several natural invariant metrics, for example the Carathéodory, Kobayashi or the Bergman metric. We define another natural metric, from generalized Hilbert metric defined in [FGW20], by considering the Borel embedding of the domain as an open subset of its dual compact Hermitian symmetric space and then its Harish-Chandra realization in projective spaces. We describe this construction on the four classical families of bounded symmetric domains and compute both this metric and its associated Finsler metric. We compare it to Carathéodory and Bergman metrics and show that, except for the complex hyperbolic space, those metrics differ.

math.DG↗

arXiv2022

Limits of Mahler measures in multiple variables

We prove that certain sequences of Laurent polynomials, obtained from a fixed Laurent polynomial P by monomial substitutions, give rise to sequences of Mahler measures which converge to the Mahler measure of P. This generalizes previous work of Boyd and Lawton, who considered univariate monomial substitutions. We provide moreover an explicit upper bound for the error term in this convergence, generalizing work of Dimitrov and Habegger, and a full asymptotic expansion for a family of 2-variable polynomials, whose Mahler measures were studied independently by the third author.

math.NT↗

arXiv2022

Slim curves, limit sets and spherical CR uniformisations

We consider here the $3$-sphere $\mathbf S^3$ seen as the boundary at infinity of the complex hyperbolic plane $\mathbf{H}^2_{\mathbf C}$. It comes equipped with a contact structure and two classes of special curves. First $\mathbf R$-circles are boundaries at infinity of totally real totally geodesic subspaces and are tangent to the contact distribution. Second, $\mathbf C$-circles, which are boundaries of complex totally geodesic subspaces and are transverse to the contact distribution. We define a quantitative notion, called slimness, that measures to what extent a continuous path in the sphere $\mathbf S^3$ is near to be an $\mathbf R$-circle. We analyze the classical foliation of the complement of an $\mathbf R$-circle by arcs of $\mathbf C$-circles. Next, we consider deformations of this situation where the $\mathbf R$-circle becomes a slim curve. We apply these concepts to the particular case where the slim curve is the limit set of a quasi-Fuchsian subgroup of $\mathrm{PU}(2,1)$. As a consequence, we describe a class of spherical CR uniformizations of certain cusped $3$-manifolds.

math.GT↗

arXiv2021

$p$-adic Directions of Primitive Vectors

Linnik type problems concern the distribution of projections of integral points on the unit sphere as their norm increases, and different generalizations of this phenomenon. Our work addresses a question of this type: we prove the uniform distribution of the projections of primitive $\mathbb{Z}^{2}$ points in the $p$-adic unit sphere, as their (real) norm tends to infinity. The proof is via counting lattice points in semi-simple $S$-arithmetic groups.

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arXiv2018

On SL(3,$\mathbb C$)-representations of the Whitehead link group

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

math.GT↗

arXiv2018

Deformation space of discrete groups of SU(2,1) in quaternionic hyperbolic plane

In this note, we study deformations of discrete and Zariski dense subgroups of SU(2, 1) in quaternionic hyperbolic space. Specifi- cally we consider two examples coming from representations of 3-manifold groups (the figure eight knot and Whitehead links complement) and show opposite behavior: one is not deformable outside U(2,1), while the other has a big space of deformations in Sp(2, 1).

math.GT↗

arXiv2018

Hilbert metric, beyond convexity

The Hilbert metric on convex subsets of $\mathbb R^n$ has proven a rich notion and has been extensively studied. We propose here a generalization of this metric to subset of complex projective spaces and give examples of applications to diverse fields. Basic examples include the classical Hilbert metric which coincides with the hyperbolic metric on real hyperbolic spaces as well as the complex hyperbolic metric on complex hyperbolic spaces.

math.MG↗

arXiv2018

Volume function and Mahler measure of exact polynomials

We study a class of 2-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the 2-dimensional torus and give a closed formula for the Mahler measure in terms of these extremal values. This formula shows that the Mahler measure of an irreducible and exact polynomial divided by $π$ is greater than the amplitude of the volume function. We also prove a $K$-theoretical criterium for a polynomial to be a factor of an $A$-polynomial and give a topological interpretation of its Mahler measure.

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arXiv2016

Yet another p-adic hyperbolic disc

We describe in this paper a geometric construction in the projective p-adic plane that gives, together with a suitable notion of p-adic convexity, some open subsets of P 2 .Q p / naturally endowed with a "Hilbert" distance and a transitive action of PGL.2; Q p / by isometries. ese open sets are natural analogues of the hyperbolic disc, more precisely of Klein's projective model. But, unlike the real case, there is not only one such hyperbolic disc. Indeed, we nd three of them if p is odd (and seven if p D 2). Let us stress out that neither the usual notion of convexity nor that of connectedness as known for the real case are meaningful in the p-adic case. us, there will be a rephrasing game for the denitions of real convexity until we reach a formulation suitable for other local elds. It will lead us to a denition of p-adic convexity by duality. Although we will not recover the beautiful behaviour of real convexity, we will still be able to dene the most important tool for our goals, namely the Hilbert distance. We construct our analogues of the hyperbolic disc (once again, via the projective model of the hyperbolic plane) in a quite geometric, even naive, way. Our construction gives 2-dimensional objects over Q p. It is very dierent, in spirit and in facts, of Drinfeld p-adic hyperbolic plane []. e possible relations between the two objects remain still unexplored. Another object often viewed as an analogue of the hyperbolic disc is the tree of PGL.2; Q p / []. We explore the relations between our discs and this tree, constructing a natural quasi-isometric projection from the discs to the tree. Eventually we explore the transformation groups of our discs. And, whereas the transformation group of the tree is huge, we prove that only PGL.2; Q p / acts on the discs preserving the convex structure.

math.GT↗

arXiv2016

Deformation of hyperbolic manifolds in $\mathrm {PGL}(n,\mathbf {C})$ and discreteness of the peripheral representations

Let $ M$ be a cusped hyperbolic $ 3$-manifold, e.g. a knot complement. Thurston showed that the space of deformations of its fundamental group in $ \mathrm {PGL}(2,\mathbf {C})$ (up to conjugation) is of complex dimension the number $ ν$ of cusps near the hyperbolic representation. It seems natural to ask whether some representations remain discrete after deformation. The answer is generically not. A simple reason for it lies inside the cusps: the degeneracy of the peripheral representation (i.e. representations of fundamental groups of the $ ν$ peripheral tori). They indeed generically become non-discrete, except for a countable set. This last set corresponds to hyperbolic Dehn surgeries on $ M$, for which the peripheral representation is no more faithful.We work here in the framework of $ \mathrm {PGL}(n,\mathbf {C})$. The hyperbolic structure lifts, via the $ n$-dimensional irreducible representation, to a representation $ ρ\_{\mathrm {geom}}$. We know from the work of Menal-Ferrer and Porti that the space of deformations of $ ρ\_{\textrm {geom}}$ has complex dimension $ (n-1)ν$.We prove here that, unlike the $ \mathrm {PGL}(2)$-case, the generic behaviour becomes the discreteness (and faithfulness) of the peripheral representation: in a neighbourhood of the geometric representation, the non-discrete peripheral representations are contained in a real analytic subvariety of codimension $ \geq 1$.

math.GT↗

arXiv2016

Equidistribution in S-arithmetic and adelic spaces

We give an introduction to adelic mixing and its applications for mathematicians knowing about the mixing of the geodesic flow on hyperbolic surfaces. We focus on the example of the Hecke trees in the modular surface.

math.DS↗

arXiv2016

Volumes of representations and birationality of the peripheral holonomy

We discuss here a generalization of a theorem by Dunfield stating that the peripheral holonomy map, from the character variety of a 3-manifold to the A-polynomial is birational. Dunfield's proof involves the rigidity of maximal volume. The volume is still an important ingredient in this paper. Unfortunately at this point no complete proof is done. Instead, a conjecture is stated about the volume function on the character variety that would imply the generalized birationality result.

math.GT↗

arXiv2015

Dimension of character varieties for $3$-manifolds

Let $M$ be a $3$-manifold, compact with boundary and $Γ$ its fundamental group. Consider a complex reductive algebraic group G. The character variety $X(Γ,G)$ is the GIT quotient $\mathrm{Hom}(Γ,G)//G$ of the space of morphisms $Γ\to G$ by the natural action by conjugation of $G$. In the case $G=\mathrm{SL}(2,\mathbb C)$ this space has been thoroughly studied. Following work of Thurston, as presented by Culler-Shalen, we give a lower bound for the dimension of irreducible components of $X(Γ,G)$ in terms of the Euler characteristic $χ(M)$ of $M$, the number $t$ of torus boundary components of $M$, the dimension $d$ and the rank $r$ of $G$. Indeed, under mild assumptions on an irreducible component $X_0$ of $X(Γ,G)$, we prove the inequality $$\mathrm{dim}(X_0)\geq t \cdot r - dχ(M).$$

math.GT↗

arXiv2014

Character varieties for SL(3,C): the figure eight knot

We give a description of several representation varieties of the fundamental group of the complement of the figure eight knot in PGL(3,C) or SL(3,C). We moreover obtain an explicit parametrization of matrices generating the representation and a description of the projection of the representation variety into the character variety of the boundary torus into SL(3,C).

math.GT↗

arXiv2013

Representations of 3-manifold groups in PGL(n,C) and their restriction to the boundary

We study here the space of representations of a fundamental group of a 3-manifold into PGL(n,C). Thurston, Neumann and Zagier initiated a strategy (in the case of PGL(2,C)) consisting in: triangulate the manifold, assign shapes to each pieces and then try to glue back. This leads to the "gluing equations" and the Neumann-Zagier symplectic space. Building on the works of Dimofte-Gabella-Goncharov and Bergeron-Falbel-Guilloux, we complete the picture in the case of PGL(n,C). We recover a situation very similar to the case of PGL(2,C). This allows for example to obtain a combinatorial proof of a local rigidity results for such representations.

math.GT↗