Searcharxiv⌕ Search

arXiv subjects

Pierre Will

Publications and source records attributed to Pierre Will.

13 recordsLinked to original sources
arXiv2026

Parabolic-preserving deformations of cusped hyperbolic lattices

We study deformations of non-cocompact lattices of ${\rm SO}(n,1)$ into ${\rm SU}(n,1)$ and ${\rm SO}(n+1,1)$. A necessary condition for these deformations to remain discrete and faithful (when $n \geqslant 3$) is for the parabolic subgroups to remain parabolic and discrete; we call such representations \emph{strongly parabolic-preserving}. We show that the figure-eight knot group admits a one-parameter family of Zariski-dense parabolic-preserving deformations into ${\rm SU}(3,1)$, with further deformations into ${\rm SU}(2,2)$. We also study the \emph{bending deformations} of the Bianchi groups (seen as subgroups of ${\rm SO}(3,1)$) along the modular surface into ${\rm SU}(3,1)$ and ${\rm SO}(4,1)$, and show that infinitely many of them are strongly parabolic-preserving in ${\rm SU}(3,1)$, while none are strongly parabolic-preserving in ${\rm SO}(4,1)$. Finally, for any $n \geqslant 3$, we show that there exist infinitely many non-commensurable cusped hyperbolic $n$-manifolds whose corresponding hyperbolic representation admits a 1-parameter family of parabolic-preserving deformations into ${\rm SU}(n,1)$.

math.GT↗
arXiv2024

A Hilbert metric for bounded symmetric domains

Bounded symmetric domains carry several natural invariant metrics, for example the Carath\'eodory, 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 Carath\'eodory and Bergman metrics and show that, except for the complex hyperbolic space, those metrics differ.

math.DG↗
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↗
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↗
arXiv2017

Rank 1 deformations of non-cocompact hyperbolic lattices

Let $X$ be a negatively curved symmetric space and $\Gamma$ a non-cocompact lattice in $\rm{Isom}(X)$. We show that small, parabolic-preserving deformations of $\Gamma$ into the isometry group of any negatively curved symmetric space containing $X$ remain discrete and faithful (the cocompact case is due to Guichard). This applies in particular to a version of Johnson-Millson bending deformations, providing for all $n$ infnitely many non-cocompact lattices in ${\rm SO}(n,1)$ which admit discrete and faithful deformations into ${\rm SU}(n,1)$. We also produce deformations of the figure-8 knot group into $\rm{SU}(3,1)$, not of bending type, to which the result applies.

math.GT↗
arXiv2016

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 $\pi$ 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 $\pi$ 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 $\pi$.

math.GT↗
arXiv2015

A complex hyperbolic Riley slice

We study subgroups of ${\rm PU}(2,1)$ generated by two non-commuting unipotent maps $A$ and $B$ whose product $AB$ is also unipotent. We call $\mathcal{U}$ the set of conjugacy classes of such groups. We provide a set of coordinates on $\mathcal{U}$ that make it homeomorphic to $\mathbb{R}^2$ . By considering the action on complex hyperbolic space $\mathbf{H}^2_{\mathbb{C}}$ of groups in $\mathcal{U}$, we describe a two dimensional disc ${\mathcal Z}$ in $\mathcal{U}$ that parametrises a family of discrete groups. As a corollary, we give a proof of a conjecture of Schwartz for $(3,3,\infty)$-triangle groups. We also consider a particular group on the boundary of the disc ${\mathcal Z}$ where the commutator $[A,B]$ is also unipotent. We show that the boundary of the quotient orbifold associated to the latter group gives a spherical CR uniformisation of the Whitehead link complement.

math.GT↗
arXiv2013

Complex hyperbolic free groups with many parabolic elements

We consider in this work representations of the of the fundamental group of the 3-punctured sphere in ${\rm PU}(2,1)$ such that the boundary loops are mapped to ${\rm PU}(2,1)$. We provide a system of coordinates on the corresponding representation variety, and analyse more specifically those representations corresponding to subgroups of $(3,3,\infty)$-groups. In particular we prove that it is possible to construct representations of the free group of rank two $\la a,b\ra$ in ${\rm PU}(2,1)$ for which $a$, $b$, $ab$, $ab^{-1}$, $ab^2$, $a^2b$ and $[a,b]$ all are mapped to parabolics.

math.GT↗
arXiv2013

Real reflections, commutators and cross-ratios in complex hyperbolic space

We provide a concrete criterion to determine whether or not two given elements of PU(2,1) can be written as products of real reflections, with one reflection in common. As an application, we show that the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ with $d=1,2,3,7,11$ are generated by real reflections up to index 1, 2, 4 or 8.

math.GT↗
arXiv2011

Bending Fuchsian representations of fundamental groups of cusped surfaces in PU(2,1)

We describe a family of representations of $π_1(Σ)$ in PU(2,1), where $Σ$ is a hyperbolic Riemann surface with at least one deleted point. This family is obtained by a bending process associated to an ideal triangulation of $Σ$. We give an explicit description of this family by describing a coordinates system in the spirit of shear coordinates on the Teichmüller space. We identify within this family new examples of discrete, faithful and type-preserving representations of $π_1(Σ)$. In turn, we obtain a 1-parameter family of embeddings of the Teichmüller space of $Σ$ in the PU(2,1)-representation variety of $π_1(Σ)$. These results generalise to arbitrary $Σ$ the results obtained in a previous paper for the 1-punctured torus.

math.GT↗
arXiv2007

A la Fock-Goncharov coordinates for PU(2,1)

We describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface $S$ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperpolic plane. We establish a bijection between a set of decorations of an ideal triangulation of $S$ and a subset of the PU(2,1)-representation variety of $π_1(S)$.

math.DG↗