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Lorenzo Ruffoni

Publications and source records attributed to Lorenzo Ruffoni.

16 recordsLinked to original sources
arXiv2026

Convex cocompact groups with three-dimensional limit sets

We provide a general construction of convex cocompact hyperbolic reflection groups with three-dimensional limit sets. More precisely, our construction takes as input an arbitrary simplicial complex L of dimension 3 on n vertices, and outputs a convex cocompact right-angled reflection group acting on real hyperbolic n-space whose nerve is precisely the Przytycki-\'Swi\k{a}tkowski subdivision of L. Moreover, the output reflection group is a thin subgroup of an n-dimensional cocompact arithmetic hyperbolic lattice. This answers affirmatively a question of M. Kapovich concerning the existence of a convex cocompact group acting on some real hyperbolic space with limit set a \v{C}ech cohomology sphere other than the standard sphere.

math.GR↗
arXiv2026

Incubulable hyperbolic 3-pseudomanifold groups

We construct compact hyperbolic 3-manifolds with totally geodesic boundary, such that the closed 3-pseudomanifolds obtained by coning off the boundary components are negatively curved and contain locally convex subspaces whose fundamental groups have property (T). In particular, the fundamental groups of these 3-pseudomanifolds are word hyperbolic but not cubulable. We deduce that in any relative cubulation of one of these hyperbolic 3-manifold groups some hyperplane stabilizer has infinite intersection with the fundamental group of some boundary component.

math.GT↗
arXiv2025

Convex cocompact groups in real hyperbolic spaces with limit set a Pontryagin sphere

We exhibit two examples of convex cocompact subgroups of the isometry groups of real hyperbolic spaces with limit set a Pontryagin sphere: one generated by $50$ reflections of $\mathbb{H}^4$, and the other by a rotation of order $21$ and a reflection of $\mathbb{H}^6$. For each of them, we also locate convex cocompact subgroups with limit set a Menger curve.

math.GT↗
arXiv2023

Infinite circle packings on surfaces with conical singularities

We show that given an infinite triangulation $K$ of a surface with punctures (i.e., with no vertices at the punctures) and a set of target cone angles smaller than $\pi$ at the punctures that satisfy a Gauss-Bonnet inequality, there exists a hyperbolic metric that has the prescribed angles and supports a circle packing in the combinatorics of $K$. Moreover, if $K$ is very symmetric, then we can identify the underlying Riemann surface and show that it does not depend on the angles. In particular, this provides examples of a triangulation $K$ and a conformal class $X$ such that there are infinitely many conical hyperbolic structures in the conformal class $X$ with a circle packing in the combinatorics of $K$. This is in sharp contrast with a conjecture of Kojima-Mizushima-Tan in the closed case.

math.GT↗
arXiv2023

Relative cubulation of relative strict hyperbolization

We prove that many relatively hyperbolic groups obtained by relative strict hyperbolization admit a cocompact action on a CAT(0) cubical complex. Under suitable assumptions on the peripheral subgroups, these groups are residually finite and even virtually special. We include some applications to the theory of manifolds, such as the construction of new non-positively curved Riemannian manifolds with residually finite fundamental group, and the existence of non-triangulable aspherical manifolds with virtually special fundamental group.

math.GR↗
arXiv2022

A graphical description of the BNS-invariants of Bestvina-Brady groups and the RAAG recognition problem

A finitely presented Bestvina-Brady group (BBG) admits a presentation involving only commutators. We show that if a graph admits a certain type of spanning trees, then the associated BBG is a right-angled Artin group (RAAG). As an application, we obtain that the class of BBGs contains the class of RAAGs. On the other hand, we provide a criterion to certify that certain finitely presented BBGs are not isomorphic to RAAGs (or more general Artin groups). This is based on a description of the Bieri-Neumann-Strebel invariants of finitely presented BBGs in terms of separating subgraphs, analogous to the case of RAAGs. As an application, we characterize when the BBG associated to a 2-dimensional flag complex is a RAAG in terms of certain subgraphs.

math.GR↗
arXiv2022

Manifolds without real projective or flat conformal structures

In any dimension at least five we construct examples of closed smooth manifolds with the following properties: 1) they have neither real projective nor flat conformal structures; 2) their fundamental group is a non-elementary Gromov hyperbolic group. These examples are obtained via relative strict hyperbolization.

math.DG↗
arXiv2022

Special cubulation of strict hyperbolization

We prove that the Gromov hyperbolic groups obtained by the strict hyperbolization procedure of Charney and Davis are virtually compact special, hence linear and residually finite. Our strategy consists in constructing an action of a hyperbolized group on a certain dual CAT(0) cubical complex. As a result, all the common applications of strict hyperbolization are shown to provide manifolds with virtually compact special fundamental group. In particular, we obtain examples of closed negatively curved Riemannian manifolds whose fundamental groups are linear and virtually algebraically fiber.

math.GR↗
arXiv2021

Tame and relatively elliptic $\mathbb{CP}^1$-structures on the thrice-punctured sphere

Suppose a relatively elliptic representation $\rho$ of the fundamental group of the thrice-punctured sphere $S$ is given. We prove that all projective structures on $S$ with holonomy $\rho$ and satisfying a tameness condition at the punctures can be obtained by grafting certain circular triangles. The specific collection of triangles is determined by a natural framing of $\rho$. In the process, we show that (on a general surface $\Sigma$ of negative Euler characteristics) structures satisfying these conditions can be characterized in terms of their M\"obius completion, and in terms of certain meromorphic quadratic differentials.

math.GT↗
arXiv2020

Graphical splittings of Artin kernels

We study Artin kernels, i.e. kernels of discrete characters of right-angled Artin groups, and we show that they decompose as graphs of groups in a way that can be explicitly computed from the underlying graph. When the underlying graph is chordal we show that every such subgroup either surjects to an infinitely generated free group or is a generalized Baumslag-Solitar group of variable rank. In particular for block graphs (e.g. trees), we obtain an explicit rank formula, and discuss some features of the space of fibrations of the associated right-angled Artin group.

math.GR↗
arXiv2020

Local deformations of branched projective structures: Schiffer variations and the Teichmüller map

We study a class of continuous deformations of branched complex projective structures on closed surfaces of genus $g\geq 2$, which preserve the holonomy representation of the structure and the order of the branch points. In the case of non-elementary holonomy we show that when the underlying complex structure is infinitesimally preserved the branch points are necessarily arranged on a canonical divisor, and we establish a partial converse for hyperelliptic structures.

math.CV↗
arXiv2019

Complex projective structures with maximal number of Möbius transformations

We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More generally we show that Galois Bely\uı curves are precisely those Riemann surfaces for which the Fuchsian uniformization is the unique complex projective structure invariant under the full group of biholomorphisms.

math.GT↗
arXiv2019

Bubbling complex projective structures with quasi-Fuchsian holonomy

For a given quasi-Fuchsian representation $ρ:π_1(S)\to$ PSL$_2\mathbb{C}$ of the fundamental group of a closed surface $S$ of genus $g\geq 2$, we prove that a generic branched complex projective structure on $S$ with holonomy $ρ$ and two branch points is obtained by bubbling some unbranched structure on $S$ with the same holonomy.

math.GT↗
arXiv2019

Multi(de)grafting quasi-Fuchsian complex projective structures via bubbles

We show that the simultaneous (de)grafting of a complex projective structure with quasi-Fuchsian holonomy along a multicurve can be performed by a simple sequence of one bubbling and one debubbling. As a consequence we obtain that any complex projective structure with quasi-Fuchsian holonomy $ρ:π_1(S)\to$ PSL$_2\mathbb{C}$ can be joined to the corresponding uniformizing structure $σ_ρ$ by a simple sequence of one bubbling and one debubbling, with a stopover in the space of branched complex projective structures.

math.GT↗
arXiv2018

Extending an example by Colding and Minicozzi

Extending an example by Colding and Minicozzi, we construct a sequence of properly embedded minimal disks $Σ_i$ in an infinite Euclidean cylinder around the $x_3$-axis with curvature blow-up at a single point. The sequence converges to a non smooth and non proper minimal lamination in the cylinder. Moreover, we show that the disks $Σ_i$ are not properly embedded in a sequence of open subsets of $\mathbb{ R}^3$ that exhausts $\mathbb{ R}^3$.

math.DG↗