New hyperbolic 4-manifolds of low volume

We prove that there are at least 2 commensurability classes of minimal-volume hyperbolic 4-manifolds. Moreover, by applying a well-known technique due to Gromov and Piatetski-Shapiro, we build the smallest known non-arithmetic hyperbolic 4-manifold.Comment: 21 pages, 6 figures. Added the Coxeter diagrams of the commensurability classes of the manifolds. New and better proof of Lemma 2.2. Modified statements and proofs of the main theorems: now there are two commensurabilty classes of minimal volume manifolds. Typos correcte

Geometric transition from hyperbolic to Anti-de Sitter structures in dimension four

We provide the first examples of geometric transition from hyperbolic to Anti-de Sitter structures in dimension four, in a fashion similar to Danciger's three-dimensional examples. The main ingredient is a deformation of hyperbolic 4-polytopes, discovered by Kerckhoff and Storm, eventually collapsing to a 3-dimensional ideal cuboctahedron. We show the existence of a similar family of collapsing Anti-de Sitter polytopes, and join the two deformations by means of an opportune half-pipe orbifold structure. The desired examples of geometric transition are then obtained by gluing copies of the polytope.Comment: 50 pages, 27 figures. Part 3 of the previous version has been removed and will be part of a new preprint to appear soo

Counting cusped hyperbolic 3-manifolds that bound geometrically

We show that the number of isometry classes of cusped hyperbolic $3$-manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and non-arithmetic settings.Comment: 17 pages, 7 figures; to appear in Transactions AM

Hyperbolic Dehn filling in dimension four

We introduce and study some deformations of complete finite-volume hyperbolic four-manifolds that may be interpreted as four-dimensional analogues of Thurston's hyperbolic Dehn filling. We construct in particular an analytic path of complete, finite-volume cone four-manifolds $M_t$ that interpolates between two hyperbolic four-manifolds $M_0$ and $M_1$ with the same volume $\frac {8}3\pi^2$. The deformation looks like the familiar hyperbolic Dehn filling paths that occur in dimension three, where the cone angle of a core simple closed geodesic varies monotonically from $0$ to $2\pi$. Here, the singularity of $M_t$ is an immersed geodesic surface whose cone angles also vary monotonically from $0$ to $2\pi$. When a cone angle tends to $0$ a small core surface (a torus or Klein bottle) is drilled producing a new cusp. We show that various instances of hyperbolic Dehn fillings may arise, including one case where a degeneration occurs when the cone angles tend to $2\pi$, like in the famous figure-eight knot complement example. The construction makes an essential use of a family of four-dimensional deforming hyperbolic polytopes recently discovered by Kerckhoff and Storm.Comment: 60 pages, 23 figures. Final versio

Sintesi in situ e caratterizzazione di nanocompositi polistirenici contenenti fillosilicati lamellari modificati con un liquido ionico polimerizzabile e coloranti perilenici

In questo lavoro di tesi sono stati preparati compositi a matrice polistirenica contenenti fillosilicati (Montmorillonite modificata con due diversi coloranti ed un liquido ionico polimerizzabile) con una concentrazione del 2,5% in peso di argilla modificata. Le cariche inorganiche modificate sono state preparate mediante una reazione di scambio cationico in soluzione acquosa sia per via indiretta con l’intercalazione prima del colorante e successivamente del liquido ionico, che per via diretta, in un singolo stadio. I coloranti utilizzati in questo lavoro di tesi appartengono alla classe dei derivati perilenici: • N,N’-bis-(propilendimetildecilammonio)–3,4,9,10-perilendiimmide bromuro • N,N’-bis-(propilentrimetileammonio)–3,4,9,10-perilendiimmide ioduro Il liquido ionico utilizzato è l’1-Dodecil-3-(4-vinilbenzil)-Imidazolio Cloruro. Le cariche modificate preparate sono state analizzate attraverso analisi FT-IR, per una valutazione qualitativa dell’avvenuta reazione di scambio, analisi TGA per valutare le quantità effettivamente intercalate di colorante e di liquido ionico. Mediante XRD è stato inoltre possibile determinare l’aumento di distanza interlamellare dovuto alla presenza delle specie intercalate. La sintesi dei compositi è stata effettuata per polimerizzazione in situ, mediante dispersione della carica modificata nel monomero liquido (stirene) e successiva polimerizzazione radicalica in massa. I compositi preparati, sono stati caratterizzati tramite FT-IR, XRD e TEM per valutare il grado di intercalazione o esfoliazione, TGA e DSC, per valutare le proprietà termiche. L’utilizzo dei coloranti ha permesso di effettuare uno studio delle loro proprietà ottiche (UV-Vis) in assorbimento ed in emissione di fluorescenza, in funzione dei diversi intorni chimici impiegati (soluzione, sospensione e dispersione nei compositi). Si è studiata la tendenza o meno dei coloranti a formare aggregati ed in particolare è stato osservato un comportamento peculiare di ciascun colorante quando si trovano dispersi nella matrice polimerica nel composito, suggerendo un possibile utilizzo futuro di questi sistemi nel campo dei sensori polimerici fotoresponsivi

A small cusped hyperbolic 4-manifold

By gluing some copies of a polytope of Kerckhoff and Storm's, we build the smallest known orientable hyperbolic 4-manifold that is not commensurable with the ideal 24-cell or the ideal rectified simplex. It is cusped and arithemtic, and has twice the minimal volume.Comment: 9 pages, 5 figures. Final version, to appear in the Bulletin LM

Compact hyperbolic manifolds without spin structures

We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n \geq 4$. The core of the argument is the construction of a compact orientable hyperbolic $4$-manifold $M$ that contains a surface $S$ of genus $3$ with self intersection $1$. The $4$-manifold $M$ has an odd intersection form and is hence not spin. It is built by carefully assembling some right angled $120$-cells along a pattern inspired by the minimum trisection of $\mathbb{C}\mathbb{P}^2$. The manifold $M$ is also the first example of a compact orientable hyperbolic $4$-manifold satisfying any of these conditions: 1) $H_2(M,\mathbb{Z})$ is not generated by geodesically immersed surfaces. 2) There is a covering $\tilde{M}$ that is a non-trivial bundle over a compact surface.Comment: 23 pages, 16 figure

Character varieties of a transitioning Coxeter 4-orbifold

In 2010, Kerckhoff and Storm discovered a path of hyperbolic 4-polytopes eventually collapsing to an ideal right-angled cuboctahedron. This is expressed by a deformation of the inclusion of a discrete reflection group (a right-angled Coxeter group) in the isometry group of hyperbolic 4-space. More recently, we have shown that the path of polytopes can be extended to Anti-de Sitter geometry so as to have geometric transition on a naturally associated 4-orbifold, via a transitional half-pipe structure. In this paper, we study the hyperbolic, Anti-de Sitter, and half-pipe character varieties of Kerckhoff and Storm's right-angled Coxeter group near each of the found holonomy representations, including a description of the singularity that appears at the collapse. An essential tool is the study of some rigidity properties of right-angled cusp groups in dimension four.Comment: 51 pages, 7 figures, 4 tables. The content overlaps with Part 3 of arXiv:1908.05112v1, of which this paper constitutes a self-contained and enlarged versio

Plusieurs variétés hyperboliques en dimension trois aux bouts cuspides ne sont pas des bords géodésiques

In this note, we show that there exist cusped hyperbolic 3-manifolds that embed geodesically, but cannot bound geometrically. Thus, being a geometric boundary is a non-trivial property for such manifolds. Our result complements the work by Long and Reid on geometric boundaries of compact hyperbolic 4-manifolds, and by Kolpakov, Reid and Slavich on embedding arithmetic hyperbolic manifolds.Nous montrons que plusieurs variétés hyperboliques en dimension trois aux bouts cuspides ne sont pas des bords géométriques. Alors, cette propriété est en fait suffisamment rare. Nos résultats augmentent le travail par Long et Reid dans le cas de variétés hyperboliques compactes en dimension trois qui fournissent des bords géodésiques pour les variétés hyperboliques compactes en dimensions quatre, et aussi le travail par Kolpakov, Reid et Slavich sur plongements géodésiques pour les variétés hyperboliques arithmétiques

Convex plumbings in closed hyperbolic 4-manifolds

We show that every plumbing of disc bundles over surfaces whose genera satisfy a simple inequality may be embedded as a convex submanifold in some closed hyperbolic four-manifold. In particular its interior has a geometrically finite hyperbolic structure that covers a closed hyperbolic four-manifold.Comment: 18 pages, 11 figure