30 research outputs found
Multiplicité des valeurs propres du laplacien sur les surfaces hyperboliques triangulaires
No full textCe mémoire porte sur l’étude du laplacien sur des surfaces de Riemann. En particulier, nous nous intéressons à ses valeurs propres qui représentent les notes que jouerait la surface si elle était un tambour. Les valeurs les plus étudiées sont la première valeur propre non nulle λ1 ainsi que sa multiplicité m1 (la dimension de l’espace propre). Notamment, Colin de Verdière conjecturait que m1 est toujours borné supérieurement par le nombre chromatique moins 1. Des travaux de Fortier Bourque et Petri ont montré que parmi toutes les surfaces hyperboliques de genre 3, c’est la quartique de Klein qui maximise la multiplicité et atteint la borne supérieure conjecturée par Colin de Verdière. Cette surface est la première d’une suite de surfaces hautement symétriques, les surfaces de Hurwitz. Nous montrons à l’aide de la formule des traces de Selberg que pour la prochaine surface dans la suite, la surface de Fricke–Macbeath F, nous avons m1(F) = 7. Une recherche indépendante menée par Chul-hee Lee arrive au même résultat à propos de la multiplicité.
Le chapitre 1 introduit des notions géométriques comme la géométrie hyperbolique, les surfaces hyperboliques et triangulaires ainsi que le théorème de Hurwitz. Le chapitre 2 présente des concepts de base de théorie spectrale ainsi que des outils comme la formule des traces de Selberg et la théorie de la représentation. Le chapitre 3 est dédié à l’étude de la surface de Fricke–Macbeath et à la preuve de notre résultat principal à l’aide des outils des chapitres précédents. Dans le chapitre 4, nous discutons de nouvelles techniques de calcul de m1 qui ont été utilisées pour montrer l’existence de contre-exemples à la conjecture de Colin de Verdière dans des travaux conjoints avec Fortier Bourque, Gruda-Mediavilla et Petri.This master’s thesis studies the Laplace operator on Riemann surfaces. We are especially interested in its eigenvalues, which correspond to the notes that the surface would play if it were a drum. In particular, the first non-zero eigenvalue λ1 and its multiplicity m1 (the dimension of the corresponding eigenspace) have been well studied. For instance, Colin de Verdière conjectured that m1 is bounded above by the chromatic number minus 1 based on a few examples. Later work by Fortier Bourque and Petri showed that among hyperbolic surfaces of genus 3, the Klein quartic maximizes the multiplicity, and attains the upper bound conjectured by Colin de Verdière. This surface is the first of a sequence of highly symmetrical surfaces named Hurwitz surfaces. We will show using the Selberg trace formula that for the next surface in the sequence, the Fricke–Macbeath surface F, we have m1(F) = 7. This result was also obtained independently by Chul-hee Lee.
Chapter 1 introduces some geometric notions including hyperbolic geometry, hyperbolic surfaces, and triangular surfaces, followed by Hurwitz’s automorphism theorem. Chapter 2 covers some basic concepts in spectral theory as well as some useful tools like the Selberg trace formula and a bit of representation theory. Chapter 3 focuses on the study of the Fricke–Macbeath surface and the proof of our main result using the techniques introduced in previous chapters. Finally, Chapter 4 discusses new methods for computing m1 which were used to show the existence of counterexamples to Colin de Verdière’s conjecture in joint work with Fortier Bourque, Gruda-Mediavilla, and Petri
CONFORMAL GRAFTING AND CONVERGENCE OF FENCHEL-NIELSEN TWIST COORDINATES
No full textAbstract. We cut a hyperbolic surface of finite area along some analytic simple closed curves, and glue in cylinders of varying moduli. We prove that as the moduli of the glued cylinders go to infinity, the Fenchel-Nielsen twist coordinates for the resulting surface around those cylinders converge. Content
THE CONVERSE OF THE SCHWARZ LEMMA IS FALSE
No full textAbstract. Let h: X → Y be a homeomorphism between hyperbolic surfaces with finite topology. If h is homotopic to a holomorphic map, then every closed geodesic in X is at least as long as the corresponding geodesic in Y, by the Schwarz Lemma. The converse holds trivially when X and Y are disks or annuli, and it holds when X and Y are closed surfaces by a theorem of W. Thurston. We prove that the converse is false in all other cases, strengthening a result of Masumoto. 1
The holomorphic couch theorem
No full textWe prove that if two conformal embeddings between Riemann surfaces with finite topology are homotopic, then they are isotopic through conformal embeddings. Furthermore, we show that the space of all conformal embeddings in a given homotopy class is homotopy equivalent to a point, a circle, a torus, or the unit tangent bundle of the codomain, depending on the induced homomorphism on fundamental groups. Quadratic differentials play a central role in the proof
The dimension of Thurston's spine
No full textWe show that for every , there exists some such that
the set of closed hyperbolic surfaces of genus whose systoles fill has
dimension at least . In particular, the dimension of this
set -- proposed as a spine for moduli space by Thurston -- is larger than the
virtual cohomological dimension of the mapping class group.Comment: Updated to published versio
Failure of the well-rounded retract for Outer space and Teichm\"uller space
No full textThe well-rounded retract for is defined as the
set of flat tori of unit volume and dimension whose systoles generate a
finite-index subgroup in homology. This set forms an equivariant spine of
minimal dimension for the space of flat tori.
For both the Outer space of metric graphs of rank and the
Teichm\"uller space of closed hyperbolic surfaces of genus ,
we show that the literal analogue of the well-rounded retract does not contain
an equivariant spine. We also prove that the sets of graphs whose systoles fill
either topologically or geometrically (two analogues of a set proposed as a
spine for by Thurston) are spines for but that their
dimension is larger than the virtual cohomological dimension of
in general.Comment: v1: 8 pages. v2: Clarified that the Teichm\"uller space result is
with respect to the extended mapping class group. v3: Modified the proof to
work for the mapping class group too. v4: Modified terminology and made minor
correction
A divergent horocycle in the horofunction compactification of the Teichmüller metric
No full textWe give an example of a horocycle in the Teichmüller space of the five-times-punctured sphere that does not converge in the Gardiner--Masur compactification, or equivalently in the horofunction compactification of the Teichmüller metric. As an intermediate step, we exhibit a simple closed curve whose extremal length is periodic but not constant along the horocycle. The example lifts to any Teichmüller space of complex dimension greater than one via covering constructions
The space of immersed polygons
No full textWe use the Schwarz-Christoffel formula to show that for every , the
space of labelled immersed -gons in the plane up to similarity is
homeomorphic to . We then prove that all immersed triangles,
quadrilaterals, and pentagons are embedded, from which it follows that the
space of labelled simple -gons up to similarity is homeomorphic to
if . This was first shown by Gonz\'ales and
L\'opez-L\'opez for and conjectured to be true for every by
Gonz\'alez and Sedano-Mendoza.Comment: v1: 7 pages, 2 figures. v2: fixed some typo
The holomorphic couch theorem
No full textWe prove that if two conformal embeddings between Riemann surfaces with finite topology are homotopic, then they are isotopic through conformal embeddings. Furthermore, we show that the space of all conformal embeddings in a given homotopy class deformation retracts into a point, a circle, a torus, or the unit tangent bundle of the codomain, depending on the induced homomorphism on fundamental groups. Quadratic differentials play a central role in the proof
