Hyperbolic isometries of the fine curve graph of higher genus surfaces

Cancellative dimer algebras on a torus have many nice algebraic and homological properties. However, these nice properties disappear for dimer algebras on higher genus surfaces. We consider a new class of quiver algebras on surfaces, called ‘geodesic ghor algebras’, that reduce to cancellative dimer algebras on a torus, yet continue to have nice properties on higher genus surfaces. These algebras exhibit a rich interplay between their central geometry and the topology of the surface. We show that (nontrivial) geodesic ghor algebras do in fact exist, and give explicit descriptions of their central geometry. This article serves a companion to the article ‘A generalization of cancellative dimer aglebras to hyperbolic surfaces’, where the main statement is proven.

We study a new class of path algebras with relations on surfaces, called ‘geodesic ghor algebras’. These algebras generalize cancellative dimer algebras on a torus to higher genus surfaces, where the relations come from perfect matchings rather than a potential. Although cancellative dimer algebras on a torus are noncommutative crepant resolutions, the center of any dimer algebra on a higher genus surface is just the polynomial ring in one variable, and so the center and surface are unrelated. In contrast, we establish a rich interplay between the central geometry of geodesic ghor algebras and the topology of the surface in which they are embedded. Furthermore, we show that the localizations of these algebras over the noetherian locus are endomorphism rings of modules over their centers.

Adolescent Idiopathic Scoliosis (AIS) characterized by the 3D spine deformity affects about 4% schoolchildren worldwide. One of the prominent theories of the etiopathogenesis of AIS was proposed to be the poor postural balance control due to the impaired vestibular function. Thus, the morphometry of the vestibular system (VS) is of great importance for studying AIS. The VS is a genus-3 structure situated in the inner ear and consists of three semicircular canals lying perpendicular to each other. The high-genus topology of the surface poses great challenge for shape analysis. In this work, we propose an effective method to analyze shapes of high-genus surfaces by considering their geodesic spectra. The key is to compute the canonical hyperbolic geodesic loops of the surface, using the Ricci flow method. The Fuchsian group generators are then computed which can be used to determine the geodesic spectra. The geodesic spectra effectively measure shape differences between high-genus surfaces up to the hyperbolic isometry. We applied the proposed algorithm to the VS of 12 normal and 15 AIS subjects. Experimental results show the effectiveness of our algorithm and reveal statistical shape difference in the VS between right-thoracic AIS and normal subjects.

We study the variance of a linear statistic of the Laplace eigenvalues on a hyperbolic surface, when the surface varies over the moduli space of all surfaces of fixed genus, sampled at random according to the Weil–Petersson measure. The ensemble variance of the linear statistic was recently shown to coincide with that of the corresponding statistic in the Gaussian orthogonal ensemble (GOE) of random matrix theory, in the double limit of first taking large genus and then shrinking size of the energy window. In this note, we show that in this same limit, the (smooth) energy variance for a typical surface is close to the GOE result, a feature called “ergodicity” in the random matrix theory literature.

Earlier work on Delaunay triangulation of point sets on the 2D flat torus, which is locally isometric to the Euclidean plane, was based on lifting the point set to a locally isometric 9-sheeted covering space of the torus. Under mild conditions the Delaunay triangulation of the lifted point set, consisting of 9 copies of the input set, projects to the Delaunay triangulation of the input set. We improve and generalize this work. First we present a new construction based on an 8-sheeted covering space, which shows that eight copies suffice for the standard flat torus. Then we generalize this construction to the context of compact orientable surfaces of higher genus, which are locally isometric to the hyperbolic plane. We investigate more thoroughly the Bolza surface, homeomorphic to a sphere with two handles, both because it is the hyperbolic surface with lowest genus, and because triangulations on the Bolza surface have applications in various fields such as neuromathematics and cosmological models. While the general properties (existence results of appropriate covering spaces) show similarities with the results for the flat case, explicit constructions and their proofs are much more complex, even in the case of the apparently simple Bolza surface. One of the main reasons is the fact that two hyperbolic translations do not commute in general. To the best of our knowledge, the results in this paper are the first ones of this kind. The interest of our contribution lies not only in the results, but most of all in the construction of covering spaces itself and the study of their properties.

Embedding graphs on surfaces is a widely studied topic. Mathematicians first studied planar graphs, which are graphs embedded on the plane or sphere. After Kuratowski, the focus shifted to graphs cellular embedded in orientable surfaces of higher genus. Even though many results have been obtained for the cellular embedding possibilities of abstract graphs, little is known about the cellular embeddability of spatial graphs. In this thesis, we focus on analyzing the cellular embedding possibilities in case the surface needs to be embedded in 3-space up to ambient isotopy. This restriction results in the most relevant setting for applications taking place in the 3-dimensional Euclidean world. In the first chapter, we introduce the leveled spatial graphs. We explore the cellular embedding possibilities of this newly introduced family of spatial graphs. We show that every leveled embedding with at most four levels can be cellular embedded and we give a construction for the surface where the spatial graph cellular embeds, together with a formula to compute the genus of the surface. Moreover, we provide an algorithm that, if successful, builds a surface where the levelled spatial graph given as input cellular embeds. Finally, we discuss the limits of our algorithm. In the second chapter, we analyze the family of leveled spatial graph in the broader context of topological graph theory. Namely, we compare the leveled embeddings with other two spatial graph properties: freeness and paneledness. We show that leveled spatial graphs are a subfamily of free embeddings and we find a sufficient condition for a paneled embedding to be leveled. Moreover, we introduce a new graph invariant related to levelled embeddings: the level number. We compare it with two existing similar graph invariants: the thickness and the book thickness of a graph. Finally, we characterize completely the level number of complete graphs and complete bipartite graphs. In the third chapter, we investigate a particular class of leveled spatial graphs: polyhedra. We analyze local symmetry-preserving operations on polyhedra, introduced by Brinkmann, Goetschalckx and Schein, and focus on which of these operations increase the symmetry of polyhedral maps. We give a complete solution to this problem for Goldberg-Coxeter operations and for local symmetry-preserving operations with inflation factor up to 6. For these operations, we find out in which genera they can increase symmetry. In the fourth chapter, we compare two well-known graph operations, the line graph and the edge-complement graph. After listing some of their properties, we show that there exist only two graphs up to isomorphism which have isomorphic line graph and edge-complement graph. We provide an alternative proof to the original one by Aigner.

The aim of this note is to give the simplest possible proof that Mapping Class Groups of closed hyperbolic surfaces are acylindrically hyperbolic, and more specifically that their curve graphs are hyperbolic and that pseudo-Anosovs act on them as loxodromic WPDs.

In this paper we show that the Bers map of the asymptotic Teichmuller space AT(X) of an arbitrary hyperbolic Riemann surface X is injective. We prove further that AT(X) and the fibers of the quotient may from T(X) to AT(X) are contractible and that every point in the fiber over the basepoint of AT(X) is represented by a quasiconformal map that is an asymptotic hyperbolic isometry. The barycentric extension operators plays a central role in our proofs.

We consider the relationship between hyperbolic cone-manifold structures on surfaces, and algebraic representations of the fundamental group into a group of isometries. A hyperbolic cone-manifold structure on a surface, with all interior cone angles being integer multiples of 2π, determines a holonomy representation of the fundamental group. We ask, conversely, when a representation of the fundamental group is the holonomy of a hyperbolic cone-manifold structure. In this paper we build upon previous work with punctured tori to prove results for higher genus surfaces. Our techniques construct fundamental domains for hyperbolic cone-manifold structures, from the geometry of a representation. Central to these techniques are the Euler class of a representation, the group $${\widetilde{PSL_{2}\mathbb{R}}}$$ , the twist of hyperbolic isometries, and character varieties. We consider the action of the outer automorphism and related groups on the character variety, which is measure-preserving with respect to a natural measure derived from its symplectic structure, and ergodic in certain regions. Under various hypotheses, we almost surely or surely obtain a hyperbolic cone-manifold structure with prescribed holonomy.

We study geometric and spectral properties of typical hyperbolic surfaces of high genus, excluding a set of small measure for the Weil-Petersson probability measure. We first prove Benjamini-Schramm convergence to the hyperbolic plane H as the genus g goes to infinity. An estimate for the number of eigenvalues in an interval [a,b] in terms of a, b and g is then proven using the Selberg trace formula. It implies the convergence of spectral measures to the spectral measure of H as g $\rightarrow$+$\infty$, and a uniform Weyl law as b $\rightarrow$+$\infty$. We deduce a bound on the number of small eigenvalues, and the multiplicity of any eigenvalue.

Spectral Distribution of Twisted Laplacian on Typical Hyperbolic Surfaces of High Genus

In this article, we study the Dirac spectrum of typical hyperbolic surfaces of finite area, equipped with a nontrivial spin structure (so that the Dirac spectrum is discrete). For random Weil–Petersson surfaces of large genus g with o(g) cusps, we prove convergence of the spectral density to the spectral density of the hyperbolic plane, with quantitative error estimates. This result implies upper bounds on spectral counting functions and multiplicities, as well as a uniform Weyl law, true for typical hyperbolic surfaces equipped with any nontrivial spin structure.

The exponent of convergence of a non-elementary discrete group of hyperbolic isometries measures the Hausdorff dimension of the conical limit set. In passing to a non-trivial regular cover the resulting limit sets are point-wise equal though the exponent of convergence of the cover uniformization may be strictly less than the exponent of convergence of the base. We show in this paper that, for closed hyperbolic surfaces, the previously established lower bound of one half on the exponent of convergence of “small” regular covers is sharp but is not attained. We also consider “large” (non-regular) covers. Here large and small are descriptive of the size of the exponent of convergence. We show that a Kleinian group that uniformizes a manifold homeomorphic to a surface fibering over a circle contains a Schottky subgroup whose exponent of convergence is arbitrarily close to two.