Geometriae Dedicata最新文献

We prove a quantitative version of the classical Tits’ alternative for discrete groups acting on packed Gromov-hyperbolic spaces supporting a convex geodesic bicombing. Some geometric consequences, as uniform estimates on systole, diastole, algebraic entropy and critical exponent of the groups, will be presented. Finally we will study the behaviour of these group actions under limits, providing new examples of compact classes of metric spaces.

We show that every cross ratio preserving homeomorphism between boundaries of Hadamard manifolds extends to a map, called circumcenter extension, provided that the manifolds satisfy certain visibility conditions. We describe regions on which this map is Hölder-continuous. Furthermore, we show that this map is a rough isometry, whenever the manifolds admit cocompact group actions by isometries and we improve previously known quasi-isometry constants, provided by Biswas, in the case of 2-dimensional (mathrm {CAT(-1)}) manifolds. Finally, we provide a sufficient condition for this map to be an isometry in the case of Hadamard surfaces.

This article develops a method to compute the action induced on the fundamental group by an automorphism of a real rational surface. Then, it uses this method to compute the induced actions for a certain family of basic quadratic real automorphisms. By utilizing an invariant set in the fundamental group, we introduce a method to estimate a lower bound of the action’s growth rate on the fundamental group. The growth rate of this induced action provides a lower bound for the entropy of the real surface automorphism. These calculations are carried out for an important family of real surface automorphisms, and new lower bounds are obtained for the entropy of these automorphisms.

A finite-dimensional CAT(0) cube complex X is equipped with several well-studied boundaries. These include the Tits boundary (partial _TX) (which depends on the CAT(0) metric), the Roller boundary ({partial _R}X) (which depends only on the combinatorial structure), and the simplicial boundary (partial _triangle X) (which also depends only on the combinatorial structure). We use a partial order on a certain quotient of ({partial _R}X) to define a simplicial Roller boundary ({mathfrak {R}}_triangle X). Then, we show that (partial _TX), (partial _triangle X), and ({mathfrak {R}}_triangle X) are all homotopy equivalent, (text {Aut}(X))-equivariantly up to homotopy. As an application, we deduce that the perturbations of the CAT(0) metric introduced by Qing do not affect the equivariant homotopy type of the Tits boundary. Along the way, we develop a self-contained exposition providing a dictionary among different perspectives on cube complexes.

Let (mathbb {P}Omega ^dmathcal {M}_{0,n}(kappa )), where (kappa =(k_1,dots ,k_n)), be a stratum of (projectivized) d-differentials in genus 0. We prove a recursive formula which relates the volume of (mathbb {P}Omega ^dmathcal {M}_{0,n}(kappa )) to the volumes of other strata of lower dimensions in the case where none of the (k_i) is divisible by d. As an application, we give a new proof of the Kontsevich’s formula for the volumes of strata of quadratic differentials with simple poles and zeros of odd order, which was originally proved by Athreya–Eskin–Zorich. In another application, we show that up to some power of (pi ), the volume of the moduli spaces of flat metrics on the sphere with prescribed cone angles is a continuous piecewise polynomial with rational coefficients function of the angles, provided none of the angles is an integral multiple of (2pi ). This generalizes the results of Koziarz and Nguyen (Ann Sci l’Éc Normale Supér 51(6):1549–1597, 2018) and McMullen (Am J Math 139(1):261–291, 2017).

In this paper we provide a way of taking (L^p), (p > frac{m}{2}) bounds on a (m-) dimensional Riemannian metric and transforming that into Hölder bounds for the corresponding distance function. One can think of this new estimate as a type of Morrey inequality for Riemannian manifolds where one thinks of a Riemannian metric as the gradient of the corresponding distance function so that the (L^p), (p > frac{m}{2}) bound analogously implies Hölder control on the distance function. This new estimate is then used to state a compactness theorem, another theorem which guarantees convergence to a particular Riemmanian manifold, and a new scalar torus stability result. We expect these results to be useful for proving geometric stability results in the presence of scalar curvature bounds when Gromov–Hausdorff convergence can be achieved.

We consider the existence of cohomogeneity one solitons for the isometric flow of ${\text{G}}_2$ -structures on the following classes of torsion-free ${\text{G}}_2$ -manifolds: the Euclidean $R^7$ with its standard ${\text{G}}_2$ -structure, metric cylinders over Calabi-Yau 3-folds, metric cones over nearly Kähler 6-manifolds, and the Bryant-Salamon ${\text{G}}_2$ -manifolds. In all cases we establish existence of global solutions to the isometric soliton equations, and determine the asymptotic behaviour of the torsion. In particular, existence of shrinking isometric solitons on $R^7$ is proved, giving support to the likely existence of type I singularities for the isometric flow. In each case, the study of the soliton equation reduces to a particular nonlinear ODE with a regular singular point, for which we provide a careful analysis. Finally, to simplify the derivation of the relevant equations in each case, we first establish several useful Riemannian geometric formulas for a general class of cohomogeneity one metrics on total spaces of vector bundles which should have much wider application, as such metrics arise often as explicit examples of special holonomy metrics.

We introduce a hierarchy of equivalence relations on the set of separated nets of a given Euclidean space, indexed by concave increasing functions $\varphi :(0,\infty )\to (0,\infty )$. Two separated nets are called $\varphi$-displacement equivalent if, roughly speaking, there is a bijection between them which, for large radii R, displaces points of norm at most R by something of order at most $\varphi \left(R\right)$. We show that the spectrum of $\varphi$-displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded $\varphi$, to the indiscrete equivalence relation, corresponding to $\varphi \left(R\right)\in \Omega \left(R\right)$, in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of $\varphi$-displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of $\varphi \left(R\right)$ for $R\to \infty$. We further undertake a comparison of our notion of $\varphi$-displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of $\varphi$-displacement equivalence with that of bilipschitz equivalence.

This paper gives a new way of characterizing L-space 3-manifolds by using orderability of quandles. Hence, this answers a question of Clay et al. (Question 1.1 of Can Math Bull 59(3):472–482, 2016). We also investigate both the orderability and circular orderability of dynamical extensions of orderable quandles. We give conditions under which the conjugation quandle on a group, as an extension of the conjugation of a bi-orderable group by the conjugation of a right orderable group, is right orderable. We also study the right circular orderability of link quandles. We prove that the n-quandle (Q_n(L)) of the link quandle of a link L in the 3-sphere is not right circularly orderable and hence it is not right orderable. But on the other hand, we show that there are infinitely many links for which the p-enveloping group of the link quandle is right circularly orderable for any prime integer p.