arXiv research

Growth rates of geodesics on modular orbifolds are studied.

Abstract Coxeter groups have growth rates that are Perron numbers.

The paper calculates the volume growth of hyperbolic surfaces with short geodesics.

CAT(0) spaces with small volume growth are homeomorphic to Euclidean space.

Upper bound on geodesic ball volume in Riemannian manifolds.

Study on geodesics and dihedral groups in lattices.

We give lower bounds for the growth of the number of Reeb chords and for the volume growth of Reeb flows on spherizations over closed manifolds M that are not of finite type, have virtually polycyclic fundamental group, and satisfy a mild assumption on the homology of the based loop space. For the special case of geode…

We prove that a reversible Berwaldian Finsler structure $\left(M,F\right)$ with base-independent non-negative radial flag curvature and large volume growth does not have any closed geodesics.

We give a quantification of residual finiteness for the fundamental groups of hyperbolic manifolds that admit a totally geodesic immersion to a compact, right-angled Coxeter orbifold of dimension 3 or 4. Specifically, we give explicit upper bounds on residual finiteness that are linear in terms of geodesic length. We t…

The study provides volume growth estimates for specific types of manifolds.

Geodesics grow infinitely in certain Finsler manifolds.

In the recent paper \cite{LoD1}, we classified closed geodesics on Finsler manifolds into rational and irrational two families, and gave a complete understanding on the index growth properties of iterates of rational closed geodesics. This study yields that a rational closed geodesic can not be the only closed geodesic…

The study examines the normal growth exponent of submanifolds in negatively curved manifolds.

The paper studies the growth of closed geodesics on hyperbolic surface amalgams.

Exponential growth of stable subgroups in Morse geodesics.

We prove exponential growth rate of contractible closed geodesics for an arbitrary bumpy metric on manifolds of the form X#Y, where the fundamental group of X has a subgroup of finite index at least 3 and Y is simply connected and not a homotopy sphere.

The study calculates the growth rate of reciprocal hyperbolic elements in Hecke groups.

We prove a Liouville property for any $f$-harmonic function with polynomial growth on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ when the Bakry-Émery Ricci curvature is nonnegative and its diameter of geodesic sphere has sublinear growth.

In section 1 we reformulate a theorem of Blichfeldt in the framework of manifolds of nonpositive curvature. As a result we obtain a lower bound on the number of homotopically distinct geodesic loops emanating from a common point q whose length is smaller than a fixed constant. This bound depends only on the volume grow…

Minimal geodesics on hyperbolic surfaces are long.

In this article, we consider the geodesic flow on a compact rank $1$ Riemannian manifold $M$ without focal points, whose universal cover is denoted by $X$. On the ideal boundary $X(\infty)$ of $X$, we show the existence and uniqueness of the Busemann density, which is realized via the Patterson-Sullivan measure. Based …

The paper explores curvature-free effects in manifolds with volume growth and ends-counting.

Study on higher-dimensional quasigeodesics in metric spaces.

We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior.

For any geodesic current we associated a quasi-metric space. For a subclass of geodesic currents, called filling, it defines a metric and we study the critical exponent associated to this space. We show that is is equal to the exponential growth rate of the intersection function for closed curves.

We relate the existence of many infinite geodesics on Alexandrov spaces to a statement about the average growth of volumes of balls. We deduce that the geodesic flow exists and preserves the Liouville measure in several important cases. The developed analytic tool has close ties to integral geometry.

Study geometric and analytical properties of $ρ$-Einstein solitons.

Study describes frequencies of geodesics on hyperbolic surfaces as genus grows.

The paper proves inequalities and growth rates for Schouten solitons.

We give an effective upper bound, for certain arithmetic hyperbolic 3-manifold groups obtained from a quadratic form construction, on the minimal index of a subgroup that embeds in a fixed 6-dimensional right-angled reflection group, stabilizing a totally geodesic subspace. In particular, for manifold groups in any fix…

The paper shows how contracting elements in groups lead to large quotients with specific growth rates.

We define a norm on homology of punctured tori equipped with a complete hyperbolic metric of finite volume and use it to find asymptotics on the growth of the number of simple geodesics of bounded length.

The aim of this paper is to state and prove polynomial analogues of the classical Manning inequality relating the topological entropy of a geodesic flow with the growth rate of the volume of balls in the universal covering. To this aim we use two numerical conjugacy invariants, the {\em strong polynomial entropy $h_{po…

Let $Γ$ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold $X$. We show that a normal subgroup $Γ_0$ has critical exponent equal to the critical exponent of $Γ$ if and only if $Γ/ Γ_0$ is amenable. We prove a similar result for the exponential growth rate of closed geodesics on $…

Study growth rates of harmonic functions on curved surfaces.

Study harmonic function growth on curved spaces, proving inequalities.

The study extends stochastic completeness to landmark spaces with any number of landmarks.

Study growth rates of subgroups in groups with a constricting element.

Let (M,g) be a compact Riemannian manifold of hyperbolic type, i.e M is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the universal covering. In particular for surfaces we show that the topological ent…

We show that, on a complete and possibly non-compact Riemannian manifold of dimension at least 2 without close conjugate points at infinity, the existence of a closed geodesic with local homology in maximal degree and maximal index growth under iteration forces the existence of infinitely many closed geodesics. For clo…

On a surface with a Finsler metric, we investigate the asymptotic growth of the number of closed geodesics of length less than $L$ which minimize length among all geodesic multicurves in the same homology class. An important class of surfaces which are of interest to us are hyperbolic surfaces.

Study geodesics in curved spaces, counts ambiguous paths, confirms number theory conjectures.

Uniform Poincaré inequalities established for various metric spaces.

The existence of two geometrically distinct closed geodesics on an $n$-dimensional sphere $S^n$ with a non-reversible and bumpy Finsler metric was shown independently by Duan--Long [7] and the author [27]. We simplify the proof of this statement by the following observation: If for some $N \in \mathbb{N}$ all closed ge…

A singularity theorem based on asymptotic volume growth

In this paper, we consider the formal power series whose n-th coefficient is the number of copies of a given finite graph in the ball of radius n centred at the identity element in the Cayley graph of a finitely generated group and call it the growth function. Epstein, Iano-Fletcher and Uri Zwick proved that the growth…

Enhanced Bishop-Gromov theorem for homogeneous and inhomogeneous spaces.

In this article we have studied some properties of subharmonic functions in a strongly symmetric Riemannian manifold with a pole. As a generalization of polynomial growth of a function we have introduced the notion of polynomial growth of some degree of a function with respect to a real function and proved that any non…