Related papers: Systoles of hyperbolic 4-manifolds

We show that for every $n\geq 2$ and any $\epsilon>0$ there exists a compact hyperbolic $n$-manifold with a closed geodesic of length less than $\epsilon$. When $\epsilon$ is sufficiently small these manifolds are non-arithmetic, and they…

Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be a link such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in $M$, in this article, we establish an upper bound of the length of an $n^{th}$ shortest…

This paper shows that the complex projective plane $\mathbb{P}^2$ can be realized as the underlying space for a closed hyperbolic $4$-orbifold. This is the first example of a closed hyperbolic $4$-orbifold whose underlying space is…

We show that cusped finite-volume hyperbolic 3-manifolds contain infinitely many simple closed geodesics.

We consider the existence of simple closed geodesics or "geodesic knots" in finite volume orientable hyperbolic 3-manifolds. Previous results show that at least one geodesic knot always exists [Bull. London Math. Soc. 31(1) (1999) 81-86],…

In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds.

In this article, for any $n\geq 4$ we construct a sequence of compact hyperbolic $n$-manifolds $\{M_i\}$ with number of systoles at least as $\mathrm{vol}(M_i)^{1+\frac{1}{3n(n+1)}-\epsilon}$ for any $\epsilon>0$. In dimension 3, the bound…

We prove that every complete finite-volume hyperbolic 3-manifold $M$ that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable…

A closed connected hyperbolic $n$-manifold bounds geometrically if it is isometric to the geodesic boundary of a compact hyperbolic $(n+1)$-manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many…

We bound two global invariants of cusped hyperbolic manifolds: the length of the shortest closed geodesic (the systole), and the radius of the biggest embedded ball (the inradius). We give an upper bound for the systole, expressed in terms…

We show that for each $n \geq 2$, the systoles of closed hyperbolic $n$-manifolds form a dense subset of $(0, +\infty)$. We also show that for any $n\geq 2$ and any Salem number $\lambda$, there is a closed arithmetic hyperbolic…

We provide an explicit lower bound for the sytole in principal congruence covers of compact quaternionic hyperbolic manifolds. We also prove the optimality of this lower bound.

We develop a way of seeing a complete orientable hyperbolic $4$-manifold $\mathcal{M}$ as an orbifold cover of a Coxeter polytope $\mathcal{P} \subset \mathbb{H}^4$ that has a facet colouring. We also develop a way of finding totally…

It is known that the volume function for hyperbolic manifolds of dimension $\geq 3$ is finite-to-one. We show that the number of nonhomeomorphic hyperbolic 4-manifolds with the same volume can be made arbitrarily large. This is done by…

We prove that every closed oriented 3-manifold admits a hyperbolic cone-manifold structure with cone-angle arbitrarily close to 2pi.

In all dimensions $n \ge 5$, we prove the existence of closed orientable hyperbolic manifolds that do not admit any $\text{spin}^c$ structure, and in fact we show that there are infinitely many commensurability classes of such manifolds.…

Kerckhoff and Storm conjectured that compact hyperbolic n-orbifolds with totally geodesic boundary are infinitesimally rigid when n>3. This paper verifies this conjecture for a specific example based on the 4-dimensional hyperbolic…

We prove an upper bound for the number of shortest closed geodesics in a closed hyperbolic manifold of any dimension in terms of its volume and systole, generalizing a theorem of Parlier for surfaces. We also obtain bounds on the number of…

We prove that if a closed hyperbolic 3-manifold M contains infinitely many totally geodesic surfaces, then M is arithmetic.

Let $M$ be a compact hyperbolic $3$-manifold with volume $V$. Let $L$ be a link such that $M\setminus L$ is hyperbolic. For any hyperbolic link $L$ in $M$, in this article, we try to establish an upper bound of the length of $n^{th}$…