Related papers: Systole Length in Hyperbolic $n$-Manifolds

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.…

Extending earlier work of Tian, we show that if a manifold admits a metric that is almost hyperbolic in a suitable sense, then there exists an Einstein metric that is close to the given metric in the $C^{2,\alpha}$-topology. In dimension…

We consider the ortho spectrum of hyperbolic surfaces with totally geodesic boundary. We show that in general the ortho spectrum does not determine the systolic length but that there are only finitely many possibilities. As a corollary we…

Let N be a compact, orientable hyperbolic 3-manifold with connected, totally geodesic boundary of genus 2. If N has Heegaard genus at least 5, then its volume is greater than 6.89. The proof of this result uses the following dichotomy:…

We show that any two geometric triangulations of a closed hyperbolic, spherical or Euclidean manifold are related by a sequence of Pachner moves and barycentric subdivisions of bounded length. This bound is in terms of the dimension of the…

We introduce the stable presentation length of a finitely presented group. The stable presentation length of the fundamental group of a 3-manifold can be considered as an analogue of the simplicial volume. We show that the stable…

This paper proves explicit bilipschitz bounds on the change in metric between the thick part of a cusped hyperbolic 3-manifold N and the thick part of any of its long Dehn fillings. Given a bilipschitz constant J > 1 and a thickness…

In this paper we consider the analytic torsion of a closed hyperbolic 3-manifold associated with the m-th symmetric power of the standard representation of SL(2,C) and we study its asymptotic behavior if m tends to infinity. The leading…

We show that there is an upper bound on the injectivity radius of a hyperbolic 3-manifold in terms of the the number of generators of its fundamental group.

It is known that any two triangulations of a compact 3-manifold are related by finite sequences of certain local transformations. We prove here an upper bound for the length of a shortest transformation sequence relating any two…

Let $M$ be a hyperbolic 3-manifold with no rank two cusps admitting an embedding in $\mathbb S^3$. Then, if $M$ admits an exhaustion by $\pi_1$-injective sub-manifolds there exists cantor sets $C_n\subset \mathbb S^3$ such that $N_n=\mathbb…

We prove uniform linear bounds on the volume variation under drilling and filling operations on finite volume hyperbolic 3-manifolds.

For a hyperbolic 3-manifold $M$ with a torus boundary component,all but finitely many Dehn fillings yield hyperbolic 3-manifolds. In this paper, we will focus on the situation where $M$ has two exceptional Dehn fillings: an annular filling…

We study the long time behaviour of Ricci flow with bubbling-off on a possibly noncompact $3$-manifold of finite volume whose universal cover has bounded geometry. As an application, we give a Ricci flow proof of Thurston's hyperbolisation…

Let $M$ be an irreducible, compact, connected, orientable 3-manifold whose boundary is a torus. We show that if $M$ is hyperbolic, then it admits at most six finite/cyclic fillings of maximal distance 5. Further, the distance of a…

We show there exist tunnel number one hyperbolic 3-manifolds with arbitrarily long unknotting tunnel. This provides a negative answer to an old question of Colin Adams.

We establish combinatorial versions of various classical systolic inequalities. For a smooth triangulation of a closed smooth manifold, the minimal number of edges in a homotopically non-trivial loop contained in the $1$-skeleton gives an…

We calculate the volumes of the hyperbolic twist knot cone-manifolds using the Schl\"{a}fli formula. Even though general ideas for calculating the volumes of cone-manifolds are around, since there is no concrete calculation written, we…

If a contact form on a (2n+1)-dimensional closed contact manifold admits closed Reeb orbits, then its systolic ration is defined to be the quotient of (n+1)th power of the shortest period of Reeb orbits by the contact volume. We prove that…

We consider closed orientable 3-dimensional hyperbolic manifolds which are cyclic branched coverings of the 3-sphere, with branching set being a two-bridge knot (or link). We establish two-sided linear bounds depending on the order of the…