arXiv research

Researchers decompose hyperbolic n-manifolds with totally geodesic boundaries into polyhedral cells.

New surfaces defy traditional pants decompositions.

Given a Delaunay decomposition of a compact hyperbolic surface, one may record the topological data of the decomposition, together with the intersection angles between the `empty disks' circumscribing the regions of the decomposition. The main result of this paper is a characterization of when a given topological decom…

Study of quaternionic hyperbolic space bisectors and their decompositions.

Find simple geodesics in hyperbolic surfaces with bounded diameter.

Optimal discrete harmonic maps between hyperbolic surfaces are found via minimizing energy.

Classifies CR submanifolds in complex hyperbolic spaces.

The paper solves curvature problems on infinite hyperbolic surfaces.

Geometrically describes hyperbolic structures on link complements using quantum groups.

Every cusped, finite-volume hyperbolic three-manifold has a canonical decomposition into ideal polyhedra. We study the canonical decomposition of the hyperbolic manifold obtained by filling some (but not all) of the cusps with solid tori: in a broad range of cases, generic in an appropriate sense, this decomposition ca…

The aim of this paper is to provide some new tools to aid the study of decomposition complexity, a notion introduced by Guentner, Tessera and Yu. In this paper, three equivalent definitions for decomposition complexity are established. We prove that metric spaces with finite hyperbolic dimension have finite (weak) deco…

Cooper and Long generalised Epstein and Penner's Euclidean cell decomposition of cusped hyperbolic manifolds of finite volume to non-compact strictly convex projective manifolds of finite volume. We show that Weeks' algorithm to compute this decomposition for a hyperbolic surface generalises to strictly convex projecti…

Proves hyperbolic 3-manifolds have angle structures under certain conditions.

Flattenings of knotted surfaces help define new invariants.

The study proves properties of 4D projective manifolds and builds non-hyperbolic examples.

The article describes how decorations on hyperbolic surfaces lead to unique tessellations and decompositions.

The paper finds lower bounds on hyperbolic 3-manifold volumes.

We present a construction of sequences of closed hyperbolic surfaces that have long systoles which form pants decompositions of these surfaces. The length of the systoles of these surfaces grows logarithmically as a function of their genus.

Hyperbolic links in thickened torus decompose into angled tetrahedra.

Proof confirms volume conjecture for a specific knot.

In this paper, we determine the canonical polyhedral decomposition of every hyperbolic once-punctured torus bundle over the circle. In fact, we show that the only ideal polyhedral decomposition that is straight in the hyperbolic structure and that is invariant under a certain involution is the ideal triangulation defin…

In this note, we consider the rigidity of the focal decomposition of closed hyperbolic surfaces. We show that, generically, the focal decomposition of a closed hyperbolic surface does not allow for non-trivial topological deformations, without changing the hyperbolic structure of the surface. By classical rigidity theo…

We present an algorithm that computes Bowditch's canonical JSJ decomposition of a given one-ended hyperbolic group over its virtually cyclic subgroups. The algorithm works by identifying topological features in the boundary of the group. As a corollary we also show how to compute the JSJ decomposition of such a group o…

This paper uses combinatorial Ricci flow to tackle Thurston's triangulation conjecture.

Proves Farrell--Jones Conjecture for hyperbolic groups and their automorphisms.

We determine the lowest volume hyperbolic Coxeter polyhedron whose corresponding hyperbolic polyhedral 3-orbifold contains an essential 2-suborbifold, up to a canonical decomposition along essential hyperbolic triangle 2-suborbifolds.

Outer automorphism group of hyperbolic groups is HHG under certain conditions.

Surveying recent progress on hyperbolic 3-manifold rigidity.

Study continuous paths in discrete subgroups of hyperbolic space, proving combination and decomposition theorems.

The study finds infinite meridional essential surfaces in hyperbolic knot exteriors of various genera.

Our goal is to show, in two different contexts, that "random" surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus $g$ for which any pants decomposition requires curves of total length at least $g^{7/6 - ε}$. Moreover, we prove that this bound holds for most metrics in the…

New formula for knot group representations and hyperbolic structures.

New hyperbolic 4-manifolds found with special functions.

In the present paper, we give a necessary and sufficient condition for a Riemannian manifold $(M,g)$ to have a reducible action of a hyperbolic analogue of the holonomy group. This condition amounts to a decomposition of $(M,g)$ as a warped product of a special form, in analogy to the classical de Rham decomposition th…

The paper derives upper bounds on eigenvalues of Laplace-Beltrami operator on hyperbolic surfaces.

This work classifies belted sum decompositions of fully augmented links.

New tiling algorithm for hyperbolic 3-manifolds, characterizing cusp areas.

New insights into X-ray transform on hyperbolic disk, with functional relations and range characterizations.

The study characterizes slopes for hyperbolic knots and Whitehead doubles.

We introduce Fenchel-Nielsen coordinates on Teicmüller spaces of surfaces of infinite type. The definition is relative to a given pair of pants decomposition of the surface. We start by establishing conditions under which any pair of pants decomposition on a hyperbolic surface of infinite type can be turned into a geom…

We study canonical decompositions of postcritically finite branched coverings of the 2-sphere, as defined by K. Pilgrim. We show that every hyperbolic cycle in the decomposition does not have a Thurston obstruction. It is thus Thurston equivalent to a rational map.

Let S be a surface with genus g and n boundary components and let d(S) = 3g-3+n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions CP(S) prove that the Weil-Petersson metric on Teichmuller space Teich(S) is Gromov-hyperbolic if and only if d(S) …

Study of hyperbolic 3-manifolds via fractional Dehn twists and cusp geometry.

We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on the hyperbolic spaces of higher dimension. We also describe the corresponding general Ham…

We present an algorithm to construct the JSJ decomposition of one-ended hyperbolic groups which are fundamental groups of graphs of free groups with cyclic edge groups. Our algorithm runs in double exponential time, and is the first algorithm on JSJ decompositions to have an explicit time bound. Our methods are combina…

L. Paoluzzi constructed a family of compact orientable three-dimensional hyperbolic manifolds with totally geodesic boundary, which were, by construction, closely related to the three-dimensional torus. This paper gives their complete classification up to isometry, and also their isometry groups. The key tool is the so…

Let M be a complete hyperbolic 3-manifold of finite volume that admits a decomposition into right-angled ideal polyhedra. We show that M has a deformation retraction that is a virtually special square complex, in the sense of Haglund and Wise and deduce that such manifolds are virtually fibered. We generalise a theorem…

We study decompositions of complex hyperbolic isometries as products of involutions. We show that PU(2,1) has involution length 4 and commutator length 1, and that for all $n \geqslant 3$ PU($n$,1) has involution length at most 8.