The paper proves rigidity of length identities for simple closed curves on hyperbolic surfaces.
problem Characterizing hyperbolic surfaces by their simple length spectra.
method Proving rigidity of length identities over Teichmüller spaces.
result Simple length spectra can be used as moduli for generic hyperbolic surfaces.
Upper bound established for the length of shortest closed geodesics in hyperbolic link complements.
problem Finding bounds on the length of shortest closed geodesics in hyperbolic link complements.
method Established an upper bound for the length of an nth shortest closed geodesic as a logarithmic function of the volume of the manifold.
result An upper bound of the length of an nth shortest closed geodesic is established as a logarithmic function of the volume of the manifold.
Study compares hyperbolic and extremal lengths for shortest curves.
problem Comparing hyperbolic and extremal lengths for shortest curves.
method Lower bounds for widths of collars and upper bounds for renormalized volume of Schottky manifolds.
result Upper bounds of renormalized volume in terms of hyperbolic length of compressible curves.
Study on length spectrum of random hyperbolic 3-manifolds.
problem Understanding the length spectrum of random hyperbolic 3-manifolds.
method Modeling random hyperbolic 3-manifolds using truncated tetrahedra and analyzing their length spectrum as volume tends to infinity.
result The length spectrum converges in distribution to a Poisson point process with a computable intensity λ as volume increases.
We prove a strong multiplicity one theorem for the length spectrum of compact even dimensional hyperbolic spaces i.e. if all but finitely many closed geodesics for two compact even dimensional hyperbolic spaces have the same length, then all closed geodesics have the same length.
Characterizes curves with short representatives on hyperbolic surfaces.
problem Inequalities on lengths of curves on hyperbolic surfaces.
method Characterization of topological types of curves and multicurves with short representatives.
result Characterizes which topological types of curves and multicurves always have a short representative.
This paper discovers new identities linking geodesic and orthogeodesic lengths on hyperbolic surfaces.
problem Understanding relationships between geodesic and orthogeodesic lengths on hyperbolic surfaces.
method Investigates a broad family of identities involving lengths of all closed geodesics and orthogeodesics.
result Introduces new identities that include lengths of all closed geodesics, contrasting with previous identities.
Study shows strong multiplicity one property for 3D hyperbolic spaces.
problem Understanding the spectrum of length-holonomy in 3D hyperbolic spaces.
method Analyzing Selberg-Gangolli-Wakayama zeta functions.
result Established a strong multiplicity one type property for length-holonomy spectrum.
Study confirms conjecture on extremal length of hyperbolic metrics.
problem Determining the extremal length of hyperbolic metrics on Riemann surfaces.
method Analyzes the topology of closed hyperbolic Riemann surfaces to find extremal lengths.
result Extremal length is topology-dependent and has a specific upper bound.
New theorem shows metrics of certain groups are close if their lengths are identical.
problem Identifying metrics of relatively hyperbolic groups from their lengths.
method Proved rigidity for relatively hyperbolic groups using coarse marked length spectrum.
result Metrics of relatively hyperbolic groups are uniformly close if lengths are identical.
Characterizes Kähler-hyperbolicity of bounded symmetric domains based on rank and genus.
problem Understanding the Kähler-hyperbolicity of bounded symmetric domains.
method Defines Kähler-hyperbolicity length by rank and genus, and characterizes it through a special Bergman potential.
result Establishes a unique constant for Kähler-hyperbolicity based on gradient length of a Bergman potential.
New lattice extensions of Schottky groups in hyperbolic space.
problem Understanding complex translation lengths in hyperbolic manifolds.
method Produced systolic lattice extensions of Schottky subgroups.
result Density of complex translation lengths in closed hyperbolic manifolds.
Study intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.
problem Understanding the relationship between intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.
method Analyzing the asymptotic behavior of interaction strength I(X) as X approaches infinity in the moduli space of compact hyperbolic surfaces.
result Determined the asymptotic behavior of interaction strength I(X) in terms of the length of the shortest geodesic sys(X).
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 ⩾ 3 n \geqslant 3 n ⩾ 3 PU( n n n ,1) has involution length at most 8.
For hyperbolic surfaces, primitive lengths are bounded below by a specific formula.
problem Understanding the distribution of primitive closed-geodesic lengths on hyperbolic surfaces.
method Analyzing Teichmüller space and proving a lower bound on the number of distinct primitive lengths.
result There exists a lower bound on the number of distinct primitive closed-geodesic lengths for hyperbolic surfaces.
In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently construct…
Lower bounds on geodesic length with few intersections on hyperbolic surfaces.
problem Finding the minimum length of geodesics with at least 2 intersections.
method Analyzing geodesics on hyperbolic surfaces with at least 2 self-intersections.
result The minimum length of such geodesics is 2 log ( 5 + 2 6 ) 2\log(5+2\sqrt6) 2 log ( 5 + 2 6 ) , and this bound is sharp. Study pants' terms on hyperbolic surfaces, proving length dominance implies isometry and pants bounds.
problem Understanding terms in hyperbolic surfaces' identities and their implications.
method Analyzing properties of terms and applying convexity to deduce theorems.
result Proved that if a simple length spectrum dominates another, the surfaces are isometric, and found bounds on pants numbers.
Study of special elliptic isometries and their lengths in complex hyperbolic plane.
problem Classifying lengths of special elliptic isometries in complex hyperbolic plane.
method Classification and description of relative SU(2,1)-character varieties.
result Fully classified lengths of special elliptic isometries (2, 3, 4).
It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well known result of Reid, for instance, shows that the geodesic length spectrum of an arithmetic hyperbolic surface determines the surface's commensurabilit…
The study of isospectral surfaces in Euclidean and hyperbolic geometries.
problem Existence of non-isometric surfaces with identical chord length distributions.
method Construction of isospectral pairs of hyperbolic surfaces without common covers.
result Found isospectral pairs of hyperbolic surfaces with no common cover.
In this paper we obtain a bound on the number of isometry classes of finite area hyperbolic surfaces which are length isospectral to a given surface depending only on the topological type of the surface and the length of the shortest closed geodesic on the surface. This will follow from a more general bound applying to…
This paper establishes the existence of a gap for the stable length spectrum on a hyperbolic manifold. If M is a hyperbolic n-manifold, for every positive e there is a positive d depending only on n and on e such that an element of pi_1(M) with stable commutator length less than d is represented by a geodesic with leng…
Study shows how to make 3D shapes hyperbolic with specific curves.
problem Understanding hyperbolic structures on 3-manifolds.
method Analyzing Heegaard splittings and using specific curves to prove hyperbolicity.
result Computed the length of a curve in terms of projection coefficients.
The paper develops formulas for hyperbolic simplices based on edge lengths.
problem Understanding the geometry of hyperbolic simplices using only edge lengths.
method Develops geometric formulas for hyperbolic simplices based on edge lengths.
result Distance and projection formulas in hyperbolic simplices.
Uniform spectral gap found for stable commutator length in hyperbolic 2-orbifolds.
problem Understanding stable commutator length in 3-manifolds.
method Explicit quasimorphisms for generic case, hyperbolic geometry for exceptional case.
result Explicit uniform gap of 1/36 for all orbifolds except a sphere with three cone points.
The study counts arcs on hyperbolic surfaces, providing asymptotic growth formulas.
problem Counting arcs on hyperbolic surfaces with boundaries and cusps.
method Asymptotic analysis of pure mapping class group orbits and arc lengths.
result The number of arcs of bounded length is asymptotically proportional to L 6 g − 6 + 2 ( n + p ) L^{6g-6+2(n+p)} L 6 g − 6 + 2 ( n + p ) . A short proof for curve lengths on hyperbolic surfaces.
problem Proving a theorem about curve lengths on hyperbolic surfaces.
method Presented a concise proof for the theorem.
result A pair of curves has length at least half the perimeter of a specific polygon.
Formula for projecting geodesics in hyperbolic 3-manifolds, relating lengths to subsurface projections.
problem Relating lengths of geodesics to projections in hyperbolic 3-manifolds.
method Formula with explicit constants relating subsurface projections to geodesic lengths.
result Effective and computable large projections versus short curves relation.
The study finds infinite commensurability classes of hyperbolic manifolds with Salem number lengths.
problem Understanding commensurability classes of arithmetic hyperbolic manifolds.
method Analyzing Salem numbers and their relation to arithmetic hyperbolic manifolds.
result Infinitely many commensurability classes of arithmetic hyperbolic manifolds with geodesic lengths equal to logarithms of Salem numbers.
Minimal geodesics on hyperbolic surfaces are long.
problem Finding the shortest closed geodesics on hyperbolic surfaces.
method Analyzing the self-intersection number to estimate geodesic lengths.
result The minimal length of geodesics grows logarithmically with the self-intersection number.
New rigidity result for hyperbolic surfaces based on curve lengths.
problem Determining hyperbolic metrics on surfaces from curve lengths.
method Investigating oriented graphs on curve complexes and Dehn quasi-homothetic functions.
result Knowing which curve is longer suffices to determine the hyperbolic metric on a surface.
Study causal structure of warped spacetimes using novel pre-length spaces.
problem Understanding the causal structure of warped spacetimes.
method Novel notion of Lorentzian pre-length spaces and proof of causal completion as globally hyperbolic pre-length space.
result Causal completion of GRW spacetime is a globally hyperbolic pre-length space under Hausdorff chronological topology.
Study on lengths of random multicurves on hyperbolic surfaces.
problem Distribution of lengths of random multicurves on closed hyperbolic surfaces.
method Using Margulis' thesis and Mirzakhani's equidistribution theorem for horospheres.
result Distribution of lengths admits a polynomial density, with coefficients expressible in terms of intersection numbers of psi-classes.
We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani's identities on orientable hyperbolic surfaces with boundary.
The systole length of hyperbolic n-manifolds is bounded by a function of n and t.
problem Bounding the systole length of hyperbolic n-manifolds.
method Relating the number of simplices to the diameter, and using bounds on Margulis tubes.
result The systole length is bounded by a function of n and t.
This paper connects spinors to horospheres in hyperbolic space.
problem Understanding geometric relationships between spinors and horospheres in hyperbolic space.
method Explicit bijective correspondence between spinors and horospheres, using bilinear forms and complex-valued distances.
result Derived applications include Ptolemy equations and Plücker coordinates connections.
The study proves limitations on isospectral hyperbolic surfaces with discrete length spectra.
problem Characterizing isospectral hyperbolic surfaces with discrete length spectra.
method Utilizing Sunada's method and topological self-duplicating ends, the study explores isospectral families and their cardinality.
result Finite groups can be realized as full isometry groups of hyperbolic structures with discrete spectrum on surfaces with self-duplicating ends.
Minimal surfaces in hyperbolic space have a sharp area bound.
problem Bounding the renormalized area of minimal surfaces.
method Proving an inequality using conformal length of ideal boundary.
result Sharp isoperimetric property of renormalized area.
Constructs non-isometric iso-length-spectral surfaces.
problem Creating non-isometric surfaces with identical geodesic lengths.
method Combining Sunada's construction with amalgams of hyperbolic surfaces.
result Found non-isometric surfaces with the same geodesic lengths.
We derive bounds on the length of the meridian and the cusp volume of hyperbolic knots in terms of the topology of essential surfaces spanned by the knot. We provide an algorithmically checkable criterion that guarantees that the meridian length of a hyperbolic knot is below a given bound. As applications we find knot …
Paper proves a conjecture about the minimum length of filling pairs on hyperbolic surfaces.
problem Finding the minimum length of filling pairs on hyperbolic surfaces.
method Proves a generalized isoperimetric inequality for disconnected regions.
result Proves the Aougab-Huang conjecture about the minimum length of filling pairs.
The study of geodesics on hyperbolic surfaces with angle and side bounds.
problem Understanding the geometric properties of geodesics on hyperbolic surfaces.
method Analyzing the angles and sides of polygons formed by geodesics, based on their lengths.
result Bounds for angles and sides of polygons formed by geodesics, depending only on geodesic lengths.
Synthetic proof shows globally hyperbolic Lorentzian spaces with specific curvature are warped products.
problem Synthetic proof of rigidity for globally hyperbolic Lorentzian spaces.
method Synthetic geometry and warped product analysis.
result Spaces with specific curvature and distance realizer are warped products.
Paper studies flow on hyperbolic surfaces to match boundary lengths.
problem Matching boundary lengths of hyperbolic surfaces.
method Combinatorial Yamabe flow on hyperbolic bordered surfaces.
result Flow converges exponentially to a surface with equal boundary lengths.
Let M = H 3 / Γ M = H^3/Γ M = H 3 /Γ be a hyperbolic 3-manifold, where Γ Γ Γ is a non-elementary Kleinian group. It is shown that the length spectrum of M M M is of unbounded multiplicity.
Improved bounds on geodesic intersections on hyperbolic surfaces.
problem Finding the shortest geodesic with a specific number of intersections.
method Proved a new formula for minimal length of geodesics with self-intersection number k.
result Improved the threshold for the existence of geodesics with self-intersection number k.
We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an applicat…