The paper shows that arithmetic hyperbolic 3-orbifolds have many non-commensurable pairs with similar geodesic spectra.
problem Understanding the relationship between geodesic length spectra and commensurability of arithmetic hyperbolic 3-orbifolds.
method Using a bounded gaps result for prime ideals in number fields, the paper constructs infinitely many non-commensurable pairs of arithmetic hyperbolic 3-orbifolds with similar geodesic spectra.
result Arithmetic hyperbolic 3-orbifolds can have many non-commensurable pairs with similar geodesic spectra.
We study the length, weak length and complex length spectrum of closed geodesics of a compact flat Riemannian manifold, comparing length-isospectrality with isospectrality of the Laplacian acting on p-forms. Using integral roots of the Krawtchouk polynomials, we give many pairs of p-isospectral flat manifolds having di…
The study examines how non-commensurable surfaces can share length spectra.
problem Understanding how non-commensurable arithmetic hyperbolic surfaces can share length spectra.
method Investigates quantitative results on the maximum cardinality of non-commensurable surfaces sharing a fixed set of length spectra.
result Proves a number of quantitative results about the maximum cardinality of a family of pairwise non-commensurable arithmetic hyperbolic surfaces whose length spectra contain a fixed set of nonnegative real numbers.
In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compar…
The paper connects Riemann surface length spectra to Brownian loop measures.
problem Understanding the length spectra of Riemann surfaces with additional cusps.
method Using the Brownian loop measure to relate length spectra of Riemann surfaces with and without additional cusps.
result Expressed the total mass of Brownian loops in terms of the length of geodesic representatives.
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.
Spaces with similar long paths have similar shapes.
problem Comparing shapes of Gromov hyperbolic spaces.
method Examining asymptotic marked length spectra.
result Spaces with identical spectra are roughly isometric.
Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first 3g−3 geodesics. We use the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short (<log(g)) geodesics on a …
The action of the mapping class group of the thrice-punctured projective plane on its GL(2,C) character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corres…
The study examines arithmetic orbifolds and their length spectra, proving uniform discreteness and linear dependence of geodesic lengths.
problem Uniform discreteness and linear dependence of geodesic lengths in arithmetic orbifolds.
method Analyzes Salem numbers and Lie groups to prove uniform discreteness, and uses geometric properties to show linear dependence of geodesic lengths.
result Existence of a positive constant δ(X) such that squares of lengths of closed geodesics shorter than δ must be pairwise linearly dependent over Q.
In this paper we examine the relationship between the length spectrum and the geometric genus spectrum of an arithmetic hyperbolic 3-orbifold M. In particular we analyze the extent to which the geometry of M is determined by the closed geodesics coming from finite area totally geodesic surfaces. Using a variety of tech…
The paper extends the Manhattan curve concept to complex dynamics and studies its relation to multiplier spectra.
problem Understanding the growth rate of lengths of closed geodesics in complex dynamics.
method Defining and studying the Manhattan curve for holomorphic endomorphisms of CPk and relating it to multiplier spectra. result The Manhattan curve for two holomorphic endomorphisms is related to the correlation number of their multiplier spectra.
Let H be a discrete cocompact subgroup of SL_2(C). We conjecture that the quotient manifold X=SL_2(C)/H contains infinitely many non-isogeneous elliptic curves and prove that this is indeed the case if Schanuel's conjecture holds. We also prove it in the special case where the intersection of H and SL_2(R) is cocompact…
Given a negatively curved geodesic metric space M, we study the asymptotic penetration behaviour of geodesic lines of M in small neighbourhoods of closed geodesics and of other compact convex subsets of M. We define a spiraling spectrum which gives precise information on the asymptotic spiraling lengths of geodesic lin…
Study sub-Riemannian metrics on compact Lie groups, finding same shortest loops.
problem Exploring sub-Riemannian length spectra on compact Lie groups.
method Restricting Killing form to root spaces of compact Lie groups.
result Same shortest loops in Riemannian and sub-Riemannian cases.
Study reveals uniform difference in stretch factors between genus two handlebody group and outer automorphism group.
problem Analyzing the relationship between stretch factors in genus two handlebody group and outer automorphism group.
method Examined natural homomorphism from genus g handlebody group to outer automorphism group of free groups, focusing on pseudo-Anosov mapping classes and their stretch factors.
result Minimum stretch factor in genus two handlebody group is less than ten times the stretch factor of fully irreducible outer automorphism.
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.
Study approximate marked length spectrum rigidity in non-positively curved groups.
problem Approximate rigidity of marked length spectra in non-positively curved groups.
method Compare marked length spectra of isometric actions of groups with non-positively curved features.
result Supremum of quotient of marked length spectra is approximately determined by restricted spectra.
The study quantifies how many questions are needed to determine a surface's length spectrum.
problem Quantifying the number of questions needed to determine a surface's length spectrum.
method Inverse spectral problems for hyperbolic surfaces, focusing on length spectra and their relation to surface geometry.
result A quantitative upper bound on the number of isospectral but non-isometric surfaces of a given genus.
Let M be a 2m-dimensional compact Riemannian manifold. We show that the spectrum of the Hodge Laplacian acting on m-forms does not determine whether the manifold has boundary, nor does it determine the lengths of the closed geodesics. Among the many examples are a projective space and a hemisphere that have the s…
We prove an analogue of Farb-Masur's theorem that the length-spectra metric on moduli space is "almost isometric" to a simple model V(S) which is induced by the cone metric over the complex of curves. As an application, we know that the Teichmüller metric and the length-spectra metric are "almost isometric…
In this paper we consider flat metrics (semi-translation structures) on surfaces of finite type. There are two main results. The first is a complete description of when a set of simple closed curves is spectrally rigid, that is, when the length vector determines a metric among the class of flat metrics. Secondly, we gi…
Algorithms compute length spectra of torus graphs efficiently.
problem Computing length spectra of graphs embedded on a torus.
method Preprocessing and algorithms based on polyhedral norms.
result Efficient computation of length spectra and spectrum comparison.
Paper proves non-arithmetic Teichmüller length spectra for subgroup of mapping class groups.
problem Proving non-arithmetic Teichmüller length spectra for subgroups of mapping class groups.
method Introducing cross-ratios on Teichmüller and projectable mapping classes, studying their geometric and dynamical properties.
result Every non-elementary subgroup of the mapping class group has non-arithmetic Teichmüller length spectrum.
Reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable. Building toward a negative answer to this question, we construct examples of hyperbolic 3-manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable. More precisely, for all …
Study of group actions on CAT(0) cube complexes, focusing on marked length spectra.
problem Comparing marked length spectra of group actions on CAT(0) cube complexes.
method Use of finite-state automata and thermodynamic formalism for suspension flows over subshifts of finite type.
result Prove that the Manhattan curve is analytic and convex, and a straight line if and only if marked length spectra are homothetic.
Geodesic currents in strongly hyperbolic spaces are dense.
problem Characterizing geodesic currents with strongly hyperbolic dual pseudometrics.
method Combining finite-cover argument and boundary data characterization.
result Dense subset of geodesic currents with strongly hyperbolic dual pseudometrics.
We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm ∣∇R∣ of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determi…
New spectra defined for metric spaces, extending existing covering spectrum.
problem Characterizing and comparing spectra for metric spaces.
method Defining and measuring 'entourage covers' to derive new spectra.
result New spectra (ECS, ES) extend existing covering spectrum (CS) and have useful properties.
Authors derive McShane identity for super tori.
problem Deriving McShane identity for super tori.
method Super Teichmüller theory and supergeometry.
result Asymptotic growth rate of length spectra established.
New proof shows surfaces can have identical length spectra but not simple ones.
problem Identifying when two covers of a surface have identical length spectra but not simple ones.
method Characterized isomorphism of covers and constructed surfaces with identical spectra but different simple length spectra.
result Found surfaces with identical length spectra but not simple length isospectral covers.
We study resurgence properties of partition function of SU(2) Chern-Simons theory (WRT invariant) on closed three-manifolds. We check explicitly that in various examples Borel transforms of asymptotic expansions posses expected analytic properties. In examples that we study we observe that contribution of irreducible f…
Study automorphism groups of geodesic currents and measured laminations on surfaces.
problem Prove Ivanov's meta-conjecture for automorphism groups of geodesic currents.
method Investigate two automorphism groups, Aut(C) and Aut(ML), and compare them to the extended mapping class group. result Prove Aut(ML) is isomorphic to the extended mapping class group for most cases, except a few special cases. The paper describes correlations of spectra for higher rank Anosov representations.
problem Understanding correlations of spectra for Anosov representations of higher rank groups.
method Relates correlation problem to counting projections in truncated hypertubes.
result Extends previous work on rank one representations to higher rank.
New curves generalize flat metrics from quadratic to q-differentials.
problem Determining flat metrics from curve lengths.
method Introduced q-simple curves to generalize results from quadratic to q-differentials.
result Lengths of q-simple curves uniquely determine non-positively curved Euclidean cone metrics induced by q-differentials.
The covering spectrum is a geometric invariant of a Riemannian manifold, more generally of a metric space, that measures the size of its one-dimensional holes by isolating a portion of the length spectrum. In a previous paper we demonstrated that the covering spectrum is not a spectral invariant of a manifold in dimens…
New series analyze manifold homology, proving a generalized Gromov inequality.
problem Analyzing homology spectra of manifolds and polyhedra.
method Defining Dirichlet series and investigating their properties.
result Established an inequality involving the entire homology spectrum.
We prove that the length spectrum metric and the arc-length spectrum metric are almost-isometric on the ε0-relative part of Teichmuller spaces of surfaces with boundary.
Study compares eigenvalues and moment spectra of geodesic balls in Riemannian manifolds.
problem Comparing eigenvalues and moment spectra of geodesic balls in Riemannian manifolds.
method Explicit upper and lower bounds for Poisson hierarchy and torsional rigidity.
result Equality of eigenvalues and moment spectra characterizes the model space.
Study spin chains and sigma models on flag manifolds, calculating spectra and geodesics.
problem Understanding the spectrum and geodesics of sigma models on flag manifolds.
method Connecting SU(n) spin chains to sigma models and calculating spectra and geodesics.
result Calculated the spectrum of the Laplace-Beltrami operator and geodesics for CP1 and F3. The paper studies dynamics of rational maps using barycentric extensions and Berkovich spaces.
problem Classifying rational maps with bounded length spectra.
method Geometric and algebraic constructions on R-trees.
result Two constructions for limiting dynamics on R-trees are equivalent.
The study finds sparse sets that uniquely determine metrics on negatively curved manifolds.
problem Determining metrics on negatively curved manifolds using spectral data.
method Analyzing conjugacy classes and marked length spectra.
result Sparse sets exist that uniquely determine metrics on negatively curved manifolds.
We study various covering spectra for complete noncompact length spaces with universal covers (including Riemannian manifolds and the pointed Gromov Hausdorff limits of Riemannian manifolds with lower bounds on their Ricci curvature). We relate the covering spectrum to the (marked) shift spectrum of such a space. We de…
Closed manifolds with close marked spectra are approximately isometric.
problem Closed manifolds with close marked length spectra are approximately isometric.
method Using Hamenstädt's methods and Gromov compactness theorem, we show diffeomorphism and volume equality.
result Closed manifolds with close marked spectra are approximately isometric.
Generalizing Cusick's theorem on the closedness of the classical Lagrange spectrum for the approximation of real numbers by rational ones, we prove that various approximation spectra are closed, using penetration properties of the geodesic flow in cusp neighbourhoods in negatively curved manifolds and a result of Mauco…
New proof for 2D lens rigidity from billiard trajectories.
problem Determining obstacles in 2D space from billiard trajectories.
method Separate proof for 2D case, using billiard trajectories.
result Proves rigidity for 2D strictly convex bodies.
We prove explicit upper and lower bounds for the L1-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds Pm in ambient Riemannian spaces Nn. We assume that P and N both have controlled radial curvatures (mean curvature and sectional curvature, respectively) as view…
New proof shows Fuchsian groups have irrational length spectra.
problem Irrationality of the length spectrum in Fuchsian groups.
method Elementary proof of linear independence of group elements' lengths.
result Non-elementary Fuchsian groups contain elements with linearly independent lengths over Q.