Describes geometry of positive configurations in limiting buildings.
problem Understanding positive representations and their limits in buildings.
method Uses positivity properties of Hitchin representations and Parreau's compactification.
result Explicitly describes the geometry of preferred apartments in limiting buildings.
Compactifies a component by studying metric degeneration.
problem Compactify the SO(2,3)-Hitchin component.
method Study metric degeneration on a surface in pseudo-hyperbolic space.
result Establish closure in projectivized geodesic currents space.
Let Γ be a finitely generated group and G be a noncompact semisimple connected real Lie group with finite center. We consider the space X of conjugacy classes of reductive representations of Γ into G. We define the {\it translation vector} of an element g in G, with values in a Weyl chamber, as a…
The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the …
The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the …
We define a new spectrum for compact length spaces and Riemannian manifolds called the "covering spectrum" which roughly measures the size of the one dimensional holes in the space. More specifically, the covering spectrum is a set of real numbers δ>0 which identify the distinct δ covers of the space. We investigat…
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.
Paper proves flat metrics from holomorphic quadratic differentials can be identified by length spectrum.
problem Identifying flat metrics from holomorphic quadratic differentials.
method Proved using length spectrum on closed oriented surfaces.
result Flat metrics from holomorphic quadratic differentials can be distinguished by their length spectrum.
New method proves length spectrum rigidity in various geometric settings.
problem Length spectrum rigidity in geometric settings.
method Combination of dynamical systems and geometric group theory.
result Provides concise proofs and extends classical results.
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.
New stability estimate for Anosov manifolds' metrics.
problem Locally determine the metric from the marked length spectrum.
method Anosov geodesic flow and non-positive curvature.
result Two close enough metrics with the same marked length spectrum are isometric.
The rigidity of marked length spectrum for closed hyperbolic surfaces due to Fricke-Klein [7] has been the motivation of many different rigidity results, specially for manifolds of negative curvature. From the works of Vigneras [18], Sunada [17] and many other authors this result is far from being true for the unmarked…
Anosov surfaces with same length spectrum are isometric.
problem Identifying metrics on surfaces based on their length spectrum.
method Combining microlocal tools with complex curve geometry.
result Metrics with the same length spectrum on Anosov surfaces are isometric.
New gaps found in metric curvature.
problem Negative curvature metrics with separated length spectra.
method Topology-based separation of length spectra.
result Exponential gaps in length spectra for negatively curved metrics.
Study shows surfaces with similar length spectra are smoothly deformable.
problem Quantifying how similar the length spectra of two negatively curved surfaces are.
method Analyzes marked length spectra of closed negatively curved surfaces and proves smooth deformations.
result Smooth diffeomorphisms exist between surfaces with close length spectra.
Given a surface of infinite topological type, there are several Teichmüller spaces associated with it, depending on the basepoint and on the point of view that one uses to compare different complex structures. This paper is about the comparison between the quasiconformal Teichmüller space and the length-spectrum Teichm…
A theorem proving all geodesics on a sphere are simple and of same length.
problem Characterizing Zoll Riemannian metrics on a 2-sphere.
method Analyzing the simple length spectrum of a 2-sphere.
result All geodesics on a sphere are simple and of the same length.
Study the energy spectrum of metrics on surfaces and its relation to simple length spectrum.
problem Relate the energy spectrum to the simple length spectrum of metrics on surfaces.
method Analyze the energy spectrum of metrics on surfaces and their Teichmüller spaces, considering homotopy conditions.
result The energy spectrum determines the simple length spectrum under certain conditions.
New theorem shows certain curved surfaces are uniquely identified by their geodesic lengths.
problem Identifying surfaces by their geodesic lengths.
method Analyzes metrics on simple, thick negatively curved two-dimensional P-manifolds.
result Piecewise negatively curved Riemannian metrics on simple, thick two-dimensional P-manifolds are uniquely determined by their geodesic lengths.
For a finitely generated group G, we introduce an asymmetric pseudometric on projectivized deformation spaces of G-trees, using stretching factors of G-equivariant Lipschitz maps, that generalizes the Lipschitz metric on Outer space and is an analogue of the Thurston metric on Teichmüller space. We show that in t…
We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In particular, the Neumann spectrum of the Laplace--Beltrami operator uniquely determines the…
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.
In this article, we prove that every arithmetic locally symmetric orbifold of classical type without Euclidean or compact factors has arbitrarily long arithmetic progressions in its primitive length spectrum. Moreover, we show the stronger property that every primitive length occurs in arbitrarily long arithmetic progr…
We characterize finitely generated torsion-free Kleinian groups for which the real length spectrum (without multiplicities) is discrete.
We extend Thurston's metric to projective filling currents, embedding Teichmüller space into the larger space.
problem Embedding Teichmüller space into the space of projectivized filling currents.
method Extending the symmetrized Thurston metric to PCfill(S) and studying its geometry. result There is no quasi-isometric projection back from PCfill(S) to T(S). Study of lengths of cycles in large genus random maps converging to Poisson process.
problem Understanding the distribution of cycle lengths in large genus random maps.
method Teichmüller theory approach for uniformly random metric maps (ribbon graphs).
result The length spectrum converges to a Poisson point process with an explicit intensity as genus tends to infinity.
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.
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 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.
Let X be an infinite geodesically complete hyperbolic surface which can be decomposed into geodesic pairs of pants. We introduce Thurston's boundary to the Teichmüller space T(X) of the surface X using the length spectrum analogous to Thurston's construction for finite surfaces. Thurston's boundary using the leng…
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.
Unfolding paths in Outer space accumulate on a simplex, not converge.
problem Understanding accumulation points in Outer space.
method Constructing an unfolding path in Outer space.
result Unfolding paths accumulate on a 1-simplex, not converge.
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.
Study proves rigidity of marked length spectra in contracting group actions.
problem Rigidity of marked length spectra in contracting group actions.
method Unified approach using the Extension Lemma and metric geometry.
result Orbit map is a rough isometry if marked length spectra match.
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.
The wave trace of certain convex domains can be smooth near some points in the length spectrum.
problem Understanding the relationship between the wave trace and the length spectrum of convex domains.
method Constructing silent periodic billiard orbits with the same length but different Maslov indices, using a microlocal parametrix for wave invariants.
result The wave trace can be smooth near some points in the length spectrum, showing potential limitations for inverse spectral problems.
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.
In this article, we show that a Finsler--Laplacian introduced previously can detect changes in the Finsler metric that the marked length spectrum cannot. We also construct examples of non-reversible Finsler metrics in negative curvature such that 4λ1>h2, where λ1 is the bottom of the L2-spectrum and h the…
Researchers refine local rigidity for marked length spectrum and introduce a new pressure metric.
problem Local rigidity of marked length spectrum and related metrics.
method Refined local rigidity result using geodesic stretch and Anosov flows, introduced new pressure metric.
result New pressure metric related to Weil-Peterson metric, reduces to it in Teichmüller space.
Study shows ortho spectrum doesn't fully determine systolic length but limits the number of possible structures.
problem Determining the systolic length of hyperbolic surfaces with boundary.
method Analyzing the ortho spectrum of hyperbolic surfaces with totally geodesic boundary.
result There are only finitely many possibilities for the ortho spectrum and corresponding hyperbolic structures.
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…
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.
Reformulated Markov's conjecture in combinatorial terms.
problem Markov's uniqueness conjecture in integral necklaces.
method Geometric reformulation and combinatorial description.
result Explicitly described set of lengths on modular torus.
Study on multiplicities in length spectrum of Salem numbers.
problem Understanding multiplicities in the length spectrum of Salem numbers.
method Analysis of square-rootable Salem numbers and their growth rate.
result Proved exponential growth rate for mean multiplicities in length spectrum.
New method uses short geodesics to approximate marked length spectrum.
problem Determining a metric from its marked length spectrum.
method Recovering hypotheses from previous work using dynamical tools.
result Approximate values of MLS on a large set determine the metric.
Study Ricci-Bourguignon flow on Heisenberg and quaternion Lie groups.
problem Deformation of spectrum and length spectrum on compact nilmanifolds.
method Ricci-Bourguignon flow on Heisenberg and quaternion Lie groups.
result Construct a solution of the Ricci-Bourguignon flow on Heisenberg and quaternion nilpotent Lie groups.
A general approach to proving that the length spectrum of a compact Riemannian manifold is an invariant of the Laplace spectrum comes from considering the wave trace, a spectrally determined tempered distribution. The Poisson relation states that the singularities of the wave trace can only occur at lengths of closed g…
Multivariate Poisson approximation of the length spectrum of random surfaces is studied by means of the Chen-Stein method. This approach delivers simple and explicit error bounds in Poisson limit theorems. They are used to prove that Poisson approximation applies to curves of length up to order o(loglogg) with g…