Marked length rigidity for symmetric spaces

Bo, Fang-zhen; Kim, I.

doi:10.1007/s00014-002-8346-y

Cited by 21 publications

(12 citation statements)

References 11 publications

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“…Isom.X 1 / and 2 W G ! Isom.X 2 / where X 1 ; X 2 are allowed to be higher dimensional, but there are more significant restrictions on the geometry of X 1 ; X 2 than in the results about MLSRC for surfaces (see, for example, Croke, Eberlein and Kleiner [19], Hersonsky and Paulin [34], Kim [47;48;49] and Dal'Bo and Kim [25]). However, the original version of MLSRC is still mostly open (except for rather special classes of metrics) in dimensions bigger than two.…”

Section: Introductionmentioning

confidence: 99%

Spectral rigidity of automorphic orbits in free groups

Carette¹,

Francaviglia

Kapovich

et al. 2012

Algebr. Geom. Topol.

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It is well-known that a point T 2 cv N in the (unprojectivized) Culler-Vogtmann Outer space cv N is uniquely determined by its translation length function k k T W F N ! R. A subset S of a free group F N is called spectrally rigid if, whenever T; T 0 2 cv N are such that kgk T D kgk T 0 for every g 2 S then T D T 0 in cv N . By contrast to the similar questions for the Teichmüller space, it is known that for N 2 there does not exist a finite spectrally rigid subset of F N .

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Section: Introductionmentioning

confidence: 99%

Spectral rigidity of automorphic orbits in free groups

Carette¹,

Francaviglia

Kapovich

et al. 2012

Algebr. Geom. Topol.

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“…When n ¼ 1; the theorem is independently proved by [5]. Such type of theorem for marked length spectrum is known in [8,9,3]. But Margulis invariant comes with a sign which reflects the dynamics of an action.…”

Section: Article In Pressmentioning

confidence: 93%

Affine action and Margulis invariant

Ким

2005

Journal of Functional Analysis

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“…In full generality, the conjecture is only known to hold for closed surfaces, which was independently established by Croke [Cro90] and Otal [Ota90] (see also Paulin and Hersonsky [HP97] for some extensions to singular metrics on surfaces). In the special case where one of the Riemannian metrics is locally symmetric, this result is due to Hamenstädt [Ham90] (see also Dal'bo and Kim [DK02] for analogous results in the higher rank case).…”

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confidence: 87%

Marked length rigidity for one-dimensional spaces

Constantine

Lafont

2019

J. Topol. Anal.

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In a compact geodesic metric space of topological dimension one, the minimal length of a loop in a free homotopy class is well-defined, and provides a function l : π 1 (X) −→ R + ∪ ∞ (the value ∞ being assigned to loops which are not freely homotopic to any rectifiable loops). This function is the marked length spectrum. We introduce a subset Conv(X), which is the union of all non-constant minimal loops of finite length. We show that if X is a compact, non-contractible, geodesic space of topological dimension one, then X deformation retracts to Conv(X). Moreover, Conv(X) can be characterized as the minimal subset of X to which X deformation retracts. Let X 1 , X 2 be a pair of compact, non-contractible, geodesic metric spaces of topological dimension one, and set Y i = Conv(X i ). We prove that any isomorphism φ : π 1 (X 1 ) −→ π 1 (X 2 ) satisfying l 2 • φ = l 1 , forces the existence of an isometry Φ : Y 1 −→ Y 2 which induces the map φ on the level of fundamental groups. Thus, for compact, noncontractible, geodesic spaces of topological dimension one, the marked length spectrum completely determines the subset Conv(X) up to isometry. furthermore, the length of the intersection is exactly equal to the length of the original geodesic segment. We then proceed to show that this correspondence is well-defined (i.e. does not depend on the pair of geodesic loops one constructs), and preserves concatenations of geodesic segments. This is used to construct an isometry between the sets of branch points. Using completeness, we extend this to an isometry between the closures of the sets of branch points. Finally, we consider points in Conv(X 1 ) which are not in the closure of the set of branch points. It is easy to see that each of these points lies on a unique maximal geodesic segment, with the property that the only branch points occur at the endpoints of the segment. The correspondance between geodesic segments can be used to see that there is a unique, well-defined, corresponding segment in Conv(X 2 ) of precisely the same length, allowing us to extend our isometry to all of Conv(X 1 ).We conclude this introduction with a few remarks. If the spaces X i are tame (i.e. are semi-locally simply connected), then our Main Theorem can also be deduced from some work of Culler and Morgan [CM87] (see also Alperin and Bass [AB87]). But of course, there are numerous examples of geodesic length spaces of topological dimension one which are not semi-locally simply connected (Hawaiian Earrings, Menger curves, Sierpinski curves, etc.), for which our result does not a priori follow from theirs. Another nice class of examples are Laakso spaces with Hausdorff dimension between one and two [Laa00]. These spaces have nice analytic properties, and work regarding the spectra of the Laplacian has been carried out on them [RS09]. In view of the connections between such spectra and the length spectrum in other contexts, this seems like a most interesting example.A preliminary version of this paper was posted by the second author on the arXiv ...

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Marked length rigidity for symmetric spaces

Cited by 21 publications

References 11 publications

Spectral rigidity of automorphic orbits in free groups

Spectral rigidity of automorphic orbits in free groups

Affine action and Margulis invariant

Marked length rigidity for one-dimensional spaces

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