Marked length rigidity for one-dimensional spaces

Abstract

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 ...

Non-backtracking cycles: length spectrum theory and graph mining applications

“…2 Therefore, it is logical to study the only non-trivial homotopy group, the fundamental group π 1 (G). Second, Constantine and Lafont (Constantine and Lafont 2018) showed that the length spectrum of a graph determines a certain subset of it up to isomorphism. Thus, we aim to determine when two graphs are close to each other by comparing their length spectra relying on the main theorem of (Constantine and Lafont 2018).…”

“…In this work we use cycle and closed path interchangeably. 4 In (Constantine and Lafont 2018), the authors need an isomorphism between the fundamental group of the spaces that are being compared, which is also computationally prohibitive. 5 Here we choose both EMD and r = r 0 based on the experimental evidence of Fig.…”

“…The length spectrum function can be defined on a broad class of metric spaces that includes Riemannian manifolds and graphs. The discriminatory power of the length spectrum is well known in other contexts: it can distinguish certain one-dimensional metric spaces up to isometry (Constantine and Lafont 2018), and it determines the Laplacian spectrum in the case of closed hyperbolic surfaces (Leininger et al 2007). However, it is not clear if this discriminatory power is also present in the case of complex networks.…”

Graph comparison via the nonbacktracking spectrum

“…Recent work [10,40] has shown that the two-cores extracted from two graphs are isomorphic when the set of all nonbacktracking cycles in each graph are equal (this is referred to as the length spectrum). This means that should we be able to enumerate all possible non-backtracking cycles we can effectively compare two graph structures.…”

Partial Marked Length Spectrum Rigidity Of Negatively Curved Surfaces

“…Several works extended this result to more general types of metrics: to surfaces without conjugate points in [CFF92] and [Fat89], translation surfaces in [DLR10], flat surfaces with cone points in [BL18], and the combination of the previous cases in [Con18] . The marked length spectrum rigidity question has been studied in other contexts as well: in locally symmetric spaces of negative curvature in [Ham90], projective geometries in [Kim01] and [CD10], Fuchsian buildings in [CL19b], and 1-dimensional spaces in [CL19a].…”