Measure-theoretic rigidity for Mumford curves

Cornelissen, Gunther; Kool, Janne

doi:10.1017/s0143385712000016

Cited by 5 publications

(7 citation statements)

References 28 publications

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“…If G and G are isomorphic, there is nothing to prove. For the converse direction, from [5], Theorem 2.7, we recall measure-theoretic rigidity for graphs (the case of graphs with the same covering trees was proven by Coornaert in [3]): the graphs G (corresponding to (∂T, Γ, µ)) and G (corresponding to (∂T , Γ , µ )) are isomorphic if and only if there exists a group isomorphism α : Γ → Γ and a homeomorphism ϕ : ∂T → ∂T such that (a) ϕ is α-equivariant, i.e., we have ϕ(γx) = α(γ)ϕ(x), ∀x ∈ ∂T, γ ∈ Γ; (b) the measures µ and µ have the same dimension; (c) ϕ is absolutely continuous w.r.t. µ and µ .…”

Section: Hence We Can Set αmentioning

confidence: 99%

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Rigidity and reconstruction for graphs

Cornelissen

Kool

2019

J. Fractal Geom.

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We present measure theoretic rigidity for graphs of first Betti number b > 1 in terms of measures on the boundary of a 2b-regular tree, that we make explicit in terms of the edge-adjacency and closed-walk structure of the graph. We prove that edge-reconstruction of the entire graph is equivalent to that of the "closed walk lengths". Some rigidity phenomenaA compact Riemann surface X of genus g ≥ 2 is uniquely determined by a dynamical system, namely, the action of the fundamental group Π g in genus g on the Poincaré disk ∆ by Möbius transformations. Things change when we replace ∆ by its real one-dimensional boundary ∂∆ = S 1 ; the action of Π g extends to S 1 , but this action will only depend on the topological isomorphism type of X, viz., the genus g. Rigidity re-enters the picture via the Lebesgue-measure on S 1 , in the sense that two Riemann surfaces X and Y are isomorphic if and only if there exists a Π g -equivariant absolutely continuous homeomorphism S 1 → S 1 (cf. e.g. [8]).A similar result holds for more general hyperbolic spaces. We describe a version for graphs (cf. Coornaert [3]): let G = (V, E) denote a graph with vertex set V and edge set E, consisting of unordered pairs of elements of V . Let b denote the first Betti number of G, and assume b ≥ 2.Knowing G is the same as knowing the action of a free group F b or rank b on the universal covering tree T of G. Again, the dynamical system of F b acting on the boundary ∂T of T (i.e., the space of ends of T) is topologically conjugate to a system that only depends on b (to wit, the action of F b on the boundary of its Cayley graph), but if one considers the set of Patterson-Sullivan measures on ∂T, rigidity holds; we provide an exact result in Theorem 2.3 below.The graph rigidity theorem shows the importance of the structure of the "space of closed walks" (C, F b , µ) of a graph in understanding the structure of a graph. We apply this insight to reconstruction problems for graphs. We find that for average degree > 4, we can reconstruct various ingredients of the explicit formula for the rigidifying measure. We conclude that reconstruction of a graph is intimately related with the structure of lengths of closed walks in a graph. The final section confirms this; we prove in an elementary way that the edge reconstruction conjecture is equivalent to the reconstruction of "closed walks and their length".

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Section: Hence We Can Set αmentioning

confidence: 99%

“…One may extend rigidity from graphs to curves over non-archimedean fields. In [5], we have explained how, for such a generalization, one needs the require boundary homomorphisms to respect relations between so-called harmonic measures. It would be interesting to use such insights to formulate reconstruction problems for such curves, and for Riemann surfaces.…”

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confidence: 99%

Rigidity and reconstruction for graphs

Cornelissen

Kool

2019

J. Fractal Geom.

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“…One can however specify the smallest clopen ball containing a given pair of points. The size of such a clopen ball is given by the Patterson-Sullivan measure [56,57], and is directly related to the (regulated) length of the boundaryanchored bulk geodesic joining the given pair of points.…”

Section: Entanglement In Genus Zero P-adic Backgroundmentioning

confidence: 99%

“…The properties of this Patterson-Sullivan measure are used to prove rigidity results for Mumford curves, [57]. However, notice that the Patterson-Sullivan measure lives on the limit set Λ Γ , which is the complement of the boundary region that determines the Mumford curve X Γ (K) = (P 1 (K) ∖ Λ Γ )/Γ.…”

Section: A Measure On the Tate-mumford Curvementioning

confidence: 99%

Nonarchimedean Holographic Entropy from Networks of Perfect Tensors

Heydeman¹,

Marcolli²,

Parikh³

et al. 2018

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We consider a class of holographic quantum error-correcting codes, built from perfect tensors in network configurations dual to Bruhat-Tits trees and their quotients by Schottky groups corresponding to BTZ black holes. The resulting holographic states can be constructed in the limit of infinite network size. We obtain a p-adic version of entropy which obeys a Ryu-Takayanagi like formula for bipartite entanglement of connected or disconnected regions, in both genus-zero and genus-one p-adic backgrounds, along with a Bekenstein-Hawking-type formula for black hole entropy. We prove entropy inequalities obeyed by such tensor networks, such as subadditivity, strong subadditivity, and monogamy of mutual information (which is always saturated). In addition, we construct infinite classes of perfect tensors directly from semiclassical states in phase spaces over finite fields, generalizing the CRSS algorithm, and give Hamiltonians exhibiting these as vacua.

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“…for notation and explanation see also [9]. In literature there is a graph theoretic definition of C har (Γ, n).…”

Section: Introductionmentioning

confidence: 99%