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464
Shrinkwrapping and the taming of hyperbolic 3manifolds
 J. Amer. Math. Soc
"... Thurston and many others developed the theory of geometrically finite ends of hyperbolic 3–manifolds. It remained to understand those ends which are not geometrically finite; such ends are called geometrically infinite. ..."
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Cited by 83 (0 self)
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Thurston and many others developed the theory of geometrically finite ends of hyperbolic 3–manifolds. It remained to understand those ends which are not geometrically finite; such ends are called geometrically infinite.
Tameness of hyperbolic 3–manifolds
"... Marden conjectured that a hyperbolic 3manifold M with finitely generated fundamental group is tame, i.e. it is homeomorphic to the interior of a compact manifold with boundary [42]. Since then, many consequences of this conjecture have been developed by Kleinian group theorists and 3manifold topol ..."
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Cited by 65 (5 self)
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Marden conjectured that a hyperbolic 3manifold M with finitely generated fundamental group is tame, i.e. it is homeomorphic to the interior of a compact manifold with boundary [42]. Since then, many consequences of this conjecture have been developed by Kleinian group theorists and 3manifold topologists. We prove this
A Riemannian interpolation inequality à la Borell, Brascamp and Lieb
, 2001
"... A concavity estimate is derived for interpolations between L¹(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian ..."
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Cited by 55 (6 self)
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A concavity estimate is derived for interpolations between L¹(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian versions of these theorems incorporate a volume distortion factor which can, however, be controlled via lower bounds on Ricci curvature. The method uses optimal mappings from mass transportation theory. Along the way, several new properties are established for optimal mass transport and interpolating maps on a Riemannian manifold.
Contractions in the 2Wasserstein Length Space and Thermalization of Granular Media
, 2004
"... An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flowthrough model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical ..."
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Cited by 54 (19 self)
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An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flowthrough model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinitedimensional manifold. An abstract theory is developed for gradient flows in length spaces, which shows how degenerate convexity (or even nonconvexity) — if uniformly controlled — will quantify contractivity (limit expansivity) of the flow.
Probabilistic and fractal aspects of Lévy trees
 Probab. Th. Rel. Fields
, 2005
"... We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete GaltonWatson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted Rtrees, which i ..."
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Cited by 48 (14 self)
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We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete GaltonWatson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted Rtrees, which is equipped with the GromovHausdorff distance. To construct Lévy trees, we make use of the coding by the height process which was studied in detail in previous work. We then investigate various probabilistic properties of Lévy trees. In particular we establish a branching property analogous to the wellknown property for GaltonWatson trees: Conditionally given the tree below level a, the subtrees originating from that level are distributed as the atoms of a Poisson point measure whose intensity involves a local time measure supported on the vertices at distance a from the root. We study regularity properties of local times in the space variable, and prove that the support of local time is the full level set, except for certain exceptional values of a corresponding to local extinctions. We also compute several fractal dimensions of Lévy trees, including Hausdorff and packing dimensions, in terms of lower and upper indices for the branching
Rayleigh processes, real trees, and root growth with regrafting
, 2004
"... Abstract. The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous’s Brownian continuum random tree, the random treelike object naturally associated with ..."
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Cited by 48 (11 self)
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Abstract. The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. Aldous’s Brownian continuum random tree, the random treelike object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N → ∞ of both a critical GaltonWatson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous–Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N → ∞ of the Aldous–Broder chain. A key technical ingredient in this work is the use of a pointed Gromov–
Phase Transitions on Nonamenable Graphs
, 2000
"... We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees. ..."
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Cited by 47 (8 self)
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We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees.
Nonembeddability theorems via Fourier analysis
"... Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group ac ..."
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Cited by 42 (9 self)
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Various new nonembeddability results (mainly into L1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0, 1}d has L1 distortion (log d) 12o(1). We also give new lower bounds on the L1 distortion of flat tori, quotients of the discrete hypercube under group actions, and the transportation cost (Earthmover) metric.
Markov chains in smooth Banach spaces and Gromov hyperbolic metric spaces
"... A metric space X has Markov type 2, if for any reversible finitestate Markov chain {Zt} (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfies E(D 2 t) ≤ K 2 t E(D 2 1) for some K = K(X) < ∞. This notion is d ..."
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Cited by 40 (23 self)
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A metric space X has Markov type 2, if for any reversible finitestate Markov chain {Zt} (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfies E(D 2 t) ≤ K 2 t E(D 2 1) for some K = K(X) < ∞. This notion is due to K. Ball (1992), who showed its importance for the Lipschitz extension problem. However until now, only Hilbert space (and its biLipschitz equivalents) were known to have Markov type 2. We show that every Banach space with modulus of smoothness of power type 2 (in particular, Lp for p> 2) has Markov type 2; this proves a conjecture of Ball. We also show that trees, hyperbolic groups and simply connected Riemannian manifolds of pinched negative curvature have Markov type 2. Our results are applied to settle several conjectures on Lipschitz extensions and embeddings. In particular, we answer a question posed by Johnson and Lindenstrauss in 1982, by showing that for 1 < q < 2 < p < ∞, any Lipschitz mapping from a subset of Lp to Lq has a Lipschitz extension defined on all of Lp. 1