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33
Harmonic measures versus quasiconformal measures for hyperbolic groups
, 2008
"... We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on ..."
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Cited by 30 (2 self)
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We establish a dimension formula for the harmonic measure of a finitely supported and symmetric random walk on a hyperbolic group. We also characterize random walks for which this dimension is maximal. Our approach is based on the Green metric, a metric which provides a geometric point of view on random walks and, in particular, which allows us to interpret harmonic measures as quasiconformal measures on the boundary of the group.
NONSMOOTH CALCULUS
"... Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singu ..."
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Cited by 25 (0 self)
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Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth functions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts. 1.
Conformal dimension and Gromov hyperbolic groups with 2sphere boundary
, 2003
"... Suppose G is a Gromov hyperbolic group, and ∂∞G is quasisymmetrically homeomorphic to an Ahlfors Qregular metric 2sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on H³. ..."
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Cited by 24 (5 self)
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Suppose G is a Gromov hyperbolic group, and ∂∞G is quasisymmetrically homeomorphic to an Ahlfors Qregular metric 2sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on H³.
Quasihyperbolic planes in hyperbolic groups
 Proc. Amer. Math. Soc
"... Abstract. The hyperbolic plane H 2 admits a quasiisometric embedding into every hyperbolic group which is not virtually free. The purpose of this note is to prove the following theorem which answers a question posed by P. Papasoglu: Theorem 1. The hyperbolic plane H 2 admits a quasiisometric embed ..."
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Cited by 14 (5 self)
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Abstract. The hyperbolic plane H 2 admits a quasiisometric embedding into every hyperbolic group which is not virtually free. The purpose of this note is to prove the following theorem which answers a question posed by P. Papasoglu: Theorem 1. The hyperbolic plane H 2 admits a quasiisometric embedding into a hyperbolic group if and only if the group is not virtually free. A map f: X → Y between two metric spaces (X, dX) and (Y, dY) is called a quasiisometric embedding if there exist constants λ ≥ 1 and K ≥ 0 such that 1 λ dX(x, y) − K ≤ dY (f(x), f(y)) ≤ λdX(x, y) + K for all x, y ∈ X. A group is virtually free if it contains a free subgroup of finite index. We refer to [9] for the definition of hyperbolic groups and related concepts from the theory of Gromov hyperbolic spaces. Every Gromov hyperbolic space X has a boundary ∂∞X which carries a class of canonical visual metrics. These metrics are biLipschitz equivalent to distance functions of the form dw,ǫ(a, b) = exp(−ǫ(a, b)w), a, b ∈ ∂∞X, where w ∈ X is a base point, ǫ> 0 is sufficiently small, and (a, b)w denotes the Gromov product of the points a and b with respect to w (cf. [9, Ch. 7]). Corollary 2. The boundary of a hyperbolic group (equipped with any visual metric) contains a quasicircle if and only if the group is not virtually free. By definition a quasicircle is a metric circle which admits a quasisymmetric parametrization by the unit circle S 1 ⊂ R 2 (see [10] for the definition and basic facts about quasisymmetric maps). Since the boundary of a virtually free group is totally disconnected, the “only if ” part of the corollary is obvious. One of the main ingredients in the proof of the theorem is a result by Tukia [14] which insures the existence of quasiarcs with given endpoints inside certain subsets of R n (a quasiarc is a quasisymmetric image of the interval [0, 1]). The authors
A Rigidity Property of Some Negatively Curved Solvable Lie Groups
, 2009
"... We show that for some negatively curved solvable Lie groups, all self quasiisometries are almost isometries. We prove this by showing that all self quasisymmetric maps of the ideal boundary (of the solvable Lie groups) are bilipschitz with respect to the visual metric. We also define parabolic vi ..."
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Cited by 13 (5 self)
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We show that for some negatively curved solvable Lie groups, all self quasiisometries are almost isometries. We prove this by showing that all self quasisymmetric maps of the ideal boundary (of the solvable Lie groups) are bilipschitz with respect to the visual metric. We also define parabolic visual metrics on the ideal boundary of Gromov hyperbolic spaces and relate them to visual metrics. 1
Asymptotic traffic flow in a hyperbolic network
 In International Symposium on Communications, Control, and Signal Processing (ISCCSP
"... Abstract. In this work we study the asymptotic traffic flow in Gromov’s hyperbolic graphs. We prove that under certain mild hypotheses the traffic flow in a hyperbolic graph tends to pass through a finite set of highly congested nodes. These nodes are called the “core ” of the graph. We provide a fo ..."
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Cited by 8 (2 self)
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Abstract. In this work we study the asymptotic traffic flow in Gromov’s hyperbolic graphs. We prove that under certain mild hypotheses the traffic flow in a hyperbolic graph tends to pass through a finite set of highly congested nodes. These nodes are called the “core ” of the graph. We provide a formal definition of the core in a very general context and we study the properties of this set for several graphs. 1.
Metric space inversions, quasihyperbolic distance, and uniform spaces
, 2006
"... We define a notion of inversion valid in the general metric space setting. We establish several basic facts concerning inversions; e.g., they are quasimöbius homeomorphisms and quasihyperbolically bilipschitz. In a certain sense, inversion is dual to sphericalization. We demonstrate that both inver ..."
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Cited by 8 (4 self)
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We define a notion of inversion valid in the general metric space setting. We establish several basic facts concerning inversions; e.g., they are quasimöbius homeomorphisms and quasihyperbolically bilipschitz. In a certain sense, inversion is dual to sphericalization. We demonstrate that both inversion and sphericalization preserve local quasiconvexity and annular quasiconvexity as well as uniformity.
Rigidity for quasiFuchsian actions on negatively curved spaces
 Int. Math. Res. Not
, 2004
"... Abstract. We prove that if G � X is a convex cocompact isometric group action on a CAT(−1) space, and the limit set has Hausdorff and topological dimensions equal to 1, then the action preserves a convex subset Y ⊆ X isometric to H 2. This implies a conjecture by M. Bourdon. Our methods also yield a ..."
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Cited by 8 (2 self)
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Abstract. We prove that if G � X is a convex cocompact isometric group action on a CAT(−1) space, and the limit set has Hausdorff and topological dimensions equal to 1, then the action preserves a convex subset Y ⊆ X isometric to H 2. This implies a conjecture by M. Bourdon. Our methods also yield a new proof of the ndimensional analog of this statement for n ≥ 2. 1.
Spaces and groups with conformal dimension greater than one
, 2010
"... Abstract. We show that if a complete, doubling metric space is annularly linearly connected then its conformal dimension is greater than one, quantitatively. As a consequence, we answer a question of Bonk and Kleiner: if the boundary of a oneended hyperbolic group has no local cut points, then its ..."
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Cited by 6 (4 self)
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Abstract. We show that if a complete, doubling metric space is annularly linearly connected then its conformal dimension is greater than one, quantitatively. As a consequence, we answer a question of Bonk and Kleiner: if the boundary of a oneended hyperbolic group has no local cut points, then its conformal dimension is greater than one. 1.