Geometry and dynamics in gromov hyperbolic metric spaces. A universal lipschitz extension property of gromov hyperbolic spaces brudnyi, alexander and brudnyi, yuri, revista matematica iberoamericana, 2007 on the range of the derivative of a smooth mapping between banach spaces deville, robert, abstract and applied analysis, 2005. A geodesic metric space is said to be hyperbolic if it is. Examples of gromov hyperbolic spaces in this section we present two examples of gromov hyperbolic spaces. Gromovs notion of hyperbolic spaces and hyperbolic groups have been much. To every gromov hyperbolic space x one can associate a space at in nity called the gromov boundary of x. Embeddings of gromov hyperbolic spaces springerlink. A primary question we naturally ask is whether a metric space x,d is hyperbolic in the sense of gromov or not. For a gromov hyperbolic space x there exists a boundary at infinity x. In other words, a geodesic metric space is said to be a metric tree or an rtree, or ttree if it is 0hyperbolic in the sense of gromov that all of its geodesic. It is known that for a geodesic metric space hyperbolicity in the sense of gromov implies geodesic stability.
A consequence of this result is a type of rigidity with respect to metric transformation of roughly geodesic gromov hyperbolic spaces. Configuration of groups and paradoxical decompositions rejali, a. Gromov hyperbolic space example mathematics stack exchange. Metric spaces are first countable since one can use balls with rational radius as a neighborhood base. Metametrics appear in the study of gromov hyperbolic metric spaces and their boundaries. Quasimetric spaces play an important role in the study of gromov hyperbolic metric spaces 80, final remarks, and in the study of optimal transport paths 83. We show that the gromov boundary is a quotient topological space of the metric boundary, and that therefore a wordhyperbolic group has an amenable action on the metric boundary of its cayley graph. Gromov hyperbolic space encyclopedia of mathematics. Newest gromovhyperbolicspaces questions mathematics. Markov chains in smooth banach spaces and gromov hyperbolic metric spaces assaf naor.
The gromov boundary of a hyperbolic metric space has been extensively studied, but the gromov boundary is not guaranteed to exist for nonhyperbolic metric spaces. Introduction to hyperbolic metric spaces server university of. This can be done, for example, by imposing inequalities between mutual distances of. We obtain a modulus of continuity of the identity map from a domain equipped with the scale invariant cassinian metric or the gromov hyperbolic metric onto the same domain equipped with the euclidean metric. As a particular consequence we obtain that euclidean space is a borderline case for gromov hyperbolicity in terms of the isoperimetric function. The measure d arising from this construction is called the pattersonsullivan measure. We investigate the relationship between the metric boundary and the gromov boundary of a hyperbolic metric space. More about the geometry of hyperbolic metric spaces 42 5. The definition, introduced by mikhael gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Gromov hyperbolic spaces form a robust class of metric spaces to which. Gromovhausdor convergence of metric spaces jan cristina august, 2008 1 introduction the hausdor distance was known to hausdor at least in 1927 in his book set theory, where he used it as a metric on collections of setshaus 27.
Particular emphasis is paid to the geometry of their limit sets and on behavior not found in the proper setting. Pdf metric transforms yielding gromov hyperbolic spaces. The overflow blog socializing with coworkers while social distancing. We study the asymptotic dirichlet problem for pharmonic functions in a very general setting of gromov hyperbolic metric measure spaces. It is shown that a gromov hyperbolic geodesic metric space x with bounded growth at some scale is roughly quasiisometric to a convex subset of hyperbolic. Rtrees, cat1 spaces, and gromov hyperbolic metric spaces. Diophantine approximation and the geometry of limit sets in. The gromov boundary of an algebraic hyperbolic space. Gromov showed that quasiisometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a wellde ned notion of the boundary of a hyperbolic group. If one is allowed to rescale the metric of x by some positive constant, then there is an embedding where distances are distorted by at most an additive constant. In this context he proved gromovs compactness theorem, stating that the set of compact riemannian manifolds with ricci curvature. The gromov boundary of a hyperbolic metric space has been extensively studied, but the gromov boundary is not guaranteed to exist for non hyperbolic metric spaces. To this end, we present a novel fast algorithm treerep such that, given a hyperbolic metric for any 0, the algorithm learns a tree structure that approximates the original metric.
Conversely, if a satisfies the rips condition with constant 5, then a is 45thin. This is an equivalence relation between metric spaces. This boundary is equipped in a natural way with a quasimetric with respect to a base. In this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is gromov hyperbolic.
Problems on boundaries of groups and kleinian groups misha kapovich most problems in this list were collected during the workshop \boundaries in geometric group theory, in aim, 2005. Hyperbolic groups are fundamental examples of gromov hyperbolic spaces in geometric group theory. Metric transforms yielding gromov hyperbolic spaces arxiv. Metric transforms yielding gromov hyperbolic spaces 3 x,y,z. Jan 20, 2014 gromovs argument is that for a general group with polynomial growth the sequence of metric spaces obtained by rescaling its wordmetric converges to a finitedimensional metric space, in the same sense that the rescaled lattices converged to the plane. B which together are the induced metric and second fundamental form of a minimal immersion of into a potentially incomplete hyperbolic 3. Embeddings of gromov hyperbolic spaces 1 introduction. Heinonen and koskela 2 for domains in banach spaces with the quasihyperbolic metric. The gehringhayman theorem for gromov hyperbolic spaces 39 6. In this paper it is shown that the converse is also true. Hyperbolic metric spaces cayley graph it can be showed that if two spaces are quasi.
The theory of gromov hyperbolic spaces, introduced by gromov in the 1980s, has been considered in the books,,, and in several papers, but it is often assumed that the spaces are geodesic and usually also proper closed bounded sets are compact. Dirichlet problem at infinity on gromov hyperbolic metric. Introduction to hyperbolic metric spaces november 3, 2017 25 36. An unbounded, approximately ultrametric space fails to have the rough midpoint property and so is never a rough geodesic metric space proposition 3. Oct, 2017 a consequence of this result is a type of rigidity with respect to metric transformation of roughly geodesic gromov hyperbolic spaces.
These metric spaces are intrinsic, but they need not be geodesic, and. We will give various versions of gromovs hyperbolic criterion. The principal references in this area are the original texts of gromov 7, 8, 9, but. Bounded geometry in relatively hyperbolic groups 91. Gromov hyperbolic spaces and the sharp isoperimetric. In the 1980s, gromov introduced the gromovhausdorff metric, a measure of the difference between two compact metric spaces.
Rtrees, cat1 spaces, and gromov hyperbolic metric spaces 15 4. This is a simple instance of a very fruitful philosophy, introduced by gromov, that. Harmonic maps into singular spaces and gromov and r. Schoen 167 maps are constant from the quaternionic hyperbolic space or the cayley plane. In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations depending quantitatively on a nonnegative real number. Request pdf characterizations of metric trees and gromov hyperbolic spaces in a recent paper chatterji and niblo proved that a geodesic metric space is gromov hyperbolic if and only if the. We also extend the existence theory to include the construction of finite energy equivariant maps into buildings associated to an almost simple padic algebraic group h. The authors provide a number of examples of groups which exhibit a wide range of phenomena not to be. X are said to be equivalent if there exists a constant c 2 rsuch that d. Gromov s notion of hyperbolic spaces and hyperbolic groups have been studied extensively since that time. Our monograph presents the foundations of the theory of groups and semigroups acting isometrically on gromov hyperbolic metric.
Yu proved in y the coarse baumconnes conjecture for proper metric spaces with. Heinonen and koskela 2 for domains in banach spaces with the quasi hyperbolic metric. A classical example of a gromov hyperbolic space is a simply connected riemannian manifold with sectional curvature k. Characterizations of metric trees and gromov hyperbolic spaces.
The gromov product in algebraic hyperbolic spaces 35 72. It is shown that a gromov hyperbolic geodesic metric space x with bounded growth at some scale is roughly quasiisometric to a convex subset of hyperbolic space. This summarizes and completes a long line of results by many authors, from pattersons classic 1976 paper to more recent results of hersonsky and paulin 2002, 2004, 2007. Gromov 16 in the eighties as geodesic metric spaces in which geodesic triangles are thin. Boundary behaviour of harmonic functions on hyperbolic manifolds 4 1 the function u converges nontangentially at 2 the function u is nontangentially bounded from below at 3 there exists c 0 such that u is bounded from below on 90 c. This book presents the foundations of the theory of groups and semigroups acting isometrically on gromov hyperbolic metric spaces. We are motivated in part by recent studies in networking and vision which sug. Uniform spaces are negatively curved in the quasihyperbolic metric 17 4. The essential properties of the hyperbolic plane are abstracted to obtain the notion of a hyperbolic metric space, which is due to gromov 23. Hyperbolic metric spaces mathematics stack exchange. We discuss a distortion property of the scale invariant cassinian metric under mobius maps of a punctured ball onto another punctured ball. Ciupeanu uofm introduction to hyperbolic metric spaces november 3, 2017 2 36 introduction gromovs notion of hyperbolic spaces and hyperbolic groups have. D is relatively compact in the gromovhausdorff metric. So gromov hyperbolicity and geodesic stability are equialent for geodesic metric spaces.
This result has significance for the study of lipnorms on group calgebras. Boundaries of hyperbolic groups department of mathematics. Gromov in the eighties, has been considered in the books cdp, gdh, sh, bow, bh, bbi, ro and in several papers, but it is often assumed that the spaces are geodesic and usually also proper closed bounded sets are compact. These are notes for the chennai tmgt conference on. With an emphasis on nonproper settings about this title. Browse other questions tagged metric spaces hyperbolic geometry gromov hyperbolic spaces or ask your own question. These metric spaces are intrinsic, but they need not be geodesic, and they are proper only in the finitedimensional case. Xis also an isometry, and in this case we say that the metric spaces x.
Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Geodesic metric spaces, isometries and quasiisometries. For this reason many important themes, such as cat. Uniformly perfect boundaries of gromov hyperbolic spaces. Call r word hyperbolic if there exists a constant c. Boundaries of hyperbolic groups harvard university. Markov chains in smooth banach spaces and gromov hyperbolic. In this paper, the authors provide a complete theory of diophantine approximation in the limit set of a group acting on a gromov hyperbolic metric space.
The essential properties of the hyperbolic plane are abstracted to obtain the notion of a hyperbolic metric space, which is due to gromov. In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations between points. Hyperbolic groups lecture notes james howie heriotwatt university, edinburgh eh14 4as scotland. Mikhail leonidovich gromov also mikhael gromov, michael gromov or mischa gromov. The theory of gromov hyperbolic spaces, introduced by m. This paper will focus on the latter of these applications. The theory of gromov hyperbolic spaces, introduced by gromov in the 1980s, has been considered in the books 48,10,11 and in several papers, but it is often assumed that the spaces are geodesic and usually also proper closed bounded sets are compact.
Interplay between interior and boundary geometry in gromov. Quasigeodesic segments and gromov hyperbolic spaces. The relation eabove is an equivalence relation if xis gromov hyperbolic, but this is not true in general metric spaces 4, 1. Gromov hausdor convergence of metric spaces jan cristina august, 2008 1 introduction the hausdor distance was known to hausdor at least in 1927 in his book set theory, where he used it as a metric on collections of setshaus 27. If a is 5thin, then a satisfies the rips condition with constant 5. Many wellknown groups, such as mapping class groups and.
Geometry and dynamics in gromov hyperbolic metric spaces with an emphasis on nonproper settings 10. The visual metametric on such a space satisfies dx, x 0 for points x on the boundary, but otherwise dx, x is approximately the distance from x to the boundary. Gromov 4 introduced another boundary which makes sense for any metric space, but this was little studied until marc rie. Gromov coopted it in grom 81a modifying it to study the convergence of metric spaces. Lectures on hyperbolic groups uc davis mathematics. The next chapter consists of proofs of the equivalence of these definitions of a hyperbolic metric space, and of a hyperbolic group. We prove similar results for the linear filling radius inequality. Ciupeanu uofm introduction to hyperbolic metric spaces november 3, 2017 25 36. The metric topology on a metric space m is the coarsest topology on m relative to which the metric d is a continuous map from the product of m with itself to the nonnegative real numbers. Markov chains in smooth banach spaces and gromovhyperbolic metric spaces naor, assaf, peres, yuval, schramm, oded, and sheffield, scott, duke mathematical journal, 2006.
This notion provides a uniform global approach to such objects as the hyperbolic plane, simplyconnected riemannian manifolds with pinched negative sectional curvature, spaces, and metric trees. A metric space t is called a tangent subcone of rat infinity if it. Markov chains in smooth banach spaces and gromov hyperbolic metric spaces naor, assaf, peres, yuval, schramm, oded, and sheffield, scott, duke mathematical journal, 2006. Extending the gromov product to the boundary 29 66.
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