Outline 1 a claim due to gromov 2 arzhantsevaolshanskiis proof 3 a new point of view 4 stallings graphs 5 counting stallings graphs. On hyperfiniteness of boundary actions of hyperbolic groups. 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 particular this theorem holds for subgroups of gromov hyperbolic groups, for kleinian groups, or more generally for any countable group acting properly discon. Gromov given a hyperbolic group g with one topological end i. Career topic essay report essay science project hook ielts essay obesity ideas pdf.
In chapter 1, we give an introduction to this subject. Epstein asked whether there is an algorithm that computes the cech cohomology of the boundary of a given hyperbolic group. A finite group is a trivial example of a hyperbolic group. This is a simple instance of a very fruitful philosophy, introduced by gromov, that. Mapping class groups admit an amenable action on an explicit boundary kida, h 09 5. In such a general setting, gis not any more gromov hyperbolic, although it is a direct limit of those in fact, of graphical small cancellation groups, see more details in section3. Gromov hyperbolicity and the kobayashi metric 5 which is a contradiction. Random groups and related topics l16 university of cambridge. These are notes for the chennai tmgt conference on. The definition, introduced by mikhael gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. What a term paper is net essay internet education relating topics.
Newest gromovhyperbolicspaces questions mathematics. Various hyperbolic properties introduced earlier mostly in the context of group theory, were summed up and further developed by m. Pdf we show that the gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by. Free group and fundamental groups of compact riemannian manifolds of negative sectional curvature cf. Amenable actions, lyapunov exponents and an idea of bowen. The subject matter is hyperbolic groups one of the main objects of study in geometric group theory.
Free and hyperbolic groups institute for advanced study. A book which appeared in birkhausers progress in mathematics in 2001. The classification of euclidean isometries leads to results on regular polyhedra and polytopes. Geometric group theory began in the 1980s with work of cannon, gromov and others, applying geometric techniques to prove algebraic properties for large classes of groups.
It considers the 4tuples in an ordering allowing to cut the search space as soon as a new lower bound is found see the modules documentation. The martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups matthieudussaule,ilyagekhtman,victorgerasimov, andleonidpotyagailo abstract. The action of a group relatively hyperbolic to a nite collection of amenable subgroups on a geometric boundary is amenable ozawa 06 4. In many ways, hyperbolic groups behave similarly to. Properties of sets of isometries of gromov hyperbolic spaces. Markov chains in smooth banach spaces and gromov hyperbolic. First observe that for every finitely presented group. One particularly natural large scale property used in this way is the gromov boundary. A nitely generated group is called hyperbolic if its cayley graph is. Postscript and pdf files of my preprints 3manifolds, geometric structures, kleinian groups.
Academic visits department of mathematics, puc, chile. The key idea is to consider actions of fundamental groups of hyperbolic manifolds on the boundary of hyperbolic space by quasiconformal maps and to obtain regularity properties of the quasiconformal actions. In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or gromov hyperbolic group, is a finitely generated group equipped with a word metric satisfying certain properties abstracted from classical hyperbolic geometry. A third set of examples live in certain hyperbolic groups embedded in the mapping class group mod. Cat0spaces, gromov hyperbolic spaces, hilbert geometries, certain pseudoconvex domains, and partially for thurstons boundary of teichmull. Hyperbolic groups are sometimes called word hyperbolic groups or negatively curved groups. In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations depending quantitatively on a nonnegative real number. The first chapter consists of a collection of alternative definitions, both of hyper bolic metric spaces and of hyperbolic groups including gromovs inner product, sli.
Yuval peresy oded schrammz scott sheeldx october 19, 2004 abstract a metric space x has markov type 2, if for any reversible. What links here related changes upload file special pages permanent link page information wikidata item cite this page. Talks at conferences and seminars the avalanche principle and negative curvature. Hyperbolicity is a largescale property, and is very useful to the study of certain infinite groups. Hyperbolic groups with homeomorphic gromov boundaries. The first half of the book introduces basic notions of group theory and studies symmetry groups in various geometries, including euclidean, projective, and hyperbolic. Hyperbolic groups are a class of groups introduced by gromov in 1987, which form an important part of geometric group theory. The notion of a hyperbolic group was introduced and developed by mikhail gromov. Such a generalization, a picture of the global structure of groups and a mix of the properties of groups from di erent classes lead us to the study of relatively hyperbolic. We moreover give a description of all gromov hyperbolic locally compact groups with a cocompact amenable subgroup.
Gromov hyperbolic groups essays in group theory character. On a small cancellation theorem of gromov yann ollivier. A countable nonelementary convergence group is primitive if and only if it contains no. Gromov, l2cohomology and group cohomology, topology, 25. This class received a lot of attention in the last three decays, and its study is in the heart of geometric group theory. Transactions of the american mathematical society volume 350, number 2, february 1998, pages 5716 s 0002994798017735 hyperbolic groups and free constructions o. Combination theorems for quasiconvex subgroups of relatively hyperbolic groups eduardo martinezpedroza university of oklahoma abstract. I will touch upon a variety of topics as random groups will serve as a pretext to discuss other important notions in modern in nite group theory, such as small cancellation theory, word hyperbolic groups, burn. Horofunctions and symbolic dynamics on gromov hyperbolic groups article pdf available in glasgow mathematical journal 4303. In chapter 2, we use the theory of complexes of groups to show that the integral homology and cohomology groups of a hyperbolic group are computable by a turing machine.
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. We moreover give a description of all gromovhyperbolic locally compact groups with a cocompact amenable subgroup. Hyperbolic groups with lowdimensional boundary numdam. The groups obtained by this process can in some cases be noticeably di. Gromovs hyperbolic groups were frequently cited as examples. Example note that if xis a tree, then it is hyperbolic for any 0 as the geodesic triangles all look like tripods in general, the smaller is, the more hyperbolic spaces \look like trees. Math 227 b w, wang, advanced topics in geometric and algebraic topology quantum invariants of 3manifolds and the volume conjecture. Thus, following olliviers terminology 31, gromovs groups are even \wilder than previously shown. Geometric group theory began in the 1980s with work of cannon, gromov and others, applying geometric techniques to prove algebraic properties for large classes of. Geometry and dynamics in gromov hyperbolic metric spaces.
In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or gromov hyperbolic group. Markov chains in smooth banach spaces and gromov hyperbolic metric spaces assaf naor. France research paper chapter 2 3 interesting topics to write essay response. A theory of groups of isometries is developed for any proper metric space. Gromov hyperbolic groups, ends and boundaries of groups and splittings and quasiconvexity. A geodesic metric space is hyperbolic if all triangles are slim. On the dynamics of isometries university of neuchatel. Gromov hyperbolic space encyclopedia of mathematics. We answer epsteins question in the affirmative for a restricted class of hyperbolic groups. The first chapter consists of a collection of alternative definitions, both of hyper bolic metric spaces and of hyperbolic groups including gromov s inner product, slim.
This extends the usual theory of hyperbolic groups and gives in particular several new results in the special case of. 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. On the other hand when x is the cayley graph of a gromov hyperbolic group g, then the. In contrast, in nitely generated groups are not considered in the theory of hyperbolic groups.
Marcin sabok on hyper niteness of boundary actions of hyperbolic groups. Gromov as a generalization of the class of fundamental groups of complete. Pdf hyperbolic groups with homeomorphic gromov boundaries. We show that the gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of. Pdf in the most recent decades, metric spaces have been studied from a variety of viewpoints. Hyperbolic groups are fundamental examples of gromov hyperbolic spaces in geometric group theory. The methods can be applied to more general gromov hyperbolic spaces and groups. If a group admits an action on a gromov hyperbolic space then large scale properties of the space give useful invariants of the group.
G is gromovhyperbolic if and only if gis convex cocompact. This is a wide class in many senses, a generic group is gromov hyperbolic consisting of groups for which the word metric roughly has a negative curvature. Detecting topological properties of boundaries of hyperbolic. Embeddings of gromov hyperbolic spaces springerlink. Let us start with three equivalent definitions of hyperbolic groups. For convergence groups we prove the following theorem. But so far not much seems to be known for arithmetic groups except the cocompact lattices in rrank 1 semisimple lie groups which are gromov hyperbolic. The class of relatively hyperbolic groups was originally introduced by m.
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