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. From max dehn to mikhael gromov, the geometry of infinite groups. Other examples of hyperbolic groups z is hyperbolic. Introduction the notion of a hyperbolic group was introduced by gromov gr. Since the characterization of gromov hyperbolic graphs seems a too ambitious task, there are many papers studying the hyperbolicity of several classes of graphs. Pdf in this paper we study asymptotic properties of symmetric and nondegenerate random walks on transient hyperbolic groups. X are said to be equivalent if there exists a constant c 2 rsuch that d. 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. The notion of a relatively hyperbolic group was originally suggested by gromov 12 and since then it has been investigated from different points of view 4,810. He constructed a hyperbolic group which is not linear. We are interested in those algebraic properties of t which aro reflected in the geometry of x tbe action of i on x.
The tits alternative for hyperbolic groups follows from the fact that a hyperbolic group either contains a cyclic subgroup of finite index so called elementary hyperbolic groups or contains a free subgroup of rank two. Gromov hyperbolic groups essays in group theory chemistry. Algebraic examples of relatively hyperbolic groups include free products and their small cancellation quotients 18, and fully residually free groups or selas limit groups 7. Hyperbolic groups and free constructions 609 proof. Mikhail leonidovich gromov also mikhael gromov, michael gromov or mischa gromov. The subject matter is hyperbolic groups one of the main objects of study in geometric group theory. For any n 0, there exist large, torsionfree, oneended, nonlinear hyperbolic groups. Outline i groups and geometry before 1900 i max dehn i in nite groups presentations, cayley graph, the word problem i geometry of groups dehn diagrams, isoparametric inequality i dehns algorithm i mikhael gromov i hyperbolic groups dave peifer from max dehn to mikhael gromov.
Hyperbolic spaces a metric space x is called hyperbolic coarse hyperbolic in. Free and hyperbolic groups institute for advanced study. Symbolic dynamics and hyperbolic groups michel coornaert. The space of metrics on gromov hyperbolic groups alex furman university of illinois at chicago northwestern university, 20101031 1. Socalled virtually free groups, or groups with a niteindex free subgroup. Existence of finite index torsion free subgroups of. On the other hand, for example, zd is nonhyperbolic for d 2. This is the case even within the class of hyperbolic freebycyclic groups.
A nitely generated group is hyperbolic if and only if all of its asymptotic cones are rtree. Notes on word hyperbolic groups chapter o some definitions and notation we assume that the reader knows what a free group is. Gromov characterized hyperbolic groups in terms of asymptotic cones. On residualing homomorphisms and gsubgroups of hyperbolic. If g is a torsionfree hyperbolic group, then g is type f. Embeddings of gromov hyperbolic spaces springerlink. Hyperbolic groups have been the subject of intensive investigation since the work of gromov 8. A finite group is a trivial example of a hyperbolic group. Career topic essay report essay science project hook ielts essay obesity ideas pdf. Sela 167 states that the isomorphism problem is also solvable for torsionfree wordhyperbolic groups.
The theory of hyperbolic spaces was introduced by m. Essay spoken language in dubai essay harvard university financial aid calculator the essay sample zusaka salesman essay topics history of science methodology in. This book is an elaboration on some ideas of gromov. We moreover give a description of all gromov hyperbolic locally compact groups with a cocompact amenable subgroup. Gromov in 1,2, and after that studied by many scientists, see, e. Here a hyperbolic group is a nitely generated group gwhose cayley graph satis es gromovs thin triangles condition see 7. Boundaries of hyperbolic groups harvard university. Gromov hyperbolic groups and the macaev norm article pdf available in pacific journal of mathematics 2231. In fact, this probability p will usually be of the form p p n.
We moreover give a description of all gromovhyperbolic locally compact groups with a cocompact amenable subgroup. The following are standard examples of hyperbolic groups. A nitely generated group is called hyperbolic if its cayley graph is. Group without torsion have finite cohomological dimension. However, after a quick flick through, i am having trouble finding this specific result. Saul%schleimer,universityofwarwick% % markus%lohrey%defined%the%compressed%word%problemfor%groups%and%gave%a%. Horofunctions and symbolic dynamics on gromov hyperbolic. Gromov hyperbolic groups essays in group theory character. If g contains a free abelian group of rank r as a subgroup of finite index, then. What a term paper is net essay internet education relating topics. Gromovs theory has also found numerous applications to and connections with. Fundamental groupsof compact riemannian manifolds with strictly negative sectional curvature are hyperbolic. Denote by hgi the set of homeomorphism types of gromov boundaries of those vertex groups in gi which are nonelementary hyperbolic i. Gromov s theory of hyperbolic groups have had a big impact in combinatorial group theory and has deep connections with many branches of mathematics suchdifferential geometry, representation theory, ergodic theory and dynamical systems.
Hyperbolic groups with homeomorphic gromov boundaries. A nilpotent group g is a group with a lower central series terminating in the identity subgroup. It was shown in dp99 that asymptotic cones of nonelementary hyperbolic groups are all isometric to the complete homogeneous rtree of valence continuum. Hyperbolic groups and free constructions hyperbolic. 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. Gromov s theory has also found numerous applications to and connections with many branches of mathematics. There are compelling practical features of groups realised as groups of. For i 1,2, let gi be graphs of groups with all vertex groups hyperbolic and all edge groups. Sela 167 states that the isomorphism problem is also solvable for torsion free word hyperbolic groups.
Hyperbolic groups lecture notes mathematical and computer. 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. Horofunctions and symbolic dynamics on gromov hyperbolic groups. If x is a finite set of generators for the group g, then there is a natural surjection. In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations depending quantitatively on a nonnegative real number. 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. Gromov gr used small cancellation theory to show that suitable quotients of a lattice as above are in nite hyperbolic groups see also ch, d, ol, and then kapovich used superrigidity results to show that any. Consider the class of all 2dimensional hyperbolic groups which are 1ended, do not split over virtually cyclic groups, are not commensurable to surface groups, are not relative pd3 groups. Mikhael gromov infinite groups as geometric objects. 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. The definition, introduced by mikhael gromov, generalizes the metric properties of classical hyperbolic geometry and of tr. A consequence of this crinkliness is that the area of a disc of radius grows exponentially with, that is, it grows like for some. Conjecture let g be weakly hyperbolic relative to a subgroup h zn f where f is free and. Then the ideal boundaries of all groups in this class are homeomorphic to the menger curve.
Jul 25, 2002 in the case where x is the cayley graph of a hyperbolic group \gamma, we show, following ideas of gromov sketched in his paper hyperbolic groups, that the space of cocycles associated to horofunctions which take integral values on the vertices is a onesided subshift of finite type. Then the renormalized variables p1 n g n 1n converge in law to a nondegenerate gaussian law. First observe that for every finitely presented group. In this paper, it is proven that every mycielskian graph g m is hyperbolic and that. Important examples of such gromov hyperbolic spaces mare i metric trees, ii gromov hyperbolic groups endowed with the leftinvariant dis. Gromov s solution of milnors problem any group of polynomial growth has a nilpotent subgroup of nite index. Furthermore, we study the extremal problems of finding the smallest and largest hyperbolicity constants of. A geodesic metric space is said to be gromov hyperbolic if every geodesic triangle is uniformly thin see definition 2. Lectures on hyperbolic groups uc davis mathematics. Nonelementary hyperbolic groups are not simple in a very strong sense.
If we denote the standard generating set for zdby fe 1 1e 1. 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. Mikhael gromov infinite groups as geometric objects let a group i faithfully and isometricajiy act on a metric space x. The rst published examples of nonlinear hyperbolic groups are due to m. Hyperbolicity is a largescale property, and is very useful to the study of certain infinite groups. Subgroups of lacunary hyperbolic groups and free products. Let us start with three equivalent definitions of hyperbolic groups. Pdf boundaries of hyperbolic groups semantic scholar.
May 16, 2011 gromov boundary of hyperbolic groups may 16, 2011 as we have seen in pervious posts, the cayley graphs of groups equipped with the word metric is a very special class of geodesic metric space they are graphs that have tons of symmetries. Note that sapirs group is residually finite ascending hnnextensions of free groups are always residually finite. On the other hand, for example, zd is non hyperbolic for d 2. To prove the assertion we need the solvability in any hyperbolic group of the word and conjugacy problems as well as the power. A word w in the alphabet j 1 is equal to 1 in g if and only if there is an equality w. It is not known 1996 whether every hyperbolic group admits a torsionfree subgroup of.
Jan 20, 2014 gromov s idea, phrased informally, was this. Proceedings of the 1978 stony brook conference state univ. We can construct a geodesic metric space called the cayley graph rx g of. Convergence of sample paths to the gromov boundary in this case is due to. 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. Discrete groups of isometriesof n dimensional hyperbolic space hn. Note that you can find a pdf of gromovs essay on his website here. In the case where x is the cayley graph of a hyperbolic group \gamma, we show, following ideas of gromov sketched in his paper hyperbolic groups, that the space of cocycles associated to horofunctions which take integral values on the vertices is a onesided subshift of finite type. Boundaries of hyperbolic groups department of mathematics. Central limit theorems for gromov hyperbolic groups.
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