We study quasiisometry invariants of gromov hyperbolic spaces, focussing on the pcohomology and closely related invariants such as the conformal dimension, combinatorial modulus, and the combinatorial loewner property. 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. Jan 20, 2014 the first is to prove a conjecture in abstract group theory. Request pdf growth of conjugacy classes in gromov hyperbolic groups let gamma be a group acting properly and cocompactly by isometrics on a proper geodesic deltahyperbolic metric space x. The first is to prove a conjecture in abstract group theory. The study of random walks on in nite groups, and speci cally the question of whether they asymptotically approach some. Symbolic dynamics and hyperbolic groups michel coornaert. For the theory of gromov hyperbolic spaces and groups we refer the reader to gromovs original paper gro1, the expositions cdp. From max dehn to mikhael gromov, the geometry of infinite. Discrete groups of isometriesof n dimensional hyperbolic space hn.
There a group is rela tively hyperbolic if its conedoff cayley graph is gromov hyperbolic, and it satisfies a. The inspiration came from various existing mathematical theories. The next chapter consists of proofs of the equivalence of these definitions of a hyperbolic metric space, and of a hyperbolic group. From max dehn to mikhael gromov, the geometry of in nite groups. Pdf boundaries of hyperbolic groups semantic scholar.
Hyperbolic groups with lowdimensional boundary numdam. 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. First observe that for every finitely presented group. Introduction the notion of a hyperbolic group was introduced by gromov gr. 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. Algorithmic properties of relatively hyperbolic groups arxiv.
Central limit theorems for gromov hyperbolic groups. For the theory of gromov hyperbolic spaces and groups we refer the reader to gromovs original paper gro1, the expositions cdp and gh, and the summary in cp. Note also that whenever kdivides l, then there is a degree l k map from s3. In contrast, in nitely generated groups are not considered in the theory of hyperbolic groups. Dave peifer university of north carolina at asheville october 28, 2011 dave peifer from max dehn to mikhael gromov. Fundamental groupsof compact riemannian manifolds with strictly negative sectional curvature are hyperbolic. In group theory, more precisely in geometric group theory, a hyperbolic group, also known as a word hyperbolic group or gromov hyperbolic group. Horofunctions and symbolic dynamics on gromov hyperbolic groups. To every gromovhyperbolic space x in particular to every wordhyperbolic group one can associate the socalled space at infinity or boundary. Random walks and the uniform measure in gromovhyperbolic groups benjamin mckenna abstract. The notion of a hyperbolic group was introduced and developed by mikhail gromov. Growth of conjugacy classes in gromov hyperbolic groups.
In this paper we study asymptotic properties of symmetric and nondegenerate random walks on transient hyperbolic groups. Gromovs theory has also found numerous applications to and connections with many branches of mathematics. A nitely generated group is called hyperbolic if its cayley graph is. Symmetry free fulltext gromov hyperbolicity in directed. The study of nite markov chains that converge to the uniform distribution is well developed. Lectures on hyperbolic groups uc davis mathematics. Gromovs 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. In this section we will describe some necessary background material and establish notation. Random walks and the uniform measure in gromov hyperbolic groups benjamin mckenna abstract. Deciding whether a group is wordhyperbolic is made easier by the following result. 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. Dave peifer university of north carolina at asheville october 28, 2011 dave peifer from max dehn to mikhael gromov, the geometry of in nite groups.
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. Originally, the theory of hyperbolic spaces was developed to be applied to finitely generated groups and, in that field, it has proved relevant competence in practical applications. The essential properties of the hyperbolic plane are abstracted to obtain the notion of a hyperbolic metric space, which is due to gromov 23. In 3 we recall with short proofs the basic results concerning random walks on gromov hyperbolic groups. In 4 we prove the complete convergence to the escape rate, the logregularity of the stationary measure on the gromov boundary, the centerability of the busemann. Let us start with three equivalent definitions of hyperbolic groups. C k z k by the rotational symmetry of c k is obtained by generalized dehn surgery on the white head link w, so the limit of 6. May 28, 2009 in this paper we study asymptotic properties of symmetric and nondegenerate random walks on transient hyperbolic groups. Champetier 18, generic finitely presentable groups are hyperbolic and have the menger curve as boundary.
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 space of metrics on gromov hyperbolic groups alex furman university of illinois at chicago northwestern university, 20101031 1. We give a complete characterization of the locally compact groups that are nonelementary gromovhyperbolic and amenable. 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. Geometric group theory began in the 1980s with work of cannon, gromov and others, applying geometric techniques to prove algebraic properties for large. 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. Then g acts discretely, cocompactly, and isometrically on. A group g is hyperbolic or wordhyperbolic or gromovhyperbolic if for some. 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. Then the ideal boundaries of all groups in this class are homeomorphic to the menger curve. We generalize this result to graphs of hyperbolic groups over. Serbin new york city, 20 andreipaul grecianu, joint work with o. The subject matter is hyperbolic groups one of the main objects of study in geometric group theory. Other examples of hyperbolic groups z is hyperbolic.
Essay about failing exams writing what is a personal essay listening social. We prove a central limit theorem and a law of iterated logarithm for the drift of a random walk, extending previous results by s. They coincide with the class of mapping tori of discrete or continuous one. 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. Sela 167 states that the isomorphism problem is also solvable for torsionfree wordhyperbolic groups. The theory of hyperbolic groups and the philosophy of considering groups as coarse geometric objects. Gromov hyperbolic groups and the macaev norm article pdf available in pacific journal of mathematics 2231. In this paper we survey the known results about boundaries of wordhyperbolic groups. This enables us to show that knapsack can be solved in logcfl for every group that belongs to the closure of hyperbolic groups under free products and direct products with z. Gromov s proof, which cleverly forces geometry on groups where no geometry seemed present, is famous not only for the result itself, but even more because of its role in starting the whole new field of geometric group theory.
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. Convergence of sample paths to the gromov boundary in this case is due to. In chapter 4, we prove successively the complete convergence towards the escape rate, the logregularity of the stationary measure on the gromov boundary, the centerability of the busemann cocycle on the busemann boundary, and the central limit theorem 4. The rst published examples of nonlinear hyperbolic groups are due to m. We give a complete characterization of the locally compact groups that are nonelementary gromov hyperbolic and amenable. 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. Groups acting on hyperbolic spaces andreipaul grecianu, joint work with o. Consider ginvariant proper, quasiconvex metric d on g.
G is quasisymmetrically homeomorphic to an ahlfors q regular metric 2sphere z with ahlfors regular conformal dimension q. Gromovs proof, which cleverly forces geometry on groups where no geometry seemed present, is famous not only for the result itself, but even more because of its role in starting the whole new field of geometric group theory. Recall that a hyperbolic group is 1ended if its gromov boundary is a nonempty connected space. For example, they were historically applied to some computer science questions, as. Note that you can find a pdf of gromovs essay on his website here. From max dehn to mikhael gromov, the geometry of infinite groups. Hyperbolic groups lecture notes mathematical and computer. The proof involves assigning a system of binary ad. Mikhail leonidovich gromov also mikhael gromov, michael gromov or mischa gromov. Gromovs invariant and the volume of a hyperbolic manifold note that the quotient space s3.
Gromov hyperbolic groups essays in group theory character. Hyperbolic groups with homeomorphic gromov boundaries. Boundaries of hyperbolic groups university of illinois at urbana. Relatively hyperbolic groups university of warwick. Gromov, l2cohomology and group cohomology, topology, 25. Boundaries of hyperbolic groups harvard university. This makes precise the notion of a group being negatively curved. It is still an open problem as to whether all hyperbolic groups act properly and cocompactly by isometries on cat0 metric spaces. The main results of this paper show that if one is trying to. This book is an elaboration on some ideas of gromov. In this paper we develop some of the foundations of the theory of relatively hyperbolic groups as originally formulated by gromov.