The notion of a hyperbolic group was introduced and developed by mikhail gromov. Random walks on hyperbolic groups i university of chicago. As commented on previously, identities for hyperbolic functions often look like those for the ordinary trigonometric functions sin, cos, tan, but there is often a change of sign. Gromov s solution of milnors problem any group of polynomial growth has a nilpotent subgroup of nite index. 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, sli. Gromov hyperbolic groups essays in group theory character. A nilpotent group g is a group with a lower central series terminating in the identity subgroup. On hyperbolic geometry structure of complex networks. A random walker performs a simple random walk on zd, and at each step adjusts the state of the lamp at hisher current site by a fair coin toss. In the geometrically in nite case, the surface is virtually the berof a bering. Amenable actions, lyapunov exponents and an idea of bowen. A second alternative is the skew students tdistribution based on order statistics, recently introduced by jones and faddy 2003. For example, the free burnside group of exponent n with two generators is given by the presentation.
The hyperbolic plane and psl 2r 10 acknowledgments references 1. Hyperbolic groups are a class of groups introduced by gromov in 1987, which form an important part of geometric group theory. Essay writing good examples very essay about movie our country nepal to essay meaning jamie waste problems essay zero. Formulas and identities of hyperbolic functions pacharapokin chanapat shinshu university nagano, japan hiroshi yamazaki shinshu university nagano, japan summary. The theory of hyperbolic groups and the philosophy of considering groups as coarse geometric objects was developed by gromov. 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. Hyperbolic groups lecture notes james howie heriotwatt university, edinburgh eh14 4as scotland j. Mitra shows that for each vertex group v inclusion v. 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. The space of metrics on gromov hyperbolic groups alex furman university of illinois at chicago northwestern university, 20101031 1.
Later, it was reformulated and studied from di erent points of view by farb far98, bowditch bow12, osin osi06c and many others. Let be a hyperbolic group relative to a collection of in. Random walks on hyperbolic groups ii steve lalley university of chicago october 10, 20. Yuval peresy oded schrammz scott sheeldx october 19, 2004 abstract a metric space x has markov type 2, if for any reversible. Mapping class groups admit an amenable action on an explicit boundary kida, h 09 5. He proved that a subgroup h of a hyperbolic group g is quasiconvex if and only if for any two disjoint closed subsets k and l of. Relative hyperbolicity and relative quasiconvexity for countable groups hruska, g christopher. The lamplighter random walk at each site x 2zd is a lamp, which can be eitheronoroff. Essay about failing exams writing what is a personal essay listening social media privacy essays essay water shortage valuable resource. Let us start with three equivalent definitions of hyperbolic groups. We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an rtree. A geometrically nite surface preserves a convex subset of hyperbolic space whose quotient by the group has nite nonzero volume.
The geometry of hyperbolic and relatively hyperbolic groups has been integral to the solutions of many algorithmic problems 40, 42, 10, 12, date. A very important fact is that the hyperbolic trigonometric functions take area as their argument called the hyperbolic angle, but this is just a name and has nothing to do with angles, as depicted below. Horofunctions and symbolic dynamics on gromov hyperbolic groups volume 43 issue 3 michel coornaert, athanase papadopoulos. The geometry of hyperbolic and relatively hyperbolic groups has been integral to the solutions of many algorithmic problems 41, 43, 10, 12, date. A metric space x is proper if closed balls of finite radius in x are compact. Bowditch bo99 characterized the quasiconvex subgroups of gromov hyperbolic groups in terms of their action on the gromov boundary of the group. Horofunctions and symbolic dynamics on gromov hyperbolic groups. But general results about the nonexistence of algorithms or their complexity play another, completely different, role in group theory. Hyperbolic geometry, surfaces, and 3manifolds bruno martelli dipartimento di matematica \tonelli, largo pontecorvo 5, 56127 pisa, italy email address. 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. Nonelementary infinite and not virtually cyclic hyperbolic groups have always exponential growth rate this is a consequence of the tits alternative. My first guess would be to take the symmetry group of a tiling, but will it cayley graph be really similar to the tiling.
Hyperbolic groups satisfy a linear isoperimetric inequality. Hyperbolic automorphisms of free groups article pdf available in geometric and functional analysis 105. Finite subgroups of hyperbolic groups springerlink. Local limit theorem for symmetric random walks in gromov. Consider ginvariant proper, quasiconvex metric d on g. Gromov called a space hyperbolic if all geodesic triangles in the space were uniformly thin, and he called a group hyperbolic if it acts properly and coboundedly on a hyperbolic. The motivating examples of relatively hyperbolic groups are the fundamental groups of complete noncompact hyperbolic manifolds of finite volume.
The class of relatively hyperbolic groups includes many examples. A finite group is a trivial example of a hyperbolic group. Other examples of hyperbolic groups z is hyperbolic. Growth of quotients of groups acting by isometries on. I wonder whether it is possible to prove most of the classifical properties of hyperbolic groups in this context, probably using van kampen diagrams. For this reason many important themes, such as cat.
A group is hyperbolic if it acts properly discontinuously and cocompactly by isometries on a proper hyperbolic metric space. The second lecture gives the thintriangles definition of hyperbolic group. For instance, any hyperbolic group is hyperbolic relative to the trivial subgroup. Denote by hgi the set of homeomorphism types of gromov boundaries of those vertex groups in gi which are nonelementary hyperbolic i.
First observe that for every finitely presented group r there exists a smooth bounded i. Kleinian groups and thurstons work 7 references 8 1. Group without torsion have finite cohomological dimension. Mar 21, 2006 on hyperbolic groups on hyperbolic groups houcine, abderezak ould 20060321 00. It is not known whether there exists an hyperbolic group which is not residually finite. It is not known 1996 whether every hyperbolic group.
Hyperbolic groups from dehn functions mathematics stack. In this paper, we generalize the classical definition of gromov hyperbolicity to the context of directed graphs and we extend one of the main results of the theory. In this article, we proved formulas of hyperbolic sine, hyper bolic cosine and hyperbolic tangent, and their identities. Hyperbolic groups lecture notes mathematical and computer. In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations depending quantitatively on a nonnegative real number.
Hyperbolic functions show up in many reallife situations. Hyperbolicity is a largescale property, and is very useful to the study of certain infinite groups. Note that a word hyperbolic group has exponential growth if and only if it is nonelementary. Oct 28, 20 among smooth dynamical systems, hyperbolic dynamics is characterized by the presence of expanding and contracting directions for the derivative. However, i have included a section on the hyperbolic boundary. Will it be hyperbolic if we started with hyperbolic space. We characterize lacunary hyperbolicgroups as direct limits of gromov hyperbolic groups satisfying certain restrictions on the hyperbolicity constants and injectivity radii. The action of a group relatively hyperbolic to a nite collection of amenable subgroups on a geometric boundary is amenable ozawa 06 4. Then the ideal boundaries of all groups in this class are homeomorphic to the menger curve. Virtual properties of 3manifolds 7 geometrically nite or geometrically in nite. First observe that for every finitely presented group. Random walks on hyperbolic groups ii university of chicago. Normal automorphisms of relatively hyperbolic groups 3 in fact, theorem 1. The core part of the course is devoted to gromovhyperbolic spaces and groups.
This is a situation where the differential alone provides strong local, semilocal or even global information about the dynamics. Convergence of sample paths to the gromov boundary in this case is due to. As the class of word hyperbolic groups is, in a geometric sense, a generalization of free groups, it is natural to conjecture that word hyperbolic groups realize their uniform exponential growth rate. The rst case includes quasifuchsian surfaces figure5.
Forms group of geometric transformations contains all circlepreserving transformations in higher. In chapter 1, we give an introduction to this subject. Algorithmic properties of relatively hyperbolic groups arxiv. In principle, one could take a similar approach to studying. In practice, i ran out of time and settled for discussing only the rips complex. The nal lecture was designed to give a glimpse of two slightly more advanced aspects of the subject, namely the rips complex and the boundary of a hyperbolic group. The study of random walks on in nite groups, and speci cally the question of whether they asymptotically approach some. 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. Hyperbolic groups may be defined as finitely generated groups admitting a linear dehn function. In his seminal paper 34, gromov introduced the class of hyperbolic groups which revolutionized the study of geometric group theory. The study of nite markov chains that converge to the uniform distribution is well developed. For example, they are related to the curve one traces out when chasing an. We moreover give a description of all gromov hyperbolic locally compact groups with a cocompact amenable subgroup.
Markov chains in smooth banach spaces and gromov hyperbolic metric spaces assaf naor. 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. We show that the gromov boundary of the free product of two in. Hyperbolic groups with homeomorphic gromov boundaries. The generalised hyperbolic skew students tdistribution. We say that the walk is aperiodic if there exists an odd integer n such that p ne,e0. We show that every finite subgroup of g is conjugate to a subgroup each element of which has length at most 2. There is a general rule for deriving an identity for hyperbolic functions from the corresponding identity for ordinary trigonometric functions. Many researchers i count myself among this number are less interested in particular groups than in classes of groups.
Horofunctions and symbolic dynamics on gromov hyperbolic. In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group. In general, a finitely presented group can have very nasty properties, but many of these properties are avoided if the group is assumed to admit a nice action by isometries on a space with a negative curvature property, such as gromov hyperbolicity. Free group and fundamental groups of compact riemannian manifolds of negative sectional curvature cf. This work is concerned with the stable norm in word hyperbolic groups as defined by gromov. We give a short elementary proof of one of its basic property, that is existence of a computable uniform non null lower bound for stable norm in a word hyperbolic group. The generalised hyperbolic skew students tdistribution 5 tdistribution with degrees of freedom.
Hyperbolic manifolds, discrete groups and ergodic theory. Fuchsian groups and closed hyperbolic surfaces 6 5. Discrete groups of isometriesof n dimensional hyperbolic space hn. Hyperbolic groups are sometimes called word hyperbolic groups or negatively curved groups. This makes precise the notion of a group being negatively curved. 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 way to use geometry to study groups is considering their isometric actions on metric spaces. On residualing homomorphisms and gsubgroups of hyperbolic. Notes on relatively hyperbolic groups and relatively quasiconvex subgroups matsuda, yoshifumi, oguni, shinichi, and yamagata, saeko, tokyo journal of mathematics, 2015.
The definition, introduced by mikhael gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. A word in sis a nite concatenation of elements of s. Moreover the preimage of any conical point is a singleton. Hyperbolic manifolds, discrete groups and ergodic theory coursenotes c. Growth gap in hyperbolic groups and amenability arxiv.
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