On hyperbolic geometry structure of complex networks. 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. Relative hyperbolicity and relative quasiconvexity for countable groups hruska, g christopher. I wonder whether it is possible to prove most of the classifical properties of hyperbolic groups in this context, probably using van kampen diagrams. Note that a word hyperbolic group has exponential growth if and only if it is nonelementary. 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. Will it be hyperbolic if we started with hyperbolic space. Convergence of sample paths to the gromov boundary in this case is due to. Gromov hyperbolic groups essays in group theory character. Growth of quotients of groups acting by isometries on. Group without torsion have finite cohomological dimension.
The geometry of hyperbolic and relatively hyperbolic groups has been integral to the solutions of many algorithmic problems 40, 42, 10, 12, date. There is a general rule for deriving an identity for hyperbolic functions from the corresponding identity for ordinary trigonometric functions. Local limit theorem for symmetric random walks in gromov. Random walks and the uniform measure in gromovhyperbolic groups benjamin mckenna abstract. Algorithmic properties of relatively hyperbolic groups arxiv. 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. Hyperbolic automorphisms of free groups article pdf available in geometric and functional analysis 105. Hyperbolic manifolds, discrete groups and ergodic theory coursenotes c. The hyperbolic plane and psl 2r 10 acknowledgments references 1. Gromov s solution of milnors problem any group of polynomial growth has a nilpotent subgroup of nite index. In this article, we proved formulas of hyperbolic sine, hyper bolic cosine and hyperbolic tangent, and their identities. The theory of hyperbolic groups and the philosophy of considering groups as coarse geometric objects was developed by gromov.
Mitra shows that for each vertex group v inclusion v. 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. Denote by hgi the set of homeomorphism types of gromov boundaries of those vertex groups in gi which are nonelementary hyperbolic i. We show that every finite subgroup of g is conjugate to a subgroup each element of. 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. Hypertree is a java phylogenetic tree viewer, with a hyperbolic fisheye view and editing abilities that help in managing very large trees new june 2011.
Bowditch bo99 characterized the quasiconvex subgroups of gromov hyperbolic groups in terms of their action on the gromov boundary of the group. We say that the walk is aperiodic if there exists an odd integer n such that p ne,e0. The motivating examples of relatively hyperbolic groups are the fundamental groups of complete noncompact hyperbolic manifolds of finite volume. Fuchsian groups and closed hyperbolic surfaces 6 5. Discrete groups of isometriesof n dimensional hyperbolic space hn. Swarup suppose g is a hyperbolic group which is a graph of hyperbolic groups such that all edge to vertex inclusions are quasiisometric embeddings. 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. Notes on relatively hyperbolic groups and relatively quasiconvex subgroups matsuda, yoshifumi, oguni, shinichi, and yamagata, saeko, tokyo journal of mathematics, 2015. 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 satisfy a linear isoperimetric inequality. 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. It is not known whether there exists an hyperbolic group which is not residually finite. In mathematics, the concept of a relatively hyperbolic group is an important generalization of the geometric group theory concept of a hyperbolic group. Moreover the preimage of any conical point is a singleton. 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. A finite group is a trivial example of a hyperbolic group. The definition, introduced by mikhael gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. For example, the free burnside group of exponent n with two generators is given by the presentation. A nilpotent group g is a group with a lower central series terminating in the identity subgroup. 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. Mapping class groups admit an amenable action on an explicit boundary kida, h 09 5.
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. This makes precise the notion of a group being negatively curved. Amenable actions, lyapunov exponents and an idea of bowen. 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. A hyperbolic group is said to be elementary if it is finite or contains an infinite cyclic subgroup of finite index.
The class of relatively hyperbolic groups includes many examples. First observe that for every finitely presented group. In mathematics, a hyperbolic metric space is a metric space satisfying certain metric relations depending quantitatively on a nonnegative real number. Hyperbolicity is a largescale property, and is very useful to the study of certain infinite groups. The generalised hyperbolic skew students tdistribution 5 tdistribution with degrees of freedom. In his seminal paper 34, gromov introduced the class of hyperbolic groups which revolutionized the study of geometric group theory. The second lecture gives the thintriangles definition of hyperbolic group. My first guess would be to take the symmetry group of a tiling, but will it cayley graph be really similar to the tiling. The notion of a hyperbolic group was introduced and developed by mikhail gromov. Formulas and identities of hyperbolic functions pacharapokin chanapat shinshu university nagano, japan hiroshi yamazaki shinshu university nagano, japan summary.
Random walks on hyperbolic groups i university of chicago. 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. Kleinian groups and thurstons work 7 references 8 1. We show that every finite subgroup of g is conjugate to a subgroup each element of which has length at most 2.
But general results about the nonexistence of algorithms or their complexity play another, completely different, role in group theory. A metric space x is proper if closed balls of finite radius in x are compact. Normal automorphisms of relatively hyperbolic groups 3 in fact, theorem 1. Hyperbolic groups are sometimes called word hyperbolic groups or negatively curved groups. We characterize lacunary hyperbolicgroups as direct limits of gromov hyperbolic groups satisfying certain restrictions on the hyperbolicity constants and injectivity radii. In chapter 1, we give an introduction to this subject. In the geometrically in nite case, the surface is virtually the berof a bering. Hyperbolic groups may be defined as finitely generated groups admitting a linear dehn function. Later, it was reformulated and studied from di erent points of view by farb far98, bowditch bow12, osin osi06c and many others.
A group is hyperbolic if it acts properly discontinuously and cocompactly by isometries on a proper hyperbolic metric space. Let us start with three equivalent definitions of hyperbolic groups. In practice, i ran out of time and settled for discussing only the rips complex. A second alternative is the skew students tdistribution based on order statistics, recently introduced by jones and faddy 2003. The study of random walks on in nite groups, and speci cally the question of whether they asymptotically approach some. 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. The space of metrics on gromov hyperbolic groups alex furman university of illinois at chicago northwestern university, 20101031 1. 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. The geometry of hyperbolic and relatively hyperbolic groups has been integral to the solutions of many algorithmic problems 41, 43, 10, 12, date. Hyperbolic groups lecture notes james howie heriotwatt university, edinburgh eh14 4as scotland j.
Hyperbolic groups are a class of groups introduced by gromov in 1987, which form an important part of geometric group theory. Consider ginvariant proper, quasiconvex metric d on g. Nonelementary infinite and not virtually cyclic hyperbolic groups have always exponential growth rate this is a consequence of the tits alternative. A geometrically nite surface preserves a convex subset of hyperbolic space whose quotient by the group has nite nonzero volume.
The lamplighter random walk at each site x 2zd is a lamp, which can be eitheronoroff. Circles and spheres in hyperbolic space correspond to circles and spheres in the model angles between curves or surfaces correspond to same angles in the model distance, straightness, convexity, circle centers, etc. 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. In principle, one could take a similar approach to studying.
For this reason many important themes, such as cat. A word in sis a nite concatenation of elements of s. The generalised hyperbolic skew students tdistribution. The rst case includes quasifuchsian surfaces figure5. We show that the gromov boundary of the free product of two in.
However, i have included a section on the hyperbolic boundary. The core part of the course is devoted to gromovhyperbolic spaces and groups. We moreover give a description of all gromov hyperbolic locally compact groups with a cocompact amenable subgroup. 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 way to use geometry to study groups is considering their isometric actions on metric spaces. Many researchers i count myself among this number are less interested in particular groups than in classes of groups.
For instance, any hyperbolic group is hyperbolic relative to the trivial subgroup. Horofunctions and symbolic dynamics on gromov hyperbolic. For i 1,2, let gi be graphs of groups with all vertex groups hyperbolic and all edge groups. We call a finitely generated group lacunary hyperbolic if one of its asymptotic cones is an rtree. Mar 21, 2006 on hyperbolic groups on hyperbolic groups houcine, abderezak ould 20060321 00. Hyperbolic groups with homeomorphic gromov boundaries. 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. Hyperbolic groups lecture notes mathematical and computer. Let be a hyperbolic group relative to a collection of in. 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. Growth gap in hyperbolic groups and amenability arxiv. Other examples of hyperbolic groups z is hyperbolic. Virtual properties of 3manifolds 7 geometrically nite or geometrically in nite. Oct 28, 20 among smooth dynamical systems, hyperbolic dynamics is characterized by the presence of expanding and contracting directions for the derivative.
Fundamental groupsof compact riemannian manifolds with strictly negative sectional curvature are hyperbolic. This is a situation where the differential alone provides strong local, semilocal or even global information about the dynamics. Geometric mapping theory of the heisenberg group, subriemannian manifolds, and hyperbolic spaces by anton lukyanenko dissertation submitted in partial ful llment of the requirements for the degree of doctor of philosophy in mathematics in the graduate college of the university of illinois at urbanachampaign, 2014 urbana, illinois doctoral. Horofunctions and symbolic dynamics on gromov hyperbolic groups. Forms group of geometric transformations contains all circlepreserving transformations in higher. Essay about failing exams writing what is a personal essay listening social media privacy essays essay water shortage valuable resource. Random walks on hyperbolic groups ii university of chicago. Then the ideal boundaries of all groups in this class are homeomorphic to the menger curve.
Finite subgroups of hyperbolic groups springerlink. Hyperbolic groups from dehn functions mathematics stack. 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. Hyperbolic functions show up in many reallife situations. Free group and fundamental groups of compact riemannian manifolds of negative sectional curvature cf. It is not known 1996 whether every hyperbolic group. Hyperbolic manifolds, discrete groups and ergodic theory. The action of a group relatively hyperbolic to a nite collection of amenable subgroups on a geometric boundary is amenable ozawa 06 4.
Horofunctions and symbolic dynamics on gromov hyperbolic groups volume 43 issue 3 michel coornaert, athanase papadopoulos. First observe that for every finitely presented group r there exists a smooth bounded i. Essay writing good examples very essay about movie our country nepal to essay meaning jamie waste problems essay zero. On residualing homomorphisms and gsubgroups of hyperbolic. Hyperbolic geometry, surfaces, and 3manifolds bruno martelli dipartimento di matematica \tonelli, largo pontecorvo 5, 56127 pisa, italy email address. One of his axioms called the parallel postulate has been the.
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