Gromov introduced geometric group theory, the study of infinite groups via the geometry of their Cayley graphs and their word metric. In 1981 he proved Gromov's theorem on groups of polynomial growth: a finitely generated group has polynomial growth (a geometric property) if and only if it is virtually nilpotent (an algebraic property).
Hyperbolic groups are sometimes called word hyperbolic groups or negatively curved groups. A finite group is a trivial example of a hyperbolic group. Free groups of finite rank (cf. Free group) and fundamental groups of compact Riemannian manifolds of negative sectional curvature (cf. Fundamental group; Riemannian manifold) are hyperbolic.The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite groups called (Gromov-) hyperbolic groups.In group theory, a hyperbolic group, also known as a word hyperbolic group, Gromov hyperbolic group, negatively curved group is a finitely generated group equipped with a word metric satisfying certain properties characteristic of hyperbolic geometry.
Hyperbolic groups in the sense of Gromov. Different definitions and equivalences.The triangle conditions, thin triangles, comparison triangles. Generalization of Dehn’s algorithm to any hyperbolic group. The fundamental characterization theorem: a finitely presented group is hyperbolic if and only if its Dehn function is linear and if and only if it admits a Dehn-like algorithm for the.
This is a hyperbolic group whose Gromov boundary is a Cantor set. Hyperbolic groups and their boundaries are important topics in geometric group theory, as are Cayley graphs. The (6,4,2) triangular hyperbolic tiling. The triangle group corresponding to this tiling has a circle as its Gromov boundary.
The primary focus of this book is to cover the foundations of geometric group theory, including coarse topology, ultralimits and asymptotic cones, hyperbolic groups, isoperimetric inequalities, growth of groups, amenability, Kazhdan's Property (T) and the Haagerup property, as well as their characterizations in terms of group actions on median spaces and spaces with walls.
M. Gromov, Hyperbolic groups, in “Essays in Group Theory”, Ed.: S.M. Gersten, M.S.R.I. Pub. 8, Springer-Verlag (1987).
Abstract 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 equivalence of the.
The proof of Gromov's theorem on groups of polynomial growth is given in full, with the theory of asymptotic cones developed on the way. Grigorchuk's first and general groups are described, as well as the proof that they have intermediate growth, with explicit bounds, and their relationship to automorphisms of regular trees and finite automata.
Hyperbolic groups are fundamental examples of Gromov hyperbolic spaces in geometric group theory.. I'm aware about the fact stated above, but I'm not able to find some references or proofs besides Gromov's Hyperbolic Groups - Essays in Group Theory. I'll state things precisely. I will consider a. reference-request metric-spaces hyperbolic-geometry gromov-hyperbolic-spaces. modified Aug.
It's known that subgroups of Gromov's hyperbolic groups are not necessarily hyperbolic. Is there any counter-example when the quotient is Abelian.
Note that Sapir's group is residually finite (ascending HNN-extensions of free groups are always residually finite). Another group to consider is a group of M. Kapovich. He constructed a hyperbolic group which is not linear. The paper, allegedly, is this one. However, after a quick flick through, I am having trouble finding this specific result.
Examples studied in detail include hyperbolic groups, Euclidean groups, braid groups, Coxeter groups, Artin groups, and automata groups such as the Grigorchuk group. This book will be a convenient reference point for established mathematicians who need to understand background material for applications, and can serve as a textbook for research students in (geometric) group theory.
A geodesic metric space is said to be (Gromov) hyperbolic if every geodesic triangle is uniformly “thin” (see Definition 2). The theory of hyperbolic spaces was introduced by M. Gromov in ( 1, 2 ), and after that studied by many scientists, see, e.g., in ( 3, 4, 5, 6 ).
Geometric ideas have played a very major role in the development of Group theory in 20th century. Coxeter groups and Coxeter complexes, Tits theory of buildings, Bass-Serre theory of Groups acting on trees, Gromov’s notion of hyperbolic groups and combinatorial methods have all been very central in several areas of mathe- matics.
Gromov Hyperbolic Groups Essays In Group Theory and perform them promptly to provide you with the required assistance on time. We even have an urgent delivery option for short essays, term papers, or research papers needed within 8 to 24 hours.
We generalize the small cancellation theory over ordinary hyperbolic groups to relatively hyperbolic settings. This generalization is then used to prove various embedding theorems for countable groups. For instance, we show that any countable torsion free group can be embedded into a finitely generated group with exactly two conjugacy classes. In particular, this gives the affirmative answer.