⭕Geometric Group Theory Unit 6 – Hyperbolic Groups & Gromov Spaces

Key Concepts and Definitions

• Hyperbolic space: a non-Euclidean geometry with constant negative curvature, where parallel lines diverge and the sum of angles in a triangle is less than 180 degrees
  • Includes the Poincaré disk and upper half-plane models
• Hyperbolic groups: finitely generated groups with word metrics that exhibit hyperbolic geometry properties
  • Satisfy a linear isoperimetric inequality and have a thin triangle condition
• Gromov hyperbolic spaces: metric spaces that satisfy the Gromov four-point condition, generalizing the concept of hyperbolic groups
• Quasi-isometry: a map between metric spaces that preserves large-scale geometry up to bounded additive and multiplicative constants
• Cayley graph: a graph representation of a group, with vertices corresponding to group elements and edges representing generators
• Geodesic: a locally length-minimizing path in a metric space, generalizing the concept of straight lines in Euclidean space
• Boundary at infinity: a topological space associated with a hyperbolic space or group, capturing the asymptotic behavior of geodesics

Historical Context and Development

• Non-Euclidean geometries emerged in the 19th century, challenging the long-held belief in the universality of Euclidean geometry
  • Lobachevsky, Bolyai, and Gauss independently developed hyperbolic geometry
• Poincaré and Klein provided models of hyperbolic space in the late 19th century, enabling a deeper understanding of its properties
• Dehn's work on the word problem and isoperimetric inequalities in the early 20th century laid the foundation for the study of hyperbolic groups
• Gromov introduced the concept of hyperbolic groups in the 1980s, unifying various examples and establishing a framework for their study
  • Extended the notion of hyperbolicity to metric spaces, leading to the development of Gromov hyperbolic spaces
• Geometric group theory emerged as a distinct field, combining ideas from geometry, topology, and group theory to study finitely generated groups
• The work of Cannon, Epstein, Ghys, de la Harpe, and others further developed the theory of hyperbolic groups and their applications

Hyperbolic Space Fundamentals

• Hyperbolic space has constant negative curvature, distinguishing it from Euclidean and spherical geometries
• The Poincaré disk model represents hyperbolic space as the interior of a unit disk, with geodesics as circular arcs perpendicular to the boundary
  • Provides a conformal model, preserving angles but distorting distances
• The upper half-plane model represents hyperbolic space as the set $\{(x, y) \in \mathbb{R}^2 : y > 0\}$, with geodesics as vertical lines or semicircles orthogonal to the real axis
• Isometries of hyperbolic space include Möbius transformations, which are compositions of inversions and reflections
• Hyperbolic trigonometry differs from Euclidean trigonometry, with formulas involving hyperbolic functions (sinh, cosh, tanh)
• The area of a hyperbolic triangle depends only on its angles, given by the Gauss-Bonnet formula: $A = \pi - (\alpha + \beta + \gamma)$
• Hyperbolic space exhibits exponential growth of volume and circumference with respect to radius, in contrast to the polynomial growth in Euclidean space

Hyperbolic Groups: Properties and Examples

• Hyperbolic groups have a Cayley graph that is Gromov hyperbolic, meaning it satisfies the thin triangle condition
  • Every geodesic triangle is $\delta$-thin: each side is contained in the $\delta$-neighborhood of the other two sides
• Satisfy a linear isoperimetric inequality: the area of a loop is linearly bounded by its perimeter
• Have a boundary at infinity that is a compact metric space, capturing the asymptotic behavior of geodesics
• Possess a unique geodesic between any two points, making them geodesic metric spaces
• Examples include free groups, fundamental groups of compact hyperbolic manifolds, and small cancellation groups
  • The fundamental group of a closed surface of genus $g \geq 2$ is hyperbolic
• Word problem is solvable in linear time for hyperbolic groups, using the geodesic automatic structure
• Closed under finite free products, finite extensions, and quasi-isometries, forming a robust class of groups

Gromov Spaces: Introduction and Characteristics

• Gromov hyperbolic spaces generalize the concept of hyperbolic groups to metric spaces
• Defined by the Gromov four-point condition: for any four points $x, y, z, w$, the largest two of the three sums $d(x, y) + d(z, w)$, $d(x, z) + d(y, w)$, and $d(x, w) + d(y, z)$ differ by at most $2\delta$
  • Captures the idea that triangles are thin and the space has a tree-like structure on a large scale
• Geodesic Gromov hyperbolic spaces are uniquely geodesic, with geodesics staying close to each other
• Possess a boundary at infinity, which is a compact metric space that encodes the asymptotic geometry of the space
• Examples include hyperbolic groups, Cayley graphs of hyperbolic groups, and metric trees
  • The hyperbolic plane and higher-dimensional hyperbolic spaces are Gromov hyperbolic
• Gromov hyperbolicity is preserved under quasi-isometries, allowing for the study of coarse geometry
• The Gromov product provides a way to measure the relative distance between points and the boundary at infinity

Geometric and Algebraic Connections

• Hyperbolic groups and spaces bridge the gap between geometry and algebra, allowing for the application of geometric techniques to group-theoretic problems
• The Cayley graph of a hyperbolic group is a geometric realization of the group, encoding its algebraic structure
  • Geodesics in the Cayley graph correspond to minimal representations of group elements
• The boundary at infinity of a hyperbolic group has a natural group action, relating the geometry of the boundary to the algebraic properties of the group
• Quasi-isometries between hyperbolic groups induce homeomorphisms between their boundaries at infinity
• The cohomology of a hyperbolic group can be studied using the geometry of its boundary at infinity
• The growth function of a hyperbolic group is exponential, reflecting the exponential growth of volume in hyperbolic space
• Automatic structures and combings of hyperbolic groups provide a way to efficiently represent and manipulate group elements using geodesics

Applications in Geometric Group Theory

• Hyperbolic groups and spaces have found numerous applications in geometric group theory, providing a rich source of examples and techniques
• The study of hyperbolic groups has led to significant advances in the understanding of finitely generated groups and their geometry
  • Has provided a framework for studying groups acting on non-positively curved spaces (CAT(0) spaces)
• Hyperbolic groups have been used to construct examples of groups with specific properties, such as groups with unsolvable conjugacy problem or groups with exotic boundaries
• The geometry of hyperbolic groups has been applied to the study of 3-manifolds and their fundamental groups
  • Thurston's geometrization conjecture, proved by Perelman, relies heavily on the theory of hyperbolic 3-manifolds
• Hyperbolic groups have connections to other areas of mathematics, such as dynamical systems, ergodic theory, and low-dimensional topology
• The study of relatively hyperbolic groups, which generalize hyperbolic groups by allowing for certain subgroups to be "parabolic," has gained attention in recent years

Advanced Topics and Current Research

• The study of hyperbolic groups and spaces continues to be an active area of research, with many open questions and ongoing developments
• The Baum-Connes conjecture, which relates the K-theory of a group's C*-algebra to its equivariant K-homology, has been studied in the context of hyperbolic groups
  • Hyperbolic groups satisfy the Baum-Connes conjecture, providing a link between their geometry and operator algebras
• The Farrell-Jones conjecture, concerning the K- and L-theory of group rings, has been investigated for hyperbolic groups
• The study of relatively hyperbolic groups has led to the development of new techniques and results, such as the Bowditch boundary and the Osin-Groves-Manning construction
• The relationship between hyperbolic groups and CAT(0) groups has been a topic of interest, with results on the boundaries and rigidity properties of these groups
• The use of probabilistic methods, such as random walks and boundary theory, has provided new insights into the geometry and asymptotic properties of hyperbolic groups
• The connection between hyperbolic groups and dynamics, particularly in the context of group actions on the boundary at infinity, has been a fruitful area of research
• The study of hyperbolic groups in higher dimensions, such as hyperbolic Coxeter groups and hyperbolic buildings, has gained attention in recent years