⭕Geometric Group Theory Unit 10 – Gromov Boundary and Compactifications

Key Concepts and Definitions

• Gromov boundary, also known as the visual boundary or the boundary at infinity, represents the "points at infinity" of a metric space
• Hyperbolic groups are finitely generated groups that exhibit properties similar to hyperbolic geometry, and their Gromov boundaries play a crucial role in understanding their structure
• Quasi-isometries are maps between metric spaces that preserve large-scale geometry, and they induce homeomorphisms between Gromov boundaries
• Geodesic rays are infinite paths in a metric space that are locally distance-minimizing, and they converge to points on the Gromov boundary
• Compactification is the process of adding a boundary to a space to make it compact, and the Gromov compactification is obtained by adding the Gromov boundary to the original space
• Topology on the Gromov boundary is defined using the notion of convergence of sequences of points or geodesic rays
• Gromov product measures the distance between two points in a hyperbolic space relative to a base point, and it is used to define the topology on the Gromov boundary

Historical Context and Development

• The concept of the Gromov boundary was introduced by Mikhail Gromov in the 1980s as part of his seminal work on hyperbolic groups and spaces
• Gromov's ideas built upon earlier work on the geometry of negatively curved spaces, such as the Poincaré disk model of hyperbolic geometry
• The development of the Gromov boundary was motivated by the need to understand the large-scale geometry of hyperbolic groups and their asymptotic behavior
• The notion of quasi-isometries, which was also introduced by Gromov, played a crucial role in the study of Gromov boundaries and their invariance properties
• The Gromov boundary has since become a fundamental tool in geometric group theory, with applications to the study of the structure and properties of various classes of groups
• The concept has been generalized to other settings, such as CAT(0) spaces and relatively hyperbolic groups, leading to further developments in the field

Geometric Intuition and Visualization

• The Gromov boundary can be thought of as the "points at infinity" of a metric space, representing the directions in which geodesic rays can escape to infinity
• In the hyperbolic plane (Poincaré disk model), the Gromov boundary corresponds to the circle at infinity, with each point on the circle representing a unique geodesic ray
• Geodesic rays in a hyperbolic space can be visualized as paths that "aim" towards a specific point on the boundary at infinity
• The topology on the Gromov boundary can be understood in terms of the convergence of geodesic rays: two points on the boundary are close if there are geodesic rays connecting them that stay close for a long time
• The Gromov product can be visualized as the distance between two points "seen" from a base point, with larger values indicating that the points are closer together relative to the base point
• In the context of hyperbolic groups, the Gromov boundary encodes information about the "directions" in which the group can grow exponentially, providing insight into its asymptotic structure

Construction and Properties

• The Gromov boundary of a hyperbolic space $X$ is constructed by considering equivalence classes of geodesic rays in $X$, where two rays are equivalent if they stay within a bounded distance of each other
• The Gromov product $(\cdot,\cdot)_o$ with respect to a base point $o$ is defined as: $(x,y)_o = \frac{1}{2}(d(o,x) + d(o,y) - d(x,y))$, where $d$ is the metric on $X$
• The topology on the Gromov boundary is defined using the Gromov product: a sequence of points $(x_n)$ in $X$ converges to a point $\xi$ on the boundary if $\lim_{n,m \to \infty} (x_n,x_m)_o = \infty$ and $\lim_{n \to \infty} (x_n,\gamma(t))_o = \infty$ for any geodesic ray $\gamma$ converging to $\xi$
• The Gromov boundary of a hyperbolic group is compact, metrizable, and has a natural action of the group by homeomorphisms
• Quasi-isometries between hyperbolic spaces induce homeomorphisms between their Gromov boundaries, making the boundary an invariant of the large-scale geometry
  • This property allows the use of the Gromov boundary to distinguish between different hyperbolic groups or spaces
• The Gromov boundary of a product of hyperbolic spaces is the join of their respective boundaries, providing a way to understand the boundaries of more complex spaces

Applications in Geometric Group Theory

• The Gromov boundary is a powerful tool for studying the asymptotic geometry and structure of hyperbolic groups
• The boundary can be used to classify hyperbolic groups up to quasi-isometry: two hyperbolic groups are quasi-isometric if and only if their Gromov boundaries are homeomorphic
• The action of a hyperbolic group on its Gromov boundary provides information about the group's dynamical and ergodic properties
  • For example, the Patterson-Sullivan measure on the boundary is a useful tool for understanding the growth and orbit structure of the group
• The Gromov boundary can be used to construct and study important objects associated with hyperbolic groups, such as the Martin boundary and the Poisson boundary
• The boundary plays a role in the study of algebraic properties of hyperbolic groups, such as their subgroup structure and splittings
• The Gromov boundary has applications to the study of 3-manifolds and their fundamental groups, as many 3-manifold groups are hyperbolic or relatively hyperbolic
• The boundary is also used in the study of random walks on hyperbolic groups and their asymptotic behavior

Connections to Other Mathematical Areas

• The Gromov boundary is closely related to the Martin boundary in potential theory, which describes the asymptotic behavior of harmonic functions on a space
• The Patterson-Sullivan measure on the Gromov boundary is analogous to the harmonic measure in complex analysis and potential theory
• The Poisson boundary of a random walk on a group, which encodes the asymptotic behavior of the walk, can be realized as a subset of the Gromov boundary in the case of hyperbolic groups
• The study of Gromov boundaries has connections to the theory of CAT(0) spaces and their boundaries, as hyperbolic spaces are a special case of CAT(0) spaces
• The Gromov boundary has been used to study the geometry and dynamics of certain classes of non-positively curved manifolds, such as rank-one symmetric spaces
• The boundary has also found applications in the study of relatively hyperbolic groups and their asymptotic geometry, generalizing the theory of hyperbolic groups
• The concepts related to the Gromov boundary, such as quasi-isometries and the Gromov product, have been used in the study of other geometric structures, such as Teichmüller spaces and mapping class groups

Examples and Counterexamples

• The hyperbolic plane (Poincaré disk model) is a classic example of a hyperbolic space, and its Gromov boundary is the circle at infinity
• Free groups are hyperbolic, and their Gromov boundary is a Cantor set, reflecting the tree-like structure of their Cayley graphs
• The fundamental group of a closed surface of genus at least 2 is hyperbolic, and its Gromov boundary is homeomorphic to the circle
• Baumslag-Solitar groups $BS(m,n)$ with $|m| \neq |n|$ are not hyperbolic, and they do not have a well-defined Gromov boundary in the classical sense
• The fundamental group of a compact 3-manifold with a hyperbolic JSJ component has a Gromov boundary that is a Sierpiński carpet, reflecting the presence of quasi-isometrically embedded copies of the hyperbolic plane
• Coxeter groups acting on hyperbolic spaces have Gromov boundaries that can be described using the combinatorics of their defining graphs
• The mapping class group of a surface is not hyperbolic but is relatively hyperbolic with respect to its curve stabilizers, and its Gromov boundary is the space of ending laminations

Advanced Topics and Open Problems

• The Cannon-Thurston map, which extends the inclusion of a hyperbolic subgroup to a continuous map between their Gromov boundaries, has been the subject of extensive research and has led to important results in geometric group theory
• The Kapovich-Kleiner conjecture, which states that the Gromov boundary of a hyperbolic group is either a sphere, a Sierpiński carpet, or a Menger curve, remains an open problem
• The Ahlfors regular conformal dimension of the Gromov boundary of a hyperbolic group is a numerical invariant that captures the conformal structure of the boundary and is related to the group's analytic and geometric properties
• The Poisson-Furstenberg boundary of a random walk on a hyperbolic group can be identified with a subset of the Gromov boundary, but the exact relationship between these boundaries is not fully understood in all cases
• The study of relatively hyperbolic groups and their Gromov boundaries has led to the development of new techniques and results in geometric group theory, such as the Bowditch boundary and the Osin-Groves-Manning construction
• The relationship between the Gromov boundary and other boundaries, such as the Floyd boundary and the Bowditch boundary, is an active area of research
• The use of the Gromov boundary in the study of 3-manifolds and their geometric structures, such as in the Ending Lamination Conjecture and the Geometrization Conjecture, continues to be a fruitful area of investigation