Boundary at infinity

Definition

The boundary at infinity refers to a concept in geometric group theory that captures the asymptotic behavior of spaces, particularly in the context of hyperbolic spaces and CAT(0) spaces. It provides a way to understand how groups act on these spaces by extending their structure to include 'points at infinity', enabling the study of their geometric properties and their relationships with various group actions.

5 Must Know Facts For Your Next Test

1. The boundary at infinity helps classify spaces by identifying points that represent directions in which the space can be infinitely extended.
2. In hyperbolic geometry, the boundary at infinity can be visualized as a circle or sphere where points correspond to equivalence classes of geodesics that escape to infinity.
3. The Gromov boundary is homeomorphic to a compact space, allowing for meaningful topological analysis of groups acting on hyperbolic spaces.
4. Groups acting on CAT(0) spaces can have rich geometrical structures, and their boundaries at infinity provide insight into their properties and behaviors.
5. Understanding the boundary at infinity is crucial for studying the asymptotic properties of groups and their actions, particularly in relation to rigidity and divergence.

Review Questions

• How does the concept of the boundary at infinity enhance our understanding of hyperbolic groups?
  • The boundary at infinity provides a way to analyze hyperbolic groups by allowing us to study their asymptotic behavior and identify how they act on their boundaries. This connection reveals important properties of the group, such as its growth and the types of actions it can perform. By examining these boundaries, we can gain insights into phenomena like convergence groups and the dynamics of group actions.
• Discuss the relationship between CAT(0) spaces and their boundaries at infinity, including examples of how this concept applies in practice.
  • CAT(0) spaces have well-defined boundaries at infinity that help illustrate their geometric properties. For instance, in Euclidean spaces (which are CAT(0)), the boundary at infinity corresponds to a flat plane extending infinitely. Conversely, in more complex CAT(0) spaces like products of trees, the boundaries can exhibit intricate structures that reflect the underlying geometric characteristics. This relationship allows for greater exploration of rigidities and other properties essential in geometric group theory.
• Evaluate how quasi-isometries relate to the concept of boundaries at infinity and what implications this has for geometric classification.
  • Quasi-isometries play a crucial role in connecting different metric spaces by demonstrating that they can share similar boundaries at infinity. When two spaces are quasi-isometric, their boundaries reflect similar asymptotic behaviors, providing a powerful tool for classifying groups and their geometric structures. This relationship has significant implications for understanding phenomena such as hyperbolicity and CAT(0) properties, as it implies that if two groups have the same boundary at infinity, they may share deeper geometric features despite potentially differing in other aspects.

Related terms

Gromov boundary: The Gromov boundary is a specific type of boundary at infinity for hyperbolic spaces, capturing the behavior of geodesics escaping to infinity.

CAT(0) space:

A CAT(0) space is a geodesic metric space that satisfies certain curvature conditions, which influences the structure of its boundary at infinity.

Quasi-isometry: A quasi-isometry is a map between metric spaces that preserves distances up to a bounded distortion, often used to show that two spaces have the same boundary at infinity.