Action on a CAT(0) space

Definition

An action on a CAT(0) space is a way for a group to interact with a geometric space that satisfies the CAT(0) curvature condition, meaning that geodesic triangles in the space are 'thinner' than their Euclidean counterparts. This type of action reveals both geometric and algebraic properties of the group, as groups acting on CAT(0) spaces often have interesting features such as the existence of a proper action, or properties related to their growth and word metrics.

5 Must Know Facts For Your Next Test

1. In a CAT(0) space, any two points can be connected by a unique geodesic segment, which is crucial for analyzing the properties of the space.
2. A proper action on a CAT(0) space implies that the group does not have too many elements moving points in compact subsets, leading to important compactness results.
3. Actions on CAT(0) spaces can be analyzed using techniques from both geometry and topology, highlighting how groups can exhibit 'flexible' behaviors.
4. If a group acts cocompactly on a CAT(0) space, it implies that the quotient space is compact, which leads to significant implications for the group's algebraic properties.
5. The fixed point property for actions on CAT(0) spaces plays an essential role; if a group acts on such a space without fixed points, it often indicates strong conditions about the group's structure.

Review Questions

• How does an action on a CAT(0) space help us understand the structure of groups?
  • An action on a CAT(0) space allows us to explore how groups can exhibit various geometric and algebraic properties. For example, when a group acts properly discontinuously on such a space, it helps reveal the group's growth type and potential rigidity. Additionally, studying how groups interact with the curvature of the space can yield insights into their overall behavior and classifications.
• Discuss the significance of proper actions on CAT(0) spaces and their implications for compactness.
  • Proper actions on CAT(0) spaces are significant because they ensure that the quotient space retains important compactness properties. This means that if a group acts properly discontinuously on a CAT(0) space, then its orbit space is compact. This result can lead to deeper understanding of group representations and potential applications in geometric topology and algebraic geometry.
• Evaluate how the fixed point property in actions on CAT(0) spaces influences our understanding of infinite groups.
  • The fixed point property is crucial when analyzing infinite groups acting on CAT(0) spaces. If an infinite group acts on such a space without fixed points, it often indicates that the group's structure may have restrictive conditions or unusual behaviors. Understanding this relationship can lead to insights about group dynamics and can help classify groups based on their actions, revealing whether they are virtually free or exhibit other intriguing traits.

Related terms

Geodesic: The shortest path between two points in a given space, particularly in CAT(0) spaces where geodesics have specific properties related to curvature.

CAT(0) space:

A type of metric space that satisfies certain curvature conditions, making it non-positively curved; this includes spaces like Euclidean spaces and certain kinds of trees.

Group Action:

A formal way of describing how a group interacts with a set, allowing the elements of the group to be represented as transformations of the set.