Discrete subgroup

Definition

A discrete subgroup is a subgroup of a topological group that is discrete as a subset of that group's topology. This means that for each element in the subgroup, there exists a neighborhood around it such that no other points from the subgroup are within that neighborhood. In the context of covering groups and the fundamental group, discrete subgroups play an important role in understanding how groups act on spaces and how these actions relate to the structure of topological spaces.

5 Must Know Facts For Your Next Test

1. Discrete subgroups are often seen in the context of Lie groups, where they correspond to 'points' in a larger topological structure.
2. The fundamental group can be viewed as a discrete subgroup when considering paths in spaces that are locally homeomorphic to Euclidean space.
3. Every finite subgroup of a topological group is a discrete subgroup because finite sets have the discrete topology.
4. In the context of covering spaces, a discrete subgroup can determine the structure of the covering space by defining how fibers over points relate to one another.
5. If a discrete subgroup acts on a topological space, it can lead to interesting quotient spaces which may reflect non-trivial topological properties.

Review Questions

• How does the concept of a discrete subgroup enhance our understanding of group actions on topological spaces?
  • Understanding discrete subgroups allows us to see how groups can act on topological spaces without affecting local topology. Since a discrete subgroup contains isolated points, this means that its action can produce distinct orbits that maintain separation from one another. This separation helps in analyzing how transformations affect the structure and properties of the underlying space.
• Discuss the implications of having a finite discrete subgroup within a topological group in terms of covering spaces and their properties.
  • Having a finite discrete subgroup within a topological group implies that every element is isolated, leading to distinct cosets in any related covering space. This isolation ensures that the covering map behaves nicely, as the fibers over points remain well-defined and separated. The structure provided by these finite subgroups can lead to straightforward calculations regarding homotopy types and help define simple covering spaces.
• Evaluate how discrete subgroups influence the fundamental group when analyzing paths in different topological contexts.
  • Discrete subgroups significantly influence the fundamental group by determining how paths can be deformed or connected within the space. If we consider paths that represent elements of a discrete subgroup, these paths can only loop around isolated points without interference from others. This characteristic leads to clearer relationships between homotopy classes and provides insight into potential coverings based on these discrete structures, thus enriching our overall understanding of both algebraic topology and geometric group theory.

Related terms

Topological group: A group that is also a topological space where the group operations (multiplication and inversion) are continuous.

Fundamental group: The group of loop classes based at a point in a space, which captures the space's topological structure in terms of its paths.

Covering space: A topological space that 'covers' another space such that there is a continuous surjective map with certain lifting properties, allowing for local homeomorphism.