Baumslag-Solitar Groups

Definition

Baumslag-Solitar groups are a family of groups defined by the presentation $$BS(m,n) = \langle a, b \mid a^m b^n a^{-1} b^{-1} \rangle$$ for integers m and n. These groups exhibit unique properties and behaviors that make them significant in the study of group theory, particularly in exploring connections to algorithmic problems such as the word problem and the conjugacy problem.

5 Must Know Facts For Your Next Test

1. Baumslag-Solitar groups can exhibit very different behaviors based on the values of m and n, leading to a wide variety of structural properties.
2. For certain pairs of m and n, Baumslag-Solitar groups can be non-Hopfian, meaning they can be embedded in themselves in a way that is not isomorphic.
3. These groups can also serve as examples of finitely presented groups that do not have solvable word or conjugacy problems, illustrating their complexity.
4. The structure of Baumslag-Solitar groups is heavily influenced by the integers m and n; for example, if both m and n are equal to 1, then BS(1,1) is isomorphic to the free group on two generators.
5. Baumslag-Solitar groups have been used to construct examples of groups with specific properties in geometric group theory, such as having infinite cyclic subgroups.

Review Questions

• How do the parameters m and n in Baumslag-Solitar groups affect their algebraic structure and properties?
  • The parameters m and n greatly influence the behavior of Baumslag-Solitar groups. For instance, if both parameters are equal to 1, BS(1,1) becomes isomorphic to a free group, which has very different properties compared to cases like BS(2,3), which may have non-Hopfian behavior. This variation demonstrates how different pairs can lead to vastly different group characteristics and highlight the complexity within this family of groups.
• Discuss the significance of Baumslag-Solitar groups in relation to algorithmic problems such as the word problem and conjugacy problem.
  • Baumslag-Solitar groups provide critical examples when examining algorithmic problems like the word problem and conjugacy problem. Certain Baumslag-Solitar groups have been shown to have non-solvable word problems, making them important cases for understanding limitations in computational group theory. The ability or inability to solve these problems within specific Baumslag-Solitar groups allows researchers to better understand the boundaries of algorithmic decidability in group theory.
• Evaluate how Baumslag-Solitar groups illustrate the concept of finitely presented groups while also providing counterexamples for various properties within geometric group theory.
  • Baumslag-Solitar groups serve as significant examples of finitely presented groups due to their concise group presentations using generators and relations. They also act as counterexamples for various properties in geometric group theory, such as exhibiting non-Hopfian characteristics or having unsolvable word problems. By analyzing these groups, one can explore deeper implications about group behavior, ultimately advancing understanding of both theoretical and computational aspects of group theory.

Related terms

Word Problem: The word problem for a group asks whether two words (or expressions) represent the same element in the group, which can be a complex computational task.

Conjugacy Problem: The conjugacy problem involves determining whether two elements in a group are conjugate to each other, meaning there exists an element in the group that transforms one into the other through conjugation.

Free Group: A free group is a type of group where there are no relations between the generators other than those necessary to satisfy the group axioms, making it a fundamental building block in group theory.