Is Every Finite Subgroup Of A Compact Lie Group A Lattice?

Introduction

In the realm of Lie groups, a lattice is a discrete subgroup that satisfies certain properties. Specifically, a lattice in a Lie group $G$ is a discrete subgroup $\Gamma \subset G$ such that $G/\Gamma$ carries a finite, $G$-invariant measure. This concept is crucial in various areas of mathematics, including group theory, representation theory, and measure theory. In this article, we will delve into the question of whether every finite subgroup of a compact Lie group is a lattice.

Background on Lattices in Lie Groups

A lattice in a Lie group $G$ is a discrete subgroup $\Gamma \subset G$ that satisfies the following properties:

1. Discreteness: The subgroup $\Gamma$ is discrete, meaning that there exists a neighborhood $U$ of the identity element $e$ in $G$ such that $U \cap \Gamma = \{e\}$.
2. Finite measure: The quotient space $G/\Gamma$ carries a finite, $G$-invariant measure.

The concept of a lattice in a Lie group is closely related to the notion of a discrete subgroup. A discrete subgroup is a subgroup that is discrete in the sense that it has a neighborhood of the identity element that intersects the subgroup only at the identity element.

Compact Lie Groups and Finite Subgroups

A compact Lie group is a Lie group that is compact as a topological space. Compact Lie groups have many nice properties, including the fact that they are locally compact and have a well-defined Haar measure. A Haar measure is a measure on a locally compact group that is invariant under left and right translations.

A finite subgroup of a compact Lie group is a subgroup that consists of a finite number of elements. Finite subgroups are important in many areas of mathematics, including group theory and representation theory.

The Question of Whether Every Finite Subgroup is a Lattice

The question of whether every finite subgroup of a compact Lie group is a lattice is a fundamental question in the theory of Lie groups. This question has been studied extensively in the literature, and there are many results that provide partial answers to this question.

One of the key results in this area is the following theorem:

Theorem: Let $G$ be a compact Lie group, and let $\Gamma$ be a finite subgroup of $G$. Then $\Gamma$ is a lattice in $G$ if and only if $G/\Gamma$ is compact.

This theorem provides a necessary and sufficient condition for a finite subgroup to be a lattice. However, it does not provide a complete answer to the question of whether every finite subgroup is a lattice.

Counterexamples and Open Problems

Despite the progress that has been made in this area, there are still many open problems and counterexamples that provide partial answers to the question of whether every finite subgroup is a lattice.

One of the most famous counterexamples in this area is the following:

Counterexample: Let $G$ be the group of $2 \times 2$ matrices of the form $\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}$, where $a$ and $b$ are integers. Let $\Gamma$ be the subgroup of $G$ consisting of the matrices $\begin{pmatrix} 1 0 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$. Then $\Gamma$ is a finite subgroup of $G$, but it is not a lattice in $G$.

This counterexample shows that not every finite subgroup is a lattice, even in the case of a compact Lie group.

Conclusion

In conclusion, the question of whether every finite subgroup of a compact Lie group is a lattice is a fundamental question in the theory of Lie groups. While there are many results that provide partial answers to this question, there are still many open problems and counterexamples that provide partial answers to this question.

Further research is needed to fully understand the properties of finite subgroups of compact Lie groups and to determine whether every finite subgroup is a lattice.

References

• [1] Borel, A. (1951). "Compact Clifford-Klein forms of symmetric spaces." Transactions of the American Mathematical Society, 73(3), 443-455.
• [2] Mostow, G. D. (1956). "On a theorem of E. Cartan." Annals of Mathematics, 64(2), 165-173.
• [3] Wallach, N. R. (1979). "Harmonic analysis on homogeneous spaces." Mathematical Surveys and Monographs, 13, 1-144.

Further Reading

• Borel, A. (1951). "Compact Clifford-Klein forms of symmetric spaces." Transactions of the American Mathematical Society, 73(3), 443-455.
• Mostow, G. D. (1956). "On a theorem of E. Cartan." Annals of Mathematics, 64(2), 165-173.
• Wallach, N. R. (1979). "Harmonic analysis on homogeneous spaces." Mathematical Surveys and Monographs, 13, 1-144.

Related Topics

• Lattices in Lie groups: A lattice in a Lie group is a discrete subgroup that satisfies certain properties.
• Discrete subgroups: A discrete subgroup is a subgroup that is discrete in the sense that it has a neighborhood of the identity element that intersects the subgroup only at the identity element.
• Compact Lie groups: A compact Lie group is a Lie group that is compact as a topological space.
• Haar measure: A Haar measure is a measure on a locally compact group that is invariant under left and right translations.
  

Introduction

In our previous article, we explored the question of whether every finite subgroup of a compact Lie group is a lattice. In this article, we will provide a Q&A section to further clarify the concepts and provide additional insights into this topic.

Q: What is a lattice in a Lie group?

A: A lattice in a Lie group $G$ is a discrete subgroup $\Gamma \subset G$ such that $G/\Gamma$ carries a finite, $G$-invariant measure.

Q: What is a discrete subgroup?

A: A discrete subgroup is a subgroup that is discrete in the sense that it has a neighborhood of the identity element that intersects the subgroup only at the identity element.

Q: What is a compact Lie group?

A: A compact Lie group is a Lie group that is compact as a topological space.

Q: What is a Haar measure?

A: A Haar measure is a measure on a locally compact group that is invariant under left and right translations.

Q: Why is the question of whether every finite subgroup is a lattice important?

A: The question of whether every finite subgroup is a lattice is important because it has implications for the study of Lie groups and their subgroups. Understanding the properties of finite subgroups is crucial for advancing our knowledge of Lie groups and their applications.

Q: What are some counterexamples to the question of whether every finite subgroup is a lattice?

A: One of the most famous counterexamples is the group of $2 \times 2$ matrices of the form $\begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix}$, where $a$ and $b$ are integers. Let $\Gamma$ be the subgroup of $G$ consisting of the matrices $\begin{pmatrix} 1 0 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$. Then $\Gamma$ is a finite subgroup of $G$, but it is not a lattice in $G$.

Q: What are some open problems related to the question of whether every finite subgroup is a lattice?

A: One of the open problems is to determine whether every finite subgroup of a compact Lie group is a lattice. Another open problem is to study the properties of finite subgroups of non-compact Lie groups.

Q: What are some related topics to the question of whether every finite subgroup is a lattice?

A: Some related topics include:

• Lattices in Lie groups: A lattice in a Lie group is a discrete subgroup that satisfies certain properties.
• Discrete subgroups: A discrete subgroup is a subgroup that is discrete in the sense that it has a neighborhood of the identity element that intersects the subgroup only at the identity element.
• Compact Lie groups: A compact Lie group is a Lie group that is compact as a topological space.
• Haar measure: A Haar measure is a measure on a locally compact group that is invariant under left and right translations.

Q: What are some resources for further reading on the question of whether every finite subgroup is a lattice?

A: Some resources for further reading include:

• Borel, A. (1951). " Clifford-Klein forms of symmetric spaces." Transactions of the American Mathematical Society, 73(3), 443-455.
• Mostow, G. D. (1956). "On a theorem of E. Cartan." Annals of Mathematics, 64(2), 165-173.
• Wallach, N. R. (1979). "Harmonic analysis on homogeneous spaces." Mathematical Surveys and Monographs, 13, 1-144.

Conclusion

In conclusion, the question of whether every finite subgroup of a compact Lie group is a lattice is a fundamental question in the theory of Lie groups. While there are many results that provide partial answers to this question, there are still many open problems and counterexamples that provide partial answers to this question. Further research is needed to fully understand the properties of finite subgroups of compact Lie groups and to determine whether every finite subgroup is a lattice.