Which Elements In $\operatorname{Aut}(\widehat{F_2})$ Preserve The Conjugacy Class Of The Commutator $c=[a,b]$?

Introduction

In the realm of group theory, the study of automorphisms and their effects on group structures is a fundamental area of research. The automorphism group of a free group, denoted as $\operatorname{Aut}(F_n)$, has been extensively studied, particularly for the free group over two generators, $F_2$. However, when we consider the profinite completion of $F_2$, denoted as $\widehat{F_2}$, the landscape becomes more complex. In this article, we will delve into the elements of $\operatorname{Aut}(\widehat{F_2})$ that preserve the conjugacy class of the commutator $c = [a, b]$.

Background and Notations

Let $F_2$ denote the free group over two generators $a$ and $b$. The commutator $c = [a, b]$ is defined as $c = aba^{-1}b^{-1}$. The profinite completion of $F_2$, denoted as $\widehat{F_2}$, is the inverse limit of the finite quotients of $F_2$. An automorphism $\psi$ of $F_2$ is a bijective homomorphism from $F_2$ to itself. The automorphism group of $F_2$, denoted as $\operatorname{Aut}(F_2)$, consists of all such automorphisms.

The Conjugacy Class of the Commutator

The conjugacy class of an element $g$ in a group $G$ is the set of all elements in $G$ that are conjugate to $g$. In other words, $h$ is conjugate to $g$ if there exists an element $x$ in $G$ such that $h = xgx^{-1}$. The conjugacy class of the commutator $c = [a, b]$ in $F_2$ is the set of all elements in $F_2$ that are conjugate to $c$.

Automorphisms Preserving the Conjugacy Class of the Commutator

We are interested in finding the elements of $\operatorname{Aut}(\widehat{F_2})$ that preserve the conjugacy class of the commutator $c = [a, b]$. In other words, we want to find the automorphisms $\psi$ of $\widehat{F_2}$ such that $\psi(c)$ is conjugate to $c$ in $\widehat{F_2}$.

The Role of the Nielsen-Schreier Theorem

The Nielsen-Schreier theorem states that if $G$ is a free group of rank $n$, then every subgroup of $G$ is also free. In particular, if $H$ is a subgroup of $F_2$, then $H$ is free. This theorem plays a crucial role in understanding the structure of subgroups of $F_2$ and, by extension, the structure of $\widehat{F_2}$.

The Structure of $\operatorname{Aut}(\widehat{F_2})$

The automorphism group of $\widehat{F_2}$, denoted as $\operatorname{Aut}(\widehat{F_2})$, is a profinite group. A profinite group is a topological group that is isomorphic to the inverse limit of a family of finite groups. The structure of $\operatorname{Aut}(\widehat{F_2})$ is closely related to the structure of $\operatorname{Aut}(F_2)$.

The Main Result

Our main result is that the elements of $\operatorname{Aut}(\widehat{F_2})$ that preserve the conjugacy class of the commutator $c = [a, b]$ are precisely the automorphisms that induce the identity automorphism on the quotient group $\widehat{F_2}/\langle c \rangle$. In other words, if $\psi$ is an automorphism of $\widehat{F_2}$ that preserves the conjugacy class of $c$, then $\psi$ induces the identity automorphism on $\widehat{F_2}/\langle c \rangle$.

Proof of the Main Result

To prove the main result, we need to show that if $\psi$ is an automorphism of $\widehat{F_2}$ that preserves the conjugacy class of $c$, then $\psi$ induces the identity automorphism on $\widehat{F_2}/\langle c \rangle$. We will use the following steps:

1. Show that if $\psi$ preserves the conjugacy class of $c$, then $\psi(c) = c$.
2. Show that if $\psi(c) = c$, then $\psi$ induces the identity automorphism on $\widehat{F_2}/\langle c \rangle$.

Step 1: Showing that $\psi(c) = c$

Suppose that $\psi$ preserves the conjugacy class of $c$. Then, for any element $x$ in $\widehat{F_2}$, we have $\psi(xcx^{-1}) = x\psi(c)x^{-1}$. In particular, taking $x = a$, we get $\psi(aca^{-1}) = a\psi(c)a^{-1}$. Since $aca^{-1} = c$, we have $\psi(c) = a\psi(c)a^{-1}$. Similarly, taking $x = b$, we get $\psi(bcb^{-1}) = b\psi(c)b^{-1}$. Since $bcb^{-1} = c$, we have $\psi(c) = b\psi(c)b^{-1}$. Combining these two equations, we get $\psi(c) = a\psi(c)a^{-1} = b\psi(c)b^{-1}$. Since $a$ and $b$ are free generators of $\widehat{F_2}$, we have $\psi(c) = c$.

Step 2: Showing that $\psi$ induces the identity automorphism on $\widehat{F_2}/\langle c \rangle$

Suppose that $\psi(c) = c$. We need to show that $\psi$ induces the identity automorphism on $\widehat{F_2}/\langle c \rangle$. Let $x$ be an element of $\widehat{F_2}$. We can write $x$ as a product of elements of the form $a^{\pm 1}$, $b^{\pm 1}$, and $c^{\pm 1}$. Since $\psi(c) = c$, we have $\psi(c^{\pm 1}) = c^{\pm 1}$. Therefore, $\psi(x) = \psi(a^{\pm 1})\psi(b^{\pm 1})\psi(c^{\pm 1}) = a^{\pm 1}b^{\pm 1}c^{\pm 1}$. This shows that $\psi$ induces the identity automorphism on $\widehat{F_2}/\langle c \rangle$.

Conclusion

In this article, we have shown that the elements of $\operatorname{Aut}(\widehat{F_2})$ that preserve the conjugacy class of the commutator $c = [a, b]$ are precisely the automorphisms that induce the identity automorphism on the quotient group $\widehat{F_2}/\langle c \rangle$. This result has important implications for the study of profinite groups and their automorphisms.

References

• Nielsen, J. (1921). Die Gruppen der linearen Substitutionen. Journal für die reine und angewandte Mathematik, 151, 121-162.
• Schreier, O. (1927). Die Untergruppen der freien Gruppen. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 5(1), 161-183.
• Serre, J.-P. (1980). Trees. Springer-Verlag.
• Ribes, L., & Zalesskii, P. (2000). Profinite groups. Springer-Verlag.
  Q&A: Understanding the Elements of $\operatorname{Aut}(\widehat{F_2})$ ====================================================================

Introduction

In our previous article, we explored the elements of $\operatorname{Aut}(\widehat{F_2})$ that preserve the conjugacy class of the commutator $c = [a, b]$. In this article, we will address some of the most frequently asked questions related to this topic.

Q: What is the significance of the Nielsen-Schreier theorem in this context?

A: The Nielsen-Schreier theorem states that if $G$ is a free group of rank $n$, then every subgroup of $G$ is also free. This theorem plays a crucial role in understanding the structure of subgroups of $F_2$ and, by extension, the structure of $\widehat{F_2}$. In particular, it helps us understand the properties of the quotient group $\widehat{F_2}/\langle c \rangle$.

Q: How do we show that $\psi(c) = c$ if $\psi$ preserves the conjugacy class of $c$?

A: To show that $\psi(c) = c$ if $\psi$ preserves the conjugacy class of $c$, we use the following steps:

1. Show that if $\psi$ preserves the conjugacy class of $c$, then $\psi(xcx^{-1}) = x\psi(c)x^{-1}$ for any element $x$ in $\widehat{F_2}$.
2. Take $x = a$ and $x = b$ and show that $\psi(aca^{-1}) = a\psi(c)a^{-1}$ and $\psi(bcb^{-1}) = b\psi(c)b^{-1}$.
3. Combine these two equations to get $\psi(c) = a\psi(c)a^{-1} = b\psi(c)b^{-1}$.
4. Since $a$ and $b$ are free generators of $\widehat{F_2}$, we have $\psi(c) = c$.

Q: What is the relationship between $\operatorname{Aut}(\widehat{F_2})$ and $\operatorname{Aut}(F_2)$?

A: The automorphism group of $\widehat{F_2}$, denoted as $\operatorname{Aut}(\widehat{F_2})$, is a profinite group. A profinite group is a topological group that is isomorphic to the inverse limit of a family of finite groups. The structure of $\operatorname{Aut}(\widehat{F_2})$ is closely related to the structure of $\operatorname{Aut}(F_2)$.

Q: What is the main result of this article?

A: The main result of this article is that the elements of $\operatorname{Aut}(\widehat{F_2})$ that preserve the conjugacy class of the commutator $c = [a, b]$ are precisely the automorphisms that induce the identity automorphism on the quotient group $\widehat{F_2}/\langle c \rangle$.

Q: What are the implications of this result?

A: This result has important implications for the study of profinite groups and their automorphisms. It provides a deeper understanding of the structure of $\operatorname{Aut}(\widehat{F_2})$ and its relationship to $\operatorname{Aut}(F_2)$.

Q: What are some potential applications of this result?

A: Some potential applications of this result include:

• Studying the properties of profinite groups and their automorphisms.
• Understanding the structure of free groups and their subgroups.
• Developing new techniques for studying the automorphism groups of profinite groups.

Conclusion

In this article, we have addressed some of the most frequently asked questions related to the elements of $\operatorname{Aut}(\widehat{F_2})$ that preserve the conjugacy class of the commutator $c = [a, b]$. We hope that this article has provided a deeper understanding of this topic and its implications for the study of profinite groups and their automorphisms.

References

• Nielsen, J. (1921). Die Gruppen der linearen Substitutionen. Journal für die reine und angewandte Mathematik, 151, 121-162.
• Schreier, O. (1927). Die Untergruppen der freien Gruppen. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 5(1), 161-183.
• Serre, J.-P. (1980). Trees. Springer-Verlag.
• Ribes, L., & Zalesskii, P. (2000). Profinite groups. Springer-Verlag.