5

Let $B_3$ be the braid group on three strands. I was looking at an element in $B_3$, which I will write in the standard presentation:

$$(\sigma_2\sigma_1\sigma_2)^{-1}\sigma_1^3\sigma_2^{-3}(\sigma_2\sigma_1\sigma_2)$$

and I was able to explicitly show it equal to $\sigma_2^{3}\sigma_1^{-3}$, which inverts the element by conjugation. I was wondering if one see this geometrically? (Via some diagram) Or rather, if there is some phenomenon that explains this, or if it is a mere coincedence.

If the following is known: what do inner automorphisms of $B_3$ look like in general? Maybe the semi direct product presentation is more promising for understanding it.

Andres Mejia
  • 21,467
  • This braid group is a central extension of the modular group $PSL(2,Z)$. In the case of the latter, it is easy to see where this phenomenon comes from: An order two rotation of the hyperbolic plane which preserves the axis of a hyperbolic element and swaps its endpoints. As for your last question, I have no idea what you want to know beyond the definition of an inner automorphism. – Moishe Kohan Nov 23 '17 at 20:06
  • @MoisheCohen I guess I was wondering what the group structure was. I'll spend some time thinking about the former half of your comment. – Andres Mejia Nov 23 '17 at 20:08
  • The group structure of $Inn(B_3)$ is very simple, it is the quotient $B_3/Z(B_3)\cong PSL(2,Z)$. – Moishe Kohan Nov 23 '17 at 20:13
  • Ah, I see. Is there an easy way to tell when an element is in the center of $B_3$? – Andres Mejia Nov 23 '17 at 20:17
  • Yes, it is, take a look at https://en.wikipedia.org/wiki/Braid_group – Moishe Kohan Nov 23 '17 at 20:23
  • Oh, so absolutely standard. Thank you. If you would like to submit an answer, I would accept it. – Andres Mejia Nov 23 '17 at 20:28

1 Answers1

2

This braid group is a central extension of the modular group $PSL(2,Z)$. In the case of the latter, it is easy to see where this phenomenon comes from: An order two rotation of the hyperbolic plane which preserves the axis of a hyperbolic element and swaps its endpoints. To see how this can happen, lift to $SL(2,Z)$. The order two rotation lifts to an order 4 rotation. It will conjugate a real-diagonalizable matrix $M\in SL(2,Z)$ to its inverse iff the eigenlines of $M$ are orthogonal, i.e. iff $M=M^T$. There are infinitely many examples of this happening.

As for your last question, for every group $G$, $Inn(G)\cong G/Z(G)$, where $Z(G)$ is the center of $G$. In the case of $B_3$, this quotient is isomorphic to $PSL(2,Z)$.

See this Wikipedia article for more detail.

Moishe Kohan
  • 111,854