### Sphere Spectrum Analogue of PGL(2,Z)

#### Posted by John Baez

Since I’ve been thinking about continued fractions I’ve been thinking about $PGL(2,\mathbb{Z})$, the group of transformations

$z \mapsto \frac{a z + b}{c z + d} , \qquad a,b,c,d \; \text{s.t.} \; a d - b c \ne 0$

mod its center. You can think of this as a group of transformations of the integral form of the projective line. When we see something like

$\frac{\sqrt{5} + 1}{2} = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \ddots}}}$

or even

$\frac{\pi}{4} = \frac{1}{1 + \frac{1^2}{2 + \frac{3^2}{2 + \frac{5^2}{2 + \ddots}}}}$

we should presumably be thinking about this group.

But in modern mathematics the sphere spectrum is what Joyal called the “true integers”: it’s the initial ring spectrum just as $\mathbb{Z}$ is the initial ring. So there should be an enhanced version of $PGL(2,\mathbb{Z})$ with the sphere spectrum taking over the role of the integers, and a lot should be known about it.

What’s it called, and what do people know about it?

## Re: Sphere Spectrum Analogue of PGL(2,Z)

People also consider $P S L(2, \mathbb{Z})$ in the context of continued fractions, such as Todd Trimble here (though there with regard to the presentation $a_1 - 1/(a_2 - 1/(a_3 - 1/...$).

What does the difference between $P G L$ and $P S L$ amount to in this context?