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August 31, 2020

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,ℤ)PGL(2,\mathbb{Z}), the group of transformations

z↦az+bcz+d,a,b,c,ds.t.ad−bc≠0 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

5+12=1+11+11+11+⋱ \frac{\sqrt{5} + 1}{2} = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \ddots}}}

or even

π4=11+1 22+3 22+5 22+⋱ \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 ℤ\mathbb{Z} is the initial ring. So there should be an enhanced version of PGL(2,ℤ)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?

Posted at August 31, 2020 5:27 AM UTC

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

People also consider PSL(2,ℤ)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/...a_1 - 1/(a_2 - 1/(a_3 - 1/...).

What does the difference between PGLP G L and PSLP S L amount to in this context?

Posted by: David Corfield on August 31, 2020 8:05 AM | Permalink | Reply to this

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

Perhaps I should have focused on PSL(2,ℤ)PSL(2,\mathbb{Z}) because that’s the one number theorists always talk about. A sphere spectrum analogue of PSL(2,ℤ)PSL(2,\mathbb{Z}) would make me perfectly happy.

James Dolan brainwashed me into thinking PGL(2,−)PGL(2,-) is more fundamental because it’s the full symmetry group of the projective line, while PSL(2,−)PSL(2,-) consists of orientation-preserving symmetries of the projective line.

I have to think about these things every time because the details depend on the commutative ring you put in the −-, but I think PSL(2,ℤ)PSL(2,\mathbb{Z}) is an index-two subgroup of PGL(2,ℤ)PGL(2,\mathbb{Z}). The point is that the determinant of an invertible integer matrix must be either 11 or −1-1, and the matrix is in SLSL if the determinant 11. For example

(1 0 0 −1) \left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right)

is a guy in GL(2,ℤ)GL(2,\mathbb{Z}) that’s not in SL(2,ℤ)SL(2,\mathbb{Z}), which gives an element of PGL(2,ℤ)PGL(2,\mathbb{Z}) that’s not in PSL(2,ℤ)PSL(2,\mathbb{Z}) .

Posted by: John Baez on August 31, 2020 6:21 PM | Permalink | Reply to this

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

Hmm, can one define ℙ 1\mathbb{P}^1 of the sphere spectrum 𝕊\mathbb{S}? This would probably be a spectral scheme. Then its automorphism group could be one form of PGL(2,𝕊)PGL(2,\mathbb{S})…

Posted by: David Roberts on August 31, 2020 11:43 AM | Permalink | Reply to this

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

With regard to the question in your first sentence, this is discussed for example in sections 5.4 and 19.2.6 of Lurie’s Spectral Algebraic Geometry.

Posted by: Richard Williamson on September 1, 2020 11:50 AM | Permalink | Reply to this

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

Here’s a thought. I claim that the functor R↦BPGL(n,R)R \mapsto B\mathrm{PGL}(n,R), from commutative rings to connected pointed spaces, is naturally the restriction of a functor defined on E ∞E_\infty ring spectra.

For a commutative ring RR, the tensor product of RR-modules induces a map of pointed spaces BGL(n,R)×BGL(m,R)→BGL(nm,R)B\mathrm{GL}(n,R) \times B\mathrm{GL}(m,R) \to B\mathrm{GL}(n m,R). This is part of a higher structure, making BGL(1,R)B\mathrm{GL}(1,R) into an E ∞E_\infty-space whose underlying E 1E_1-space acts on BGL(n,R)B\mathrm{GL}(n,R). Hence we may form the homotopy orbit space (BGL(n,R)) hBGL(1,R)(B\mathrm{GL}(n,R))_{h B\mathrm{GL}(1,R)}, which I think is naturally pointed homotopy equivalent to BPGL(n,R)B\mathrm{PGL}(n,R).

This should make sense for E ∞E_\infty ring spectra as well, for example BGL(1,R)B\mathrm{GL}(1,R) is an E ∞E_\infty-space, the basepoint component of what is nowadays called Pic(R)\mathrm{Pic}(R). It acts on the space BGL(n,R)B\mathrm{GL}(n,R) by ⊗ R\otimes_R, so we obtain a functor R↦(BGL(n,R)) hBGL(1,R)R \mapsto (B\mathrm{GL}(n,R))_{h B\mathrm{GL}(1,R)} from E ∞E_\infty ring spectra to connected pointed spaces.

It seems one could take BPGL(n,R):=(BGL(n,R)) hBGL(1,R)B\mathrm{PGL}(n,R) := (B\mathrm{GL}(n,R))_{h B\mathrm{GL}(1,R)} and PGL(n,R)=ΩBPGL(n,R)\mathrm{PGL}(n,R) = \Omega B\mathrm{PGL}(n,R). I don’t know how to justify whether those are good definitions though.

Posted by: anonymous on August 31, 2020 3:46 PM | Permalink | Reply to this

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