# Origin of the name “Torelli group”

The genus $g$ Torelli group $I_g$ is the kernel of the action of the mapping class group of a genus $g$ surface on the first homology group of the surface.

The first paper I am aware of that uses the name "Torelli group" for this group is

Johnson, Dennis A survey of the Torelli group. Low-dimensional topology (San Francisco, Calif., 1981), 165–179, Contemp. Math., 20, Amer. Math. Soc., Providence, RI, 1983. (Reviewer: J. S. Birman) 57N05 (14H15 32G15 57-02)

It had previously been studied without this name in numerous papers (e.g. earlier papers of Johnson, work of Powell, and work of Birman-Craggs). In Johnson's survey above, he says "The topologists had no name for it, but it has been known for a long time to the analysts, so I will use their label: the Torelli group".

Now, I understand the mathematical justification for calling it the Torelli group, namely Torelli's theorem. My question is instead historical : Johnson's sentence above seems to imply that he did not make up this name, and thus that there are older papers (maybe by analysts or algebraic geometers?) that use it. Is anyone aware of any?

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After Bachman-Torelli Overdrive split up, they started solo acts. My favorite is "You Ain't Seen Nothin' Yet." –  Will Jagy Jan 20 '14 at 5:58

As abx noted, the real object that algebraic geometers and complex analysts considered was not the Torelli group per se, but rather Torelli space, which is the quotient of Teichmuller space by the Torelli group. It's always hard to trace the origins of a name like this, but I believe that the terminology was introduced by Weil in his 1958 Séminaire Bourbaki lecture

A. Weil, Modules des surfaces de Riemann, in Seminaire Bourbaki; 10e annee: 1957/1958. Textes des conferences; Exposes 152 a 168; 2e ed.corrigee, Expose 168, 7 pp, Secretariat Math., Paris.

Notice that this predates the paper of Grothendieck that abx references. As evidence that this is the correct origin, Rauch explicitly attributes the terminology to Weil on page 544 of

Rauch, H. E. Weierstrass points, branch points, and moduli of Riemann surfaces. Comm. Pure Appl. Math. 12 1959 543-560.

When searching for old papers on the subject, it's also worth pointing out that before Johnson, the Torelli group was often called the Torelli modular group; similarly, the mapping class group was often called the Teichmuller modular group and the symplectic group was often called the Siegel modular group.

EDIT : Just to point out something a little confusing -- I hadn't looked through these old papers in a while, but while doing so I found a few papers that use the phrase Torelli modular group for the symplectic group! So make sure to read any of these older papers carefully to make sure you understand exactly what they are talking about...

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Well, Weil does not mention explicitly the Torelli space, but he defines "Torelli surfaces" (those which are parametrized by the Torelli space). So you have a point. My guess is that the idea was around, then naming it after Torelli was quite natural. –  abx Jan 20 '14 at 17:41
@abx : That's a good point, though Rauch does explicitly mention Torelli space. I also agree that the concept was "in the air" at that point. Most of the papers on Torelli space from the '60's and '70's that I am aware of were by people working in the complex analytic school (Rauch, Bers, Ahlfors, etc, together with their students). I'd be very interested in any papers on the topic from that era by people in the more "algebraic" side of the subject. –  Andy Putman Jan 20 '14 at 17:46

I think the terminology was coined by Grothendieck in the Cartan seminar 1960-61. He introduces the term Torelli space for the quotient of the Teichmuller space by the Torelli group. The fact that he says that the corresponding functor "pourrait s'appeler" (could be called) Torelli functor seems to indicate clearly that he is inventing the name.

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