# Frohman & Fine's proof about Bianchi groups as HNN extensions (or anyone else's)

Specifically, I am looking for a proof that for squarefree $d\in\mathbb{N}\setminus\{ 1,3\}$, there exists some Kleinian $\Gamma$ and some $\alpha\in$Aut$(PSL(2,\mathbb{Z}))$, such that the Bianchi group $\Gamma_d\cong \Gamma*_{\alpha}$. I have a reference that this proof appears in the article:

Charles Frohman and Benjamin Fine, "The amalgam structure of the Bianchi groups," 1986, Compte Rendu RSC Mathematiques, Volume 8, pp. 353-356.

But I cannot find access to this anywhere (at my university library, through Jstore, or anywhere else online). I did find an article on something very similar, by the same authors where they discuss amalgamated products instead. If I had to, I could probably go through that and alter the argument appropriately, but I expect it would take me more time than I can spare right now. Here is the info on the similar article in case that's useful:

Charles Frohman and Benjamin Fine, "Some Amalgam Structures for Bianchi Groups," 1988, Proceedings of the American Mathematical Society, Vol. 102, No. 2, pp. 221-229

From reading that and some similar things, I expect the approach would involve looking at finite-cover manifolds of the Bianchi groups, and finding incompressible embeddings of the totally real half-plane into them, but I'm lost on the details.

• Your profile says that you're at CUNY right now. I would be very surprised if the CUNY library did not have back issues of Comptes Rendus; it's a very well-known journal. Ask your librarian (they can probably get it via interlibrary loan if necessary). Another option would be to check out Columbia's library, which certainly will have it. – Andy Putman Dec 29 '13 at 6:01
• (maybe the issue is that like in your question above you spelled the journal name wrong? it can be difficult to track down foreign-language journals in some libraries, especially ones like CR which have undergone huge numbers of name changes over the years. that's why I suggest talking to a librarian.) – Andy Putman Dec 29 '13 at 6:08
• @AndyPutman Probably you're thinking of the French journal, Comptes Rendus Mathematique - but the article he mentions is in the similarly named Canadian journal, mr.math.ca – Dan Petersen Dec 29 '13 at 10:34
• @j0equ1nn No, I would not expect the CR article to include a reference to an expected full version. Without having read either paper, I think that most likely the proofs of the results announced in the CR note are exactly what is in the second article you referenced. If there is a discrepancy between what's stated in the two papers, then my guess is that either they thought of the result as an easy corollary and forgot to state it explicitly in the Proc. AMS paper, or that they found a gap in their argument when they worked out the details, and in the end stated only a weaker version. – Dan Petersen Dec 30 '13 at 20:43
• Finally, If you can't figure out what's going on then I suggest you send them an e-mail! – Dan Petersen Dec 30 '13 at 20:43

## 1 Answer

The second reference listed in the question,

Charles Frohman and Benjamin Fine, "Some Amalgam Structures for Bianchi Groups," 1988, Proceedings of the American Mathematical Society, Vol. 102, No. 2, pp. 221-229,

does in fact include a proof of the statement in question. The article is available on Jstor.