# A synopsis of Adyan’s solution to the general Burnside problem?

Where can I find a high-level overview of Adyan’s original proof of the existence of finitely generated infinite groups with finite exponent?

If possible, would an expert please write a short synopsis of this proof here on MO?

Many years ago at Illinois, Ken Appel took notes for lectures on combinatorial group theory given by Adyan. He told me once that Adyan had said that the key thing missing from Britton’s approach to the general Burnside problem was the notion of a cascade. I’d like to know, for example, why this is the case. How, precisely, is Adyan's proof related to Britton's attempt? Is the former a perturbation of the latter in which cascades resolve the inconsistency? If so, can one succinctly say how?

In an English translation of S. I. Adian, “An axiomatic method of constructing groups with given properties”, Uspekhi Mat. Nauk, 32:1(193) (1977), 3–15, we find, regarding the interlocking facts to be proved by simultaneous induction on the rank:

"...as the reader accumulates experience and gradually forms intuitive pictures of the relevant concepts, our proofs become less formal..."

What are these intuitive pictures? (Of course, I don't expect an answer for this.)

Also,

Is Adyan's notion of cascade used anywhere else in group theory today? (Or has this idea been totally abandoned for Olshanskii's diagrammatic method.)

I apologize for not including a definition of cascade, but the ability to do so concisely is implicitly part of the question!

-
Don't you mean infinite finitely generated groups? –  Qiaochu Yuan Dec 3 '10 at 16:11
Of course. Thanks for having my back, Qiaochu! –  Jon Bannon Dec 3 '10 at 16:57
Thanks again for sending the copy of S. I. Adian, “An axiomatic method of constructing groups with given properties”, Uspekhi Mat. Nauk, 32:1(193) (1977), 3–15, Theo Buehler! –  Jon Bannon Dec 3 '10 at 17:10
No problem, it cost me just a few clicks. And, sadly, it's the only thing I could do towards answering this interesting question... –  Theo Buehler Dec 3 '10 at 18:00
@Jon: I believe that only future answers will be made community wiki; in particular, it doesn't appear that Mark Sapir's answer is community wiki. –  Tom Church Dec 15 '10 at 6:15
show 1 more comment

A few things:

1. Britton did not have a proof at all.

2. The notion of cascades is central to Novikov-Adyan's proof. That concept was missing in the original announcement of Novikov, and that is why the period between the announcement and the actual proof was so long. Approximately it means the following. You start with a word W and apply relations $u^n=1$ for relatively small $u$. The word changes, of course, but certain part of it stays almost the same. So cascades reflect the stable part of the process. Without them, there is no control on what can happen when you start applying relations. The notion of cascades is not used now, as far as I know (except for the later papers by Adyan and his students): it is too connected with the Burnside problem.

3. There is no "short description" of Novikov-Adian work. One of the things simplifying reading his text is that you really need to understand the base of induction, the step of induction is much easier (the proof consists of proving hundreds of statements simultaneously using induction on some parameter). A relatively short proof is in Olshanskii's book "Geometry of defining relations in groups", and the new proof of Gromov and Delzant show much more explicitly what is going on.

4. Olshanskii's method was based on van Kampen diagrams and the key concept (as key as cascades in Novikov-Adian's proof) was the concept of contiguity subdiagram. Contiguity subdiagrams are not related to cascades, but play similar role. They help replacing the classic small cancelation condition. In Gromov-Delzant proof the key concepts are "very small cancelation", "mesoscopic curvature", and "Cartan-Hadamard"-type theorems. These are "big guns" which make proof conceptually easier.

5. If you really need more explanation, you should contact Igor Lysenok. He knows all three proofs very well.

Update. I can probably add a little bit more information. The proofs of Adyan and Olshanskii are based on the idea that if $W$ is a six-power-free word (without subwords of the form $u^6$) in the generators of the Burnside group of sufficiently large odd exponent, then $W\ne 1$ in the group. Since there are infinitely many cube-free words in two generators by Thue, there are infinitely many different six-power-free words in the free Burnside group. To prove thatfact, you start inserting words $u^n$ to $W$. Then parts of the powers $u^n$ will possibly cancel but the original word should not be completely destroyed. This simple idea turned out to be very hard to implement. As I said, in order to control what happens to $W$ when we insert the relations, Adyan and Novikov introduced the cascades.

Olshanskii's idea was to consider a possible van Kampen diagram over the presentation of the Burnside group with boundary label $W$. If that presentation satisfied a small cancelation condition, then the boundary of one of the cells would have a large intersection with the boundary of the diagram by the Greendlinger lemma, and $W$ would contain large powers, a contradiction. Unfortunately, the presentation does not satisfy the standard small cancelation condition. So Olshanskii basically introduced his own condition. Although a cell may not be attached to the boundary of the diagram, there exists a subdiagram (called the contiguity subdiagram) which has a form of a rectangle with two long sides and two short sides: one long side on the boundary of the cell, another long side on the boundary of the diagram. He proves that this subdiagram consists of cells that correspond to shorter relations. This alows him to use induction. As in the case of Adyan-Novikov, in order to prove the statement by induction, one needs to first make it stronger. In the case of Adyan-Novikov, the resulting statement consists of many parts which are proved by simultaneous induction, in the case of Olshanskii, there are very few parts.

-
Oops. I forgot to type "attempt". Your "2." answers the problem as far as I'm concerned. Regarding "3.", I wondered if you or Denis could perform a miracle. I imagined that "4" were the case, but thanks for explicitly pointing out that there is no relation of contiguity subdiagrams to cascades! Thanks for bringing Igor Lysenok to my attention. Finally, is there any obvious relation between the structure of Britton's attempt and Adyan's proof? (Not terribly mathematical, but I've always wondered.) Thanks, Mark! –  Jon Bannon Dec 3 '10 at 19:13
Oh, and I was completely unaware of Gromov-Delsant! Thanks for that! –  Jon Bannon Dec 3 '10 at 19:15
The update is very nice. It sheds some light on the origin and core of the proofs. This is great. –  Jon Bannon Dec 6 '10 at 0:40