Michael gave an excellent seminar a couple of days ago, in which he told us the story of how he and Chris Parker recently resolved the commuting graphs conjecture – that there is an absolute bound on the diameter of the commuting graph of any group. Michael posted about the original conjecture a couple of years ago, but is too busy to blog at the moment, so I’m stepping in.
He presented the seminar as a sort of “historical thriller”, tracing the origins of the conjecture, the various partial solutions that either provided supporting evidence for the conjecture or tantalisingly suggested that counterexamples were just round the corner, papers resolving the conjecture that were immediately retracted, and finally the last few weeks filled with work, to-and-fro emails, missing attachments and so on. Presenting it like this, rather than simply giving a detailed proof, was great because it really conveyed the sense of activity and excitement accompanying a research project reaching a successful conclusion. When he told us that he even worked through a West Coast Eagles finals game, we knew how serious about it he was!
So, what’s the answer? Is the conjecture true or false?
As it turns out, the conjecture is false and it is possible to give an explicit description of a family of groups, whose commuting graphs are connected, but with arbitrarily large diameter. The groups themselves are fairly straightforward, being a family of 2-groups defined by explicitly given generators and relations, and the proof that they have the required properties is not long. But of course, its the finding of the groups that’s the difficult — and fun — part.
The paper’s already written and up on the arxiv at arxiv.org/abs/1210.0348, so head over there if you want more details.