Talk:Von Neumann conjecture

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Details of the recent work, please! Charles Matthews 06:28, 18 May 2004 (UTC)

So I have found that Sapir and Olshanskii in 2000 constructed a first finitely-presented counterexample. But the first counterexample might be much older (Olshanskii 1980)? Charles Matthews 07:02, 18 May 2004 (UTC)

You are quite correct - the first counterexample (Tarski monster) was indeed due to Ol'shanskii in 1980, and it is only the fp weakening of the conjecture that is "recent". I've added something on the history, and corrected the misleading impression I gave in Thompson groups. Best wishes, Cambyses 20:08, 18 May 2004 (UTC)

[edit] Slightly confusing/ungrammatical passage

I find myself unable to improve the poorly written passage because I don't know enough about the subject. Would an expert kindly explain the following to me, and/or make an adjustment to the article?: "Although his name is popularly attached to the conjecture, it does not seem that he believed the converse to be true. Rather, this suggestion was made by a number of different authors in the 1950s and 1960s; for example a statement attributed to M. Day from 1957." Which "suggestion"--that von Neumann's name be attached to the conjecture, or that von Neumann believed the conjecture's converse to be true? The clause following the semicolon does not possess both a subject and a verb and therefore cannot stand apart from the previous separated by a semicolon. (One solution may be to add a verb and quote a relevant passage from M. Day.) Robert K S (talk) 22:23, 10 January 2008 (UTC)

The conjecture is that two properties "not amenable" and "containing a non-cyclic free group" are equivalent. Such conjectures are often proven in a mechanical process (which young mathematicians learn by rote in early courses). One first shows that every thing with property A has property B, and then shows that every thing with property B has property A. One of these steps was carried out by von Neumann. The point of the passage was that von Neumann himself probably did not intend to suggest that the other step would succeed, even though the popular belief that it might succeed was named after him, the "von Neumann conjecture".

I tried to separate the A -> B and B -> A a little, but had to insert some verbiage to make it read well at all. Future editors feel free to improve/remove the superficial similarity comment, as long as the paragraph remains clear. It was simply the first thing that came to mind when I read the article. "Oh, I guess that's a little like Tits alternative, so I can see why someone might think it is true." JackSchmidt (talk) 22:59, 10 January 2008 (UTC)

Many thanks, Jack, your edit clears up the issue considerably. I altered it a bit to remove the posing of a question, more suited to a mathematics lecture than an encyclopedia. I also added a fact tag to the sentence that contained the word "seems". Whenever that word is in an encyclopedia, it suggests the uncertainty of conjecture/supposition/speculation, which should always be attributed (i.e., who did the seeming?). Thanks again, Robert K S (talk) 05:05, 11 January 2008 (UTC)