Talk:John R. Stallings

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Bio details[edit]

I looked through the 1970 volume of the "Notices of AMS" hoping to find some more detailed bio info for Stallings, since he received the Cole Prize that year, but, unfortunately, they do not seem to have any relevant info available. There is a mention of the Cole prize being awarded in the program of the AMS annual meeting at San Antonio in Jan 1970 but no more details (I guess the names of the winners had not been announced before the meeting), and I did not find any follow-up info in the subsequent issues of the Notices. How annoying. I looked up the 1980 volume of the Notices (at that time the Cole Prize was awarded every five years; the 8-th was in 1970, the 9-th in 1975 and the 10th in 1980) and there they already give a detailed biographical sketch for the Cole Prize winners. If someone knows of any other source that can be used for verifying basic bio details such as the year of birth (which appears to be 1935 since the 65th birthday conference was in 2000), please chime in. Nsk92 (talk) 14:49, 6 August 2008 (UTC)[reply]

By the way, thanks very much for your work on the article. It is much improved and in a good state. I think biographical articles can serve as great surveys of fields, and you've introduced a nice range of topics in this article. JackSchmidt (talk) 15:02, 6 August 2008 (UTC)[reply]

Stalling's theorem - please explain[edit]

There is something I don't understand about Stalling's characterization of the finitely generated groups that have more than one end. The statement implies that $\mathbb {Z}$ can be split as a nontrivial amagamated free product or an HNN extension. Can you describe such a decomposition? (I don't see how one could be possible). Oded (talk) 15:39, 11 August 2008 (UTC)[reply]

Yes, sure. The group

\mathbb {Z}

is an HNN-extension of the trivial group with both associated subgroups trivial as well. You can write its HNN-presentation as

\langle t|t^{-1}1t=1\rangle .

In the graph-of-groups language, this HNN-extension presentation of

\mathbb {Z}

is the fundamental group of the graph of groups consisting of a single vertex a single loop-edge, where the vertex group and the edge group are trivial. Similarly, a free group of arbitrary rank is a multiple HNN-extension of the trivial group where the stable letters are the free generators. Nsk92 (talk) 15:46, 11 August 2008 (UTC)[reply]

Thanks. I thought the word nontrivial in the statement applied to both the splitting and the HNN representations. But from what you are saying, the HNN presentation can be trivial. If that's correct, perhaps it is better to clarify by writing instead "admits a splitting as an HNN-extension over a finite group or as a nontrivial amalgamated free product"? Oded (talk) 16:28, 11 August 2008 (UTC)[reply]

Yes, probably some clarification is needed here. I was using the word "nontrivial splitting" here in the sense it is usually used in Bass-Serre theory: when the action on the Bass-Serre covering tree does not have a global fixed point for the group (this definition is actually explained in the article Bass-Serre theory). In that sense, an HNN-extension splitting is always nontrivial and an amalgamated free product splitting is nontrivial exactly when the amalgamated subgroup has index > 1 in each of the factors. But you are right, probably a more explicit clarification is in order here. Nsk92 (talk) 17:10, 11 August 2008 (UTC)[reply]

OK, I have added an explicit stetement of the theorem to the article. It is probably better to create a separate article for Stallings theorem about ends of groups and to move such an explicit statement there, but it'll take me a while to get to it. Nsk92 (talk) 17:30, 11 August 2008 (UTC)[reply]

Assessment comment[edit]

The comment(s) below were originally left at Talk:John R. Stallings/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Initial WikiprojectMath rating assigned by User:CBM at 13:00, 2 December 2008=Talk%3AJohn_R._Stallings&diff=255409113&oldid=255406386. Nsk92 (talk) 05:48, 5 December 2008 (UTC)[reply]

Last edited at 13:14, 5 December 2008 (UTC). Substituted at 20:20, 29 April 2016 (UTC)

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