About Peter Scott.

A revised version of what i wrote a few days before his passing away. Reminiscences of Peter Scott: We first interacted in Liverpool during 1968-69 where we discussed mostly Kirby-Siebenmann work. Later we seem to have independently turned to 3-manifolds and there was some correspondence in 1972 about Waldhausen’s work as well as ‘finitely generated implies finitely presented’ for 3-manifold groups. There is some discussion of this in a survey article by Dani Wise and probably the neatest proof comes from Thomas Delzant which I described in ‘Delzant’s variation on Scott Complexity’ written in Peter’s honour on his 60th birthday. He had the knack of proving simple results of great applicability as well as the power to prove difficult results like his work on Seifert fibre spaces. Our next interactions were sporadic for a while. I remember Peter advising me that it is difficult to understand Thurston and that I should discuss with others. He then invited me to conference in 1984 that he organised with D.B.A. Epstein in Warwick. By this time he was quite well known and seemed to be determined to make a better mathematician out of me. We next met in Norman, Oklahoma in 1986, and he suggested two problems which we did together. One of them proved that certain finitely generated subgroups of distorted Kleinian groups are geometrically finite. Since then topic has been pursued by a few others. After that we met in a few conferences where we discussed how many results in 3-manifold theory may have analogues in group theory with annulus theorem as a possible candidate. But serious work on this started in 1994 when Peter visited Melbourne. During this visit Peter showed me a paper of Tukia and said that it should lead to a proof of annulus theorem for hyperbolic groups and he also suggested that I needed to read only a few pages in Tukia’s paper. It was like a thesis advisor advising a student. This was the start of an intense collaboration with Peter since then. We probably wrote over six hundred pages of mathematics together in about ten papers. We made mistakes and probably took up a wrong program. But after a lot of hard work we did achieve what we wanted and for both us it seemed the hardest work we have done. We visited each other several times and it was not always clear how the ideas originated but they seemed joint. The program was the main aim, we did not care who did the actual work. Some papers were completely Peter’s and some more or less mine though they appeared as joint papers. Sometimes there were other collaborators. For both us it was very satisfactory work with the last paper just nearing completion. ‘Beating it in to shape’ as Peter used to say. In retrospect, it seems to me that we were barking up the wrong tree though Peter did not agree with me. But along the way we did some work on intersection numbers and regular neighbourhoods which seem to be useful in other contexts too. Peter remained active until he could work no more. This last paper was written mostly by him. During this March-June, he developed another approach to regular neighbourhoods which is quite interesting and non-trivial. P.S. One instance of joint work: Peter always thought JSJ decomposition was a sort of regular neighbourhood. We tried to extend the idea to groups over several visits. In one of those trips we worked for a whole month and Peter was about to leave the next day. We decided that if we did not do it this time, we would give up. Towards the evening, I said may be it is a pretree. Then Peter asked for the definition. When I wrote down the conditions, Peter said ‘That’s it’. This idea of regular neighbourhood in the standard group theory presentations by Sela and others received more traction in the work of Guirardel and Levitt. References: Section 4 in [1] Dani T. Wise: An invitation to coherent groups, in What’s Next?: The Mathematical Legacy of William Thurston (AMS-205) Edited by Dylan Thurston [2] Gadde A. Swarup: Delzant’s variation on Scott Complexity, arXiv:math/0401308 2004 [3] Pekka Tukia: Homeomorphic conjugates of Fuchsian groups, Journal für die reine und angewandte Mathematik (1988) Volume: 391, page 1-54 Section 6.4 in [4] Vincent Guirardel and Gilbert Levitt : JSJ Decompositions ofGroups. Asteriaque 395, 2017