Celebratio Mathematica — Scott

Godfrey Peter Scott

by Rob Kirby

Godfrey Peter Scott was born in Cambridge, UK, on 27 November 1944 (he dropped Godfrey early on and is now known as Peter, or G. Peter Scott or GP Scott more formally). Peter’s mother, Barbara Noel Scott, née Smith, was a student at Newnham College in Cambridge and worked, surveying people for the Ministry of Food during World War II. His father, David Bernard Scott (originally Schultz, but changed to Scott in 1939) was the son of an uneducated Lithuanian man who went to South Africa at age 18 to make his fortune, and then came to London a few years later, where he did make money as a furrier.

Bernard (Peter’s father) was a Cambridge student of W. V. D. Hodge, at the time Britain declared war against Germany, and he spent 1940–42 at Bletchley Park working with Alan Turing and others on breaking the German codes. He eventually moved to an even more secret group which was breaking the codes used by the Soviet Union to communicate with Communist groups around the world. Their intelligence was sent to the OSS (Office of Secret Services of the US) under the acronym ISCOT — “I” for intelligence and “SCOT” for the name of the leader of the group. After the war, he held a Lectureship at Kings College in London (1947–1953), and in 1962 he was tapped to start the mathematics department at the University of Sussex. An obituary in the Bulletin of the London Mathematical Society1 mentions that he was also an excellent chess player who, together with B. H. Neumann, once swept the competition — and, jointly, all the prize money — at a weekend tournament in London.

Peter grew up in southwest London (East Sheen), with three brothers: Colin (now a software entrepreneur), Oliver (deceased in New Zealand) and Ed (an MD in New Zealand). He became rather good at maths at age 14. He was in a maths and physical sciences track, but failed chemistry so did a double major in maths and physics. This was at a private day school, St. Paul’s, which had a four year “4th form” which was intended to give a strong preparation for future admittance to either Cambridge or Oxford. Both Universities gave yearly entrance exams — first Cambridge, then Oxford. In his third year, Peter took the Oxford exam as a warm-up to taking the Cambridge exam in his fourth year. But Oxford awarded him a scholarship so he left St. Paul’s for Oxford at the end of his third year, entering at age 17.

At Oxford Peter did the usual undergraduate three years and finished with a first. He then went to the University of Warwick in 1965, the first year of operation for the new university, whose maths department was headed up by Christopher Zeeman. Peter finished in the normal three years under the supervision of Brian Sanderson. His Ph.D. thesis consisted of four papers, stapled together. One “proved” the striking result that $P L (3)$ was homotopy equivalent to $O (3)$ which had as a corollary (not noticed by Peter) the Smale Conjecture that $Diff (S^{3})$ is homotopy equivalent to $O (4)$ , later proved by Allen Hatcher. Peter’s proof relied on the mistaken claim that a homeomorphism of $S^{2} \times [0, 1]$ , $rel \partial$ , which took the north pole cross $I$ hemisphere cross $I$ , had the property that this arc remained unknotted. Of course this is false by the inverse “light bulb trick” (you can knot Peter’s arc by passing strands over the south pole), as was pointed out by one of his Ph.D. examiners, Raymond Lickorish.

So that paper was withdrawn and replaced by another, and Peter finished in 1968. When Andrew Casson mentioned to Peter that his wrong result proved something about sufficiently large 3-manifolds, and Peter asked what they were, Andrew suggested he read Waldhausen’s current body of work on sufficiently large 3-manifolds. This is what led Peter into 3-manifold topology, for he had just considered himself as a high-dimensional topologist, like his advisor, even though some of his work was in low dimensions.

While at Warwick, Peter married Anne Guy (daughter of Richard Guy, an active combinatorialist at the University of Calgary until his death in 2020 at age 103) and they had two daughters, Katharine and Carol. Peter was later briefly married to Rae Packer, and then to mathematician Rita Gitik, and their son David finished a BS at Harvey Mudd College in computer science in 2015.

After Warwick, Peter went to the University of Liverpool and became, in regular succession, Lecturer, Senior Lecturer and then Reader. He served as head of the department for a term, learning that one’s spare time as head was not spent thinking about math, but about administrative problems.

While at Liverpool, Peter proved the surprising fact that if a 3-manifold fundamental group is finitely generated, then it is finitely presented [2]. This follows from what is now known as the Scott Core Theorem [1], namely that if $π_{1} (M^{3})$ is finitely generated, then there is a compact submanifold $N$ in $M$ such that the inclusion induces an isomorphism $π_{1} (N) \to π_{1} (M)$ . (A compact manifold has a finitely presented fundamental group.)

In an Annals paper [4] concerning a closed, orientable, irreducible 3-manifold $M$ with infinite fundamental group, Peter showed that if $M$ is homotopy equivalent to a Seifert fiber ~~space $N$ ,~~ then $M$ is homeomorphic to $N$ . This was a significant paper, now superseded by the Perelman avalanche.

In a paper [3] that turned out to be very important, Peter proved that surface groups are locally extended residual finite (LERF). We can state the result here in brief.

A group $G$ is residually finite if for any element $g \in G$ , there exists a homomorphism of $G$ to a finite group which takes $g$ to a nontrivial element. Equivalently there exists a finite index normal subgroup $H$ of $G$ with $g \notin H$ . A group $G$ is then LERF if for all finitely generated $H \subset G$ and $g \in G - H$ , there exists another subgroup $G_{1} \subset H \subset G$ with $G_{1}$ of finite index in $G$ and $g \notin G_{1}$ .

A subgroup $S \subset π_{1} (F)$ of a surface group is geometric if it is represented by a subsurface $X \subset F$ inducing a monomorphism $i_{*} : π_{1} (X) \to π_{1} (F)$ . Peter showed that for $S$ a finitely generated subgroup, there is a finite sheeted covering $F_{1}$ of $F$ such that $π_{1} (F)$ contains $S$ as a geometric subgroup. Furthermore for an element $g \in π_{1} (F) - S$ , $F_{1}$ can be chosen so that $g \notin π_{1} (F_{1})$ . LERF follows for $π_{1} (F)$ by taking $G_{1}$ corresponding to the finite covering $F_{1}$ .

Peter’s most frequent collaborator was Joel Hass who comments in this volume on their joint work.

In 1986, Peter was awarded the Senior Berwick Prize by the London Mathematical Society, which issued this statement on the occasion:

Citation for Godfrey Peter Scott
Peter Scott is well known for his many important contributions to the geometric side of topology, particularly concerning manifolds of dimension 3 and 4. In this area there have been some spectacular new developments in the last few years, due to the success of geometric methods in solving topological problems apparently beyond the reach of purely topological methods. This revolutionary work was described by Peter Scott in his influential survey article “The geometries of 3-manifolds”, published in the Society’s Bulletin for September 1983. This confirms his established position as one of the leading figures in this exciting and challenging area of research.

Indeed this paper [5] is a brilliant exposition of Thurston’s geometries for 3-manifolds. The eight homogeneous geometries are modeled on:

hyperbolic, 3-space, $H^{3}$ ,
Euclidean 3-space, $E^{3}$ ,
$S^{3}$ ,
$S^{2} \times R$ ,
$H^{2} \times R$ ,
$\tilde{S L_{2} R}$ ,
Nil,
Sol.

The first three have constant curvature and look the same in all directions. The next two have a line along which the curvature is different than in the other two. The next two have planes twisting along the line. And finally Sol is different along all three axes, for as you travel up the $z$ -axis, the $x$ -axis shrinks and the $y$ -axis expands.

The middle six geometries, neither $H^{3}$ nor Sol, all appear as Seifert fiber spaces, 3-manifolds foliated by circles. There are myriad cases which Peter had to work out, many in original ways, so this is far from a mere exposition. Its importance is illustrated by the fact that it was translated into many languages including Russian.

The geometries paper was mostly written in 1982 while Peter was visiting the University of Michigan. Eventually, in 1987, Peter moved to Ann Arbor, and he retired there in 2018.

Peter died on September 19, 2023.

Works

[1] G. P. Scott: “Compact submanifolds of 3-manifolds,” J. Lond. Math. Soc. (2) 7 : 2 (November 1973), pp. 246–250. MR 326737 Zbl 0266.57001 article

The purpose of this note is to improve a result of the author’s in [1973]. In that paper we showed that if $ M $ is a 3-manifold and $ \pi_1(M) $ is finitely generated and not a free product, then there is a compact submanifold $ N $ of $ M $ such that inclusion induces an isomorphism \[ \pi_1(N)\to\pi_1(M) .\] This result has also been proved by P. Shalen. In this paper we show that one can remove the restriction of $ \pi_1(M) $ not being a free product. The methods used are geometrical and are basically those of [1973], i.e. simplifying a compact submanifold of $ M $ by surgery on its boundary. However we need to perform the surgery with more care in order to take into account the free product structure of the group $ \pi_1(M) $.

[2] G. P. Scott: “Finitely generated 3-manifold groups are finitely presented,” J. London Math. Soc. (2) 6 : 3 (May 1973), pp. 437–440. MR 380763 Zbl 0254.57003 article

In this paper we prove that if $ G $ is a finitely generated group which is the fundamental group of a 3-manifold then $ G $ is finitely presented. The analogous result is certainly false for 4-manifolds. For any countable group is the fundamental group of some 4-manifold. This result also gives us non-trivial information about subgroups of fundamental groups of compact 3-manifolds. For there are examples of finitely presented groups with subgroups which are finitely generated but not finitely presented [Stallings 1963].

In [1971], W. Jaco proved that if $ G $ is finitely presented then $ G $ is the fundamental group of a compact 3-manifold. And in [1973], G. A. Swarup showed in addition that if $ G $ is not a free product then $ G $ is the fundamental group of a compact submanifold of $ M $. Swarup also proved our result in the case of groups with non-trivial centre. The geometric part of our proof is exactly as in Swarup’s paper [1973], but we sketch the proof for completeness. The crux of our proof is that if $ G $ is not a free product then $ G $ is the fundamental group of a compact submanifold of $ M $ and hence is finitely presented. The main part of our paper is the group theory necessary to allow the geometry to start functioning. Our result has some obvious applications to theorems of Hempel and Jaco [1972] which use the hypothesis that a certain 3-manifold group is finitely presented.

The paper is in two sections. The first section is devoted to group theory and we prove some results on free products. We then apply these results to prove our main theorem in §2.

[3] P. Scott: “Subgroups of surface groups are almost geometric,” J. London Math. Soc. (2) 17 : 3 (June 1978), pp. 555–565. A correction to this article was published in J. London Math. Soc. 32:2 (1985). MR 494062 Zbl 0412.57006 article

[4] P. Scott: “There are no fake Seifert fibre spaces with infinite $π_{1}$ ,” Ann. Math. (2) 117 : 1 (1983), pp. 35–70. MR 683801 Zbl 0516.57006 article

[5] P. Scott: “The geometries of 3-manifolds,” Bull. London Math. Soc. 15 : 5 (September 1983), pp. 401–487. A Russian translation was published as a book in 1986. MR 705527 Zbl 0561.57001 article

The theory of 3-manifolds has been revolutionised in the last few years by work of Thurston [1982a, 1982b, 1986a, 1986b]. He has shown that geometry has an important role to play in the theory in addition to the use of purely topological methods. The basic aim of this article is to discuss the various geometries which arise and explain their significance for the theory of 3-manifolds. The idea is that many 3-manifolds admit ‘nice’ metrics which give one new insight into properties of the manifolds. For the purposes of this article, the nicest metrics are those of constant curvature. An observer in a manifold with a constant curvature metric will see the same picture wherever he stands and in whichever direction he looks. Such manifolds have special topological properties. However, we will also need to consider nice metrics which are not of constant curvature. In this article, I will explain what is meant by a ‘nice’ metric and describe their classification in dimension three which is due to Thurston. Then I will discuss some of the 3-manifolds which admit these nice metrics and the relationship between their geometric and topological properties. In this introduction all manifolds and metrics will be assumed to be smooth so that the objects of interest are all Riemannian manifolds.