Celebratio Mathematica

[1] G. P. Scott: “Homotopy links,” Abh. Math. Sem. Univ. Hamburg 32 : 3–4 (September 1968), pp. 186–190. MR 236912 Zbl 0165.57102 article

A link is the oriented image of an embedding of two spheres \( S^p \), \( S^q \) in the sphere \( S^m \). Two links are equivalent if they axe concordant oriented submanifolds of \( S^m \). In [1966–1967], Haefliger showed that, under suitable conditions–smooth or piecewise-linear (\( PL \)), and codimension three–these equivalence classes form a group \( L_{p,q}^m \).

A homotopy link is a pair of maps of two spheres \( S^p \), \( S^q \) into \( S^m \) with disjoint images. Two homotopy links are equivalent if they are homotopic through homotopy links. These equivalence classes also form a group \( HL_{p,q}^m \), provided we have codimension three. These groups are the objects of study in this paper. Our theorem determines \( HL_{p,q}^m \) in the metastable range. Of course, there is a natural homomorphism \[ \psi:L_{p,q}^m\to HL_{p,q}^m \] as concordance implies homotopy. \( \psi \) is not an isomorphism, in general, as any link with one linking class zero has zero image under \( \psi \). It is interesting that \( \psi \) is not even an epimorphism, in general.

[2] G. P. Scott: Some problems in topology. Ph.D. thesis, University of Warwick, July 1968. Advised by B. J. Sanderson. phdthesis

[3] G. P. Scott: “The space of homeomorphisms of a 2-manifold,” Topology 9 : 1 (February 1970), pp. 97–109. MR 264676 Zbl 0174.26305 article

Let \( M^2 \) be a compact, connected, piecewise-linear (\( PL \)) 2-manifold, and let \( * \) be a point of \( \operatorname{int}(M^2) \). If \( N \) is a subset of \( M^2 \), then \( H_N(M^2) \) denotes the space of \( PL \) homeomorphisms of \( M^2 \) which leave points of \( N \) fixed. This paper proves that, if \( M^2 \) is not \( S^2 \) or \( P^2 \), then the identity component (and hence each component) of \( H_{*\,\cup\,\partial M}(M^2) \) is contractible. If \( M^2 \) is \( S^2 \) or \( P^2 \), then the identity component of \( H_{*\,\cup\,\partial M}(M^2) \) has the homotopy type of a circle.

The analogous results to the above in the smooth category have been proved by Eells and Earle [1967]. In the topological category, the analogous results have been proved by Hamstrom [1966].

In §0, we make some definitions, quote some basic results and give the statement of the main theorem. In §1, we give a proof of the main theorem assuming Theorem 2.1 which is proved in §2. In §3, we consider the relationship between the spaces \( H_{*\,\cup\,\partial M}(M) \), \( H_{\partial M}(M) \), \( H_*(M) \) and \( H(M) \).

[4] G. P. Scott: “Braid groups and the group of homeomorphisms of a surface,” Proc. Cambridge Philos. Soc. 68 : 3 (November 1970), pp. 605–617. MR 268889 Zbl 0203.56302 article

[5] G. P. Scott: “A note on piecewise-linear immersions,” Quart. J. Math. Oxford Ser. (2) 21 : 3 (September 1970), pp. 257–263. MR 271953 Zbl 0204.56203 article

Let \( V \), \( X \) be piecewise-linear (\( PL \)) manifolds and \( TV \), \( TX \) denote their tangent micro-bundles. Let \( \operatorname{Im}(F,X) \) denote the space of \( PL \) immersions of \( V \) in \( X \) and \( \operatorname{R}(TV,TX) \) the space of bundle monomorphisms of \( TV \) in \( TX \). In [1964], Haefliger and Poenaru defined ‘topologies’ for these spaces, by making them into semi-simplicial complexes, and showed that they were weakly homotopy equivalent. The work of Rourke and Sanderson, in [1968] and [1967], has shown that block bundles are more natural tools for use in the \( PL \) category than micro-bundles. The purpose of this note is to prove the corresponding result to [Haefliger and Poenaru 1964], using block bundles instead of micro-bundles. To do this, we have to define a new ‘topology’ on \( \operatorname{Im}(F,X) \). A result of Haefliger’s [1967, 9.2] shows that the new space has the same number of components as the old, but, in general, the higher homotopy groups will not be the same. The proofs follow those of [Haefliger and Poenaru 1964] and use its main result.

[6] G. P. Scott: “A note on the homotopy type of \( PL_2 \),” Proc. Cambridge Philos. Soc. 69 : 2 (March 1971), pp. 257–258. MR 271954 Zbl 0212.28302 article

[7] G. P. Scott: “On sufficiently large 3-manifolds,” Quart. J. Math. Oxford Ser. (2) 23 : 2 (June 1972), pp. 159–172. A correction to this article was published in Quart. J. Math. Oxford Ser. 24:1 (1973). MR 383414 Zbl 0234.57001 article

[8] D. B. A. Epstein and G. P. Scott: “A 3-manifold which is not a product,” Proc. Cambridge Philos. Soc. 74 : 3 (November 1973), pp. 445–448. MR 326736 Zbl 0277.57001 article

[9] 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 \( \pi_1(M) \).

[10] 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.

[11] G. P. Scott: “Correction: ‘On sufficiently large 3-manifolds’,” Q. J. Math., Oxf. II. Ser. 24 : 1 (1973), pp. 527–529. Correction to an article published in Quart. J. Math. Oxford Ser. 23:2 (1972). Zbl 0272.57003 article

[12] P. Scott: An introduction to 3-manifolds. Lecture notes 11, Department of Mathematics, University of Maryland (College Park, MD), 1 June 1974. MR 413106 techreport

[13] G. P. Scott: “An embedding theorem for groups with a free subgroup of finite index,” Bull. London Math. Soc. 6 : 3 (November 1974), pp. 304–306. MR 357614 Zbl 0288.20043 article

Let \( \Phi \) denote the class of free groups and \( F \) the class of finite groups. Then \( \Phi F \) denotes the class of groups with a free normal subgroup of finite index. This class is the same as the class of groups with a free subgroup of finite index.

The purpose of this note is to prove that if \( G \) is countable and in \( \Phi F \), then \( G \) can be embedded in a finitely generated group also in \( \Phi F \). This result allows one to complete the proof of the following result conjectured in [1973] by Karrass, Pietrowski and Solitar. We state the result in terms of Serre’s theory of graphs of groups [1977].

If \( G\in \Phi F \), then \( G \) is the fundamental group of a graph of groups in which all vertex groups are finite.

If \( G \) is torsion free the result is that \( G \) is free. This was proved by Stallings [1968] in the finitely generated case and extended to the general case by Swan [1969]. The result of the theorem was proved in [Karrass et al. 1973] in the finitely generated case using Stallings’ results, and extended to the countable case by Cohen [1973], who proceeds in a similar way to Swan [1969]. Cohen shows how the embedding result of this note can be used to prove the general case, again proceeding similarly to Swan. The result is needed at the point where Swan observes that a countable free group embeds in a finitely generated free group.

[14] J. L. Dyer and G. P. Scott: “Periodic automorphisms of free groups,” Comm. Algebra 3 : 3 (1975), pp. 195–201. MR 369529 Zbl 0304.20029 article

[15] P. Scott: “Normal subgroups in 3-manifold groups,” J. London Math. Soc. (2) 13 : 1 (May 1976), pp. 5–12. MR 402751 Zbl 0321.57001 article

[16] P. Scott: “Fundamental groups of non-compact 3-manifolds,” Proc. London Math. Soc. (3) 34 : 2 (1977), pp. 303–326. MR 448333 Zbl 0341.57001 article

[17] P. Scott: “Ends of pairs of groups,” J. Pure Appl. Algebra 11 : 1–3 (December 1977), pp. 179–198. MR 487104 article

In [1943], Hopf gave a definition of the number of ends, \( e(G) \), of a finitely generated (f.g.) group \( G \). In [1950], Specker extended his definition to cover arbitrary groups. They showed that the function \( e(G) \) could only take the values \( 0,\,1,\,2 \) or \( \infty \) and they characterised those groups with two ends. In [1971], Stallings characterised those f.g. groups \( G \) with at least two ends. We will say that a group \( G \) splits over a subgroup \( C \) if either \( G \) is a HNN extension \( A*_C \) or \( G \) is an amalgamated free product \( A*_CB \) with \( A\neq C\neq B \). Then Stallings’ result says that a f.g. group \( G \) has \( e(G)\geq 2 \) if and only if \( G \) splits over some finite subgroup.

This paper arose from an attempt to generalise the above result to groups which split over infinite subgroups. There is a natural definition, due to Houghton [1974], of the number of ends, \( e(G,C) \), of a pair of groups \( (G,C) \) where \( C \) is a subgroup of \( G \). Houghton uses rather different terminology from ours and his results are stated for topological groups. For simplicity, when quoting his results, we will rewrite them in our terminology and so as to apply to discrete groups only. Presumably, the main results of this paper can be generalised to topological groups in the same way that Abels [1974] generalised Stallings’ result [1971].

[18] 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

[19] P. Scott and T. Wall: “Topological methods in group theory,” pp. 137–203 in Homological group theory (Durham, UK, September 1977). Edited by C. T. C. Wall. London Mathematical Society Lecture Note Series 36. Cambridge University Press, 1979. MR 564422 Zbl 0423.20023 incollection

[20] P. Scott: “A new proof of the annulus and torus theorems,” Am. J. Math. 102 : 2 (1980), pp. 241–277. MR 564473 Zbl 0439.57004 article

[21] P. Scott: “The classification of compact 3-manifolds,” pp. 3–7 in Low-dimensional topology (Bangor, UK, 2–5 July 1979). Edited by R. Brown and T. L. Thickstun. London Mathematical Society Lecture Note Series 48. Cambridge University Press, 1982. MR 662423 Zbl 0483.57001 incollection

[22] article M. Freedman, J. Hass, and P. Scott: “Closed geodesics on surfaces,” Bull. London Math. Soc. 14 : 5 (1982), pp. 385–391. MR 0671777 Zbl 0476.53026

Let \( M^2 \) be a closed Riemannian 2-manifold, and let \( \alpha \) denote a non-trivial element of \( \pi_1(M) \). The set of all loops in \( M \) which represent a has a shortest element \( f:\mathbb{S}^1 \rightarrow M \), which can be assumed smooth and which will be a closed geodesic. (We say a loop represents \( \alpha \) when it represents any conjugate of \( \alpha \). Such a loop need not pass through the base point of \( M \).) The map \( f \) cannot be unique, because \( f \) is not necessarily parametrised by arc length and because there is no base point. In general, even the image set of a shortest loop is not unique. In this note, we prove the following result.

Let \( M^2 \) be a closed, Riemannian 2-manifold and let \( \alpha \) denote a non-trivial element of \( \pi_1M \) which is represented by a two-sided embedded loop \( C \). Then any shortest loop \( f:\mathbb{S}^1 \rightarrow M \) representing \( \alpha \) is either an embedding or a double cover of a one-sided embedded curve \( K \). In the second case, \( C \) bounds a Moebius band in \( M \) and \( K \) is isotopic to the centre of this band.

Joel Hass	Related	Volume
M. H. Freedman	Related	Volume

[23] P. Scott: “There are no fake Seifert fibre spaces with infinite \( \pi_1 \),” Ann. Math. (2) 117 : 1 (1983), pp. 35–70. MR 683801 Zbl 0516.57006 article

[24] article M. Freedman, J. Hass, and P. Scott: “Least area incompressible surfaces in 3-manifolds,” Invent. Math. 71 : 3 (1983), pp. 609–642. MR 0695910 Zbl 0482.53045

Let \( M \) be a Riemannian manifold and let \( F \) be a closed surface. A map \( f:F\rightarrow M \) is called least area if the area of \( f \) is less than the area of any homotopic map from \( F \) to \( M \). Note that least area maps are always minimal surfaces, but that in general minimal surfaces are not least area as they represent only local stationary points for the area function.

In this paper we shall consider the possible singularities of such immersions. Our results show that the general philosophy is that least area surfaces intersect least, meaning that the intersections and self-intersections of least area immersions are as small as their homotopy classes allow, when measured correctly.

Joel Hass	Related	Volume
M. H. Freedman	Related	Volume

[25] 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.

[26] P. Scott: “Strong annulus and torus theorems and the enclosing property of characteristic submanifolds of 3-manifolds,” Quart. J. Math. Oxford Ser. (2) 35 : 4 (December 1984), pp. 485–506. MR 767777 Zbl 0589.57006 article

In [1980], I gave new proofs of the Annulus and Torus Theorems. In this paper, I show how the methods of [1980] can be slightly refined so as to obtain stronger versions of these theorems. Then I discuss how these results can be used to give an alternative approach to the theory of characteristic submanifolds of Haken 3-manifolds due to Johannson [1979] and Jaco and Shalen [1979]. The virtue of this new approach is that it does not use the existence of a hierarchy for a Haken manifold. None of the main theorems in this paper is new. But the technical results in the first two sections of the paper are new, and I think that the version of the Annulus Theorem proved here has not been stated before, although it is certainly implicit in [Johannson 1979] and [Jaco and Shalen 1979].

[27] M. Brin, K. Johannson, and P. Scott: “Totally peripheral 3-manifolds,” Pac. J. Math. 118 : 1 (1985), pp. 37–51. MR 783014 Zbl 0525.57010 article

Klaus Johannson	Related
Matthew G. Brin	Related

[28] J. Hass and P. Scott: “Intersections of curves on surfaces,” Israel J. Math. 51 : 1–2 (1985), pp. 90–120. MR 804478 Zbl 0576.57009 article

The authors consider curves on surfaces which have more intersections than the least possible in their homotopy class.

Let \( f \) be a general position arc or loop on an orientable surface \( F \) which is homotopic to an embedding but not embedded. Then there is an embedded 1-gon or 2-gon on \( F \) bounded by part of the image of \( f \).

Let \( f \) be a general position arc or loop on an orientable surface \( F \) which has excess self-intersection. Then there is a singular 1-gon or 2-gon on \( F \) bounded by part of the image of \( f \).

Examples are given showing that analogous results for the case of two curves on a surface do not hold except in the well-known special case when each curve is simple.

[29] P. Scott: “Correction to: ‘Subgroups of surface groups are almost geometric’,” J. London Math. Soc. (2) 32 : 2 (1985), pp. 217–220. Correction to an article published in J. London Math. Soc. 17:3 (1978). MR 811778 Zbl 0581.57005 article

[30] P. Scott: “Homotopy implies isotopy for some Seifert fibre spaces,” Topology 24 : 3 (1985), pp. 341–351. MR 815484 Zbl 0576.57012 article

Let \( M \) be a closed orientable, irreducible 3-manifold. It is a long standing problem to decide if homotopic homeomorphisms of \( M \) must be isotopic. No counterexamples are known. Waldhausen [1968] showed that this is true if \( M \) is Haken. See also [Laudenbach 1974]. [Hodgson and Rubinstein 1985] and Bonahon [1983] independently showed that this result holds for lens spaces. Birman and Rubinstein [1984] have proved the same result for certain Seifert fibre spaces, including the binary octahedral spaces, and for some other non-Haken 3-manifolds. Asano [1978] and Rubinstein [1979] have proved the same result for prism manifolds. It seems reasonable to conjecture that this result should hold whenever \( \pi_1(M) \) is infinite, but the case when \( \pi_1(M) \) is finite seems more doubtful. In this paper, I consider the irreducible non-Haken Seifert fibre spaces which have infinite fundamental group and prove that, for most of these manifolds, this conjecture holds.

[31] W. H. Meeks, III and P. Scott: “Finite group actions on 3-manifolds,” Invent. Math. 86 : 2 (1986), pp. 287–346. MR 856847 Zbl 0626.57006 article

[32] P. Scott: Geometrii na trekhmernykh mnogoobraziyakh [The geometries of 3-manifolds]. Matematika. Novoe v Zarubezhnoj 39. Mir (Moscow), 1986. Russian translation of an article published in Bull. London Math. Soc. 15:5 (1983). Zbl 0662.57001 book

[33]J. Hass, H. Rubinstein, and P. Scott: “Covering spaces of 3-manifolds,” Bull. Am. Math. Soc., New Ser. 16 : 1 (January 1987), pp. 117–119. MR 866028 Zbl 0624.57016 article

J. Hyam Rubinstein	Related	Volume
Joel Hass	Related	Volume

[34] R. Gulliver and P. Scott: “Least area surfaces can have excess triple points,” Topology 26 : 3 (1987), pp. 345–359. MR 899054 Zbl 0636.53011 article

[35] J. Hass and P. Scott: “The existence of least area surfaces in 3-manifolds,” Trans. Am. Math. Soc. 310 : 1 (November 1988), pp. 87–114. MR 965747 Zbl 0711.53008 article

[36]J. Hass, H. Rubinstein, and P. Scott: “Compactifying coverings of closed 3-manifolds,” J. Differ. Geom. 30 : 3 (1989), pp. 817–832. MR 1021374 Zbl 0693.57011 article

J. Hyam Rubinstein	Related	Volume
Joel Hass	Related	Volume

[37] P. Scott and T. Tucker: “Some examples of exotic noncompact 3-manifolds,” Quart. J. Math. Oxford Ser. (2) 40 : 160 (1989), pp. 481–499. MR 1033220 Zbl 0692.57006 article

[38] G. P. Scott and G. A. Swarup: “Geometric finiteness of certain Kleinian groups,” Proc. Am. Math. Soc. 109 : 3 (1990), pp. 765–768. MR 1013981 Zbl 0699.30040 article

[39] G. P. Scott and G. A. Swarup: “Least area tori in 3-manifolds,” Proc. Am. Math. Soc. 116 : 4 (December 1992), pp. 1143–1151. MR 1131040 Zbl 0818.57010 article

[40] J. Hass and P. Scott: “Homotopy equivalence and homeomorphism of 3-manifolds,” Topology 31 : 3 (July 1992), pp. 493–517. MR 1174254 Zbl 0771.57007 article

[41] J. Hass and P. Scott: “Curve flows on surfaces and intersections of curves,” pp. 415–421 in Differential geometry (Los Angeles, 8–28 July 1990), Part 3: Riemannian geometry. Edited by R. Greene and S.-T. Yau. Proceedings of Symposia in Pure Mathematics 54. Amererican Mathematical Society (Providence, RI), 1993. MR 1216633 Zbl 0793.53006 incollection

[42] J. Hass and P. Scott: “Homotopy and isotopy in dimension three,” Comment. Math. Helv. 68 : 3 (1993), pp. 341–364. MR 1236759 Zbl 0805.57008 article

[43] L. Harris and P. Scott: “Finding irreducible submanifolds of 3-manifolds,” Bull. London Math. Soc. 25 : 6 (1993), pp. 591–597. MR 1245087 Zbl 0814.57011 article

[44] J. Hass and P. Scott: “Shortening curves on surfaces,” Topology 33 : 1 (January 1994), pp. 25–43. MR 1259513 Zbl 0798.58019 article

[45] L. Harris and P. Scott: “Non-compact totally peripheral 3-manifolds,” Pac. J. Math. 167 : 1 (1995), pp. 119–127. MR 1318166 Zbl 0823.57014 article

[46] L. Harris and P. Scott: “The uniqueness of compact cores for 3-manifolds,” Pac. J. Math. 172 : 1 (1996), pp. 139–150. MR 1379290 Zbl 0865.57016 article

[47] P. Scott: “The symmetry of intersection numbers in group theory,” Geom. Topol. 2 (1998), pp. 11–29. A correction to this article was published in Geom. Topol. 2 (1998). MR 1608688 Zbl 0897.20029 ArXiv math/9712212 article

[48] P. Scott: “Correction to: ‘The symmetry of intersection numbers in group theory’,” Geom. Topol. 2 (1998), pp. 333–335. Correction to an article published in Geom. Topol. 2 (1998). MR 1639541 Zbl 1382.20048 article

[49] C. Adams, J. Hass, and P. Scott: “Simple closed geodesics in hyperbolic 3-manifolds,” Bull. London Math. Soc. 31 : 1 (January 1999), pp. 81–86. MR 1650997 Zbl 0955.53025 ArXiv math/9801071 article

Joel Hass	Related	Volume
Colin C. Adams	Related

[50] J. Hass and P. Scott: “Configurations of curves and geodesics on surfaces,” pp. 201–213 in Proceedings of the Kirbyfest (Berkeley, CA, 22–26 June 1998). Edited by J. Hass and M. G. Scharlemann. Geometry & Topology Monographs 2. Geometry & Topology Publications (Coventry, UK), 1999. MR 1734409 Zbl 1035.53053 ArXiv math/9903130 incollection

Joel Hass	Related	Volume
Robion C. Kirby	Related	Volume

[51] P. Scott and G. A. Swarup: “Splittings of groups and intersection numbers,” Geom. Topol. 4 (2000), pp. 179–218. MR 1772808 Zbl 0983.20024 ArXiv math/9906004 article

[52] B. Leeb and P. Scott: “A geometric characteristic splitting in all dimensions,” Comment. Math. Helv. 75 : 2 (2000), pp. 201–215. MR 1774701 Zbl 0979.53037 ArXiv math/9801015 article

[53] G. P. Scott and G. A. Swarup: “An algebraic annulus theorem,” Pac. J. Math. 196 : 2 (2000), pp. 461–506. MR 1800588 Zbl 0984.20028 ArXiv math/9608206 article

[54] P. Scott and G. A. Swarup: “Canonical splittings of groups and 3-manifolds,” Trans. Am. Math. Soc. 353 : 12 (2001), pp. 4973–5001. MR 1852090 Zbl 0981.57002 ArXiv math/0107232 article

[55] P. Scott and G. A. Swarup: “Regular neighbourhoods and canonical decompositions for groups,” Electron. Res. Announc. Am. Math. Soc. 8 : 3 (2002), pp. 20–28. A much longer monograph with the same title was published in 2003. MR 1928498 Zbl 1057.20032 article

[56] P. Scott and G. A. Swarup: Regular neighbourhoods and canonical decompositions for groups. Astérisque 289. 2003. A short article with the same title as this monograph was published in Electron. Res. Announc. Amer. Math. Soc. 8:3 (2002). MR 2032389 Zbl 1036.20028 ArXiv math/0110210 book

[57] G. Niblo, M. Sageev, P. Scott, and G. A. Swarup: “Minimal cubings,” Int. J. Algebra Comput. 15 : 2 (2005), pp. 343–366. MR 2142089 Zbl 1085.20026 ArXiv math/0401133 article

Gadde A. Swarup	Related
Michah El-Yakim Sageev	Related
Graham A. Niblo	Related

[58] M. Boileau, K. Johannson, and P. Scott: “Low-dimensional manifolds,” Oberwolfach Rep. 2 : 4 (2005), pp. 2519–2569. Report 45/2005. Zbl 1110.57300 article

Klaus Johannson	Related
Michel Charles Boileau	Related

[59] P. Scott and G. A. Swarup: Annulus-torus decompositions for Poincaré duality pairs. Preprint, February 2008. Dedicated to Terry Wall on his 70th birthday. techreport

Gadde A. Swarup	Related
C. T. C. Wall	Related

[60] M. Mj, P. Scott, and G. Swarup: “Splittings and \( C \)-complexes,” Algebr. Geom. Topol. 9 : 4 (2009), pp. 1971–1986. MR 2550463 Zbl 1225.20035 ArXiv 0906.1149 article

Gadde A. Swarup	Related
Mahan Mj	Related

[61] P. Scott and H. Short: “The homeomorphism problem for closed 3-manifolds,” Algebr. Geom. Topol. 14 : 4 (2014), pp. 2431–2444. MR 3331689 Zbl 1311.57025 ArXiv 1211.0264 article

[62] J. Hass and P. Scott: “Simplicial energy and simplicial harmonic maps,” Asian J. Math. 19 : 4 (2015), pp. 593–636. MR 3423736 Zbl 1332.57024 ArXiv 1206.2574 article

[63] P. Scott and G. A. Swarup: Canonical decompositions for Poincaré duality pairs. Preprint, September 2018. ArXiv math/0703890 techreport

[64] M. Neumann-Coto and P. Scott: “A property of closed geodesics on hyperbolic surfaces,” Groups Geom. Dyn 17 : 1 (2023), pp. 1–33. MR 4563327 Zbl 1515.53039 article

G. Peter Scott

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