Martin Dunwoody

Last updated

Martin John Dunwoody (born 3 November 1938) is an emeritus professor of Mathematics at the University of Southampton, England.

He earned his PhD in 1964 from the Australian National University. He held positions at the University of Sussex before becoming a professor at the University of Southampton in 1992. He has been emeritus professor since 2003. [1]

Dunwoody works on geometric group theory and low-dimensional topology. He is a leading expert in splittings and accessibility of discrete groups, groups acting on graphs and trees, JSJ-decompositions, the topology of 3-manifolds and the structure of their fundamental groups.

Since 1971 several mathematicians have been working on Wall's conjecture, posed by Wall in a 1971 paper, [2] which said that all finitely generated groups are accessible. Roughly, this means that every finitely generated group can be constructed from finite and one-ended groups via a finite number of amalgamated free products and HNN extensions over finite subgroups. In view of the Stallings theorem about ends of groups, one-ended groups are precisely those finitely generated infinite groups that cannot be decomposed nontrivially as amalgamated products or HNN-extensions over finite subgroups. Dunwoody proved the Wall conjecture for finitely presented groups in 1985. [3] In 1991 he finally disproved Wall's conjecture by finding a finitely generated group that is not accessible. [4]

Dunwoody found a graph-theoretic proof of Stallings' theorem about ends of groups in 1982, by constructing certain tree-like automorphism invariant graph decompositions. This work has been developed to an important theory in the book Groups acting on graphs, Cambridge University Press, 1989, with Warren Dicks. In 2002 Dunwoody put forward a proposed proof of the Poincaré conjecture. [5] The proof generated considerable interest among mathematicians, but a mistake was quickly discovered and the proof was withdrawn. [6] The conjecture was later proven by Grigori Perelman, following the program of Richard S. Hamilton.

Related Research Articles

In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries . In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture.

Geometric group theory

Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act.

3-manifold Space that locally looks like Euclidean 3-dimensional space

In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.

C. T. C. Wall British mathematician

Charles Terence Clegg "Terry" Wall is a British mathematician, educated at Marlborough and Trinity College, Cambridge. He is an emeritus professor of the University of Liverpool, where he was first appointed professor in 1965. From 1978 to 1980 he was the president of the London Mathematical Society.

In mathematics, the HNN extension is an important construction of combinatorial group theory.

In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups into the vertex groups. There is a unique group, called the fundamental group, canonically associated to each finite connected graph of groups. It admits an orientation-preserving action on a tree: the original graph of groups can be recovered from the quotient graph and the stabiliser subgroups. This theory, commonly referred to as Bass–Serre theory, is due to the work of Hyman Bass and Jean-Pierre Serre.

Brian Hayward Bowditch is a British mathematician known for his contributions to geometry and topology, particularly in the areas of geometric group theory and low-dimensional topology. He is also known for solving the angel problem. Bowditch holds a chaired Professor appointment in Mathematics at the University of Warwick.

Bass–Serre theory is a part of the mathematical subject of group theory that deals with analyzing the algebraic structure of groups acting by automorphisms on simplicial trees. The theory relates group actions on trees with decomposing groups as iterated applications of the operations of free product with amalgamation and HNN extension, via the notion of the fundamental group of a graph of groups. Bass–Serre theory can be regarded as one-dimensional version of the orbifold theory.

In the mathematical field of group theory, the Kurosh subgroup theorem describes the algebraic structure of subgroups of free products of groups. The theorem was obtained by Alexander Kurosh, a Russian mathematician, in 1934. Informally, the theorem says that every subgroup of a free product is itself a free product of a free group and of its intersections with the conjugates of the factors of the original free product.

In the mathematical subject of group theory, the Grushko theorem or the Grushko–Neumann theorem is a theorem stating that the rank of a free product of two groups is equal to the sum of the ranks of the two free factors. The theorem was first obtained in a 1940 article of Grushko and then, independently, in a 1943 article of Neumann.

John R. Stallings American mathematician

John Robert Stallings Jr. was a mathematician known for his seminal contributions to geometric group theory and 3-manifold topology. Stallings was a Professor Emeritus in the Department of Mathematics at the University of California at Berkeley where he had been a faculty member since 1967. He published over 50 papers, predominantly in the areas of geometric group theory and the topology of 3-manifolds. Stallings' most important contributions include a proof, in a 1960 paper, of the Poincaré Conjecture in dimensions greater than six and a proof, in a 1971 paper, of the Stallings theorem about ends of groups.

In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G has more than one end if and only if the group G admits a nontrivial decomposition as an amalgamated free product or an HNN extension over a finite subgroup. In the modern language of Bass–Serre theory the theorem says that a finitely generated group G has more than one end if and only if G admits a nontrivial action on a simplicial tree with finite edge-stabilizers and without edge-inversions.

In the mathematical subject of group theory, the Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957. In 2011, a strengthened version of the conjecture was proved independently by Joel Friedman and by Igor Mineyev.

Zlil Sela Israeli mathematician

Zlil Sela is an Israeli mathematician working in the area of geometric group theory. He is a Professor of Mathematics at the Hebrew University of Jerusalem. Sela is known for the solution of the isomorphism problem for torsion-free word-hyperbolic groups and for the solution of the Tarski conjecture about equivalence of first-order theories of finitely generated non-abelian free groups.

James W. Cannon is an American mathematician working in the areas of low-dimensional topology and geometric group theory. He was an Orson Pratt Professor of Mathematics at Brigham Young University.

Mladen Bestvina Croatian-American mathematician

Mladen Bestvina is a Croatian-American mathematician working in the area of geometric group theory. He is a Distinguished Professor in the Department of Mathematics at the University of Utah.

In mathematics, the Muller–Schupp theorem states that a finitely generated group G has context-free word problem if and only if G is virtually free. The theorem was proved by David Muller and Paul Schupp in 1983.

In the mathematical subject of group theory, the Howson property, also known as the finitely generated intersection property (FGIP), is the property of a group saying that the intersection of any two finitely generated subgroups of this group is again finitely generated. The property is named after Albert G. Howson who in a 1954 paper established that free groups have this property.

Godfrey Peter Scott, known as Peter Scott, is a British mathematician, known for the Scott core theorem.

Stephen M. Gersten was an American mathematician, specializing in finitely presented groups and their geometric properties.

References

  1. ↑ "Professor Martin Dunwoody | Mathematical Sciences | University of Southampton". www.southampton.ac.uk. Retrieved 17 June 2019.
  2. ↑ Wall, C. T. C., Pairs of relative cohomological dimension one. Journal of Pure and Applied Algebra, vol. 1 (1971), no. 2, pp. 141–154
  3. ↑ Dunwoody, M. J., The accessibility of finitely presented groups. Inventiones Mathematicae, vol. 81 (1985), no. 3, pp. 449–45
  4. ↑ Dunwoody, Martin J. An inaccessible group. Geometric group theory, Vol. 1 (Sussex, 1991), pp. 75–78, London Math. Soc. Lecture Note Ser., 181, Cambridge University Press, Cambridge, 1993. ISBN   0-521-43529-3
  5. ↑ British professor chases solution to $1m maths prize. Daily Telegraph, 14 April 2002. Accessed 23 January. 2010
  6. ↑ George G. Szpiro, The secret life of numbers: 50 easy pieces on how mathematicians work and think. National Academies Press, 2006. ISBN   0-309-09658-8; p. 19
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.