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A random walk around soluble group theory

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NPCW04 - Approximation, deformation, quasification

Co-authors: Karl Lorensen (Penn State Altoona), Armando Martino (Southampton), Conchita Martinez Perez (Zaragoza), Lison Jacoboni (Orsay)

This talk is about new developments in the theory of soluble (aka solvable) groups. In the nineteen sixties, seventies, and eighties, the theory of infinite solvable groups developed quietly and unnoticed except by experts in group theory. Philip Hall's work was a major impact and inspiration but before that there had been pioneering work of Maltsev and Hirsch. In the eighties, new vigour was brought to the subject through the work of Bieri and Strebel: the BNS invariant was born and for the first time there appeared a connection between the abstract algebra of Maltsev, Hirsch and Hall, and the topological and geometric insights of Thurston, Stallings and Dunwoody.

Nowadays, solvable groups are vital for a number of reasons. They are a primary source of examples of amenable groups, exhibiting a rich display of properties as shown in work of, for example, Erschler. There is an intimate connection with 3 manifold theory: we imagine that 3 manifolds revolve around hyperbolic geometry. But if hyperbolic geometry is the sun at the centre of the 3 manifold universe then Sol Nil S3 S2xR and R^3 (5 of the remaining 7 geometries identified by Thurstons geometrization programme must be the outlying planets: all virtually solvable and very much full of life. We might think of these solvable geometries as in some way the trivial cases. But they have also been an inspiration both in algebra and in geometry.

In this talk I will take a survey that leads in a meandering way through solvable infinite groups and culminates in a study of random walks on Cayley graphs including recent work joint with Lorensen as well as independent results of Jacoboni.

This talk is part of the Isaac Newton Institute Seminar Series series.

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