This talk is about new deve lopments in the theory of soluble (aka solvable) g roups. In the nineteen sixties\, seventies\, and e ighties\, the theory of infinite solvable groups d eveloped quietly and unnoticed except by experts i n group theory. Philip Hall'\;s work was a majo r impact and inspiration but before that there had been pioneering work of Maltsev and Hirsch. In th e eighties\, new vigour was brought to the subject through the work of Bieri and Strebel: the BNS in variant was born and for the first time there appe ared a connection between the abstract algebra of Maltsev\, Hirsch and Hall\, and the topological an d geometric insights of Thurston\, Stallings and D unwoody.

Nowadays\, solvable groups are vi tal for a number of reasons. They are a primary so urce of examples of amenable groups\, exhibiting a rich display of properties as shown in work of\, for example\, Erschler. There is an intimate conne ction with 3 manifold theory: we imagine that 3 ma nifolds revolve around hyperbolic geometry. But if hyperbolic geometry is the sun at the centre of t he 3 manifold universe then Sol Nil S^3 S^2xR and R^3 (5 of the remaining 7 geometries identified by Thurstons geometrization programme must be the ou tlying planets: all virtually solvable and very mu ch full of life. We might think of these solvable geometries as in some way the trivial cases. But t hey 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 thro ugh solvable infinite groups and culminates in a s tudy of random walks on Cayley graphs including re cent work joint with Lorensen as well as independe nt results of Jacoboni. LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR