Ian Chiswell

An ordinary tree defines an integer-valued metric on its set of vertices, just by taking the shortest path between two vertices, and defining their distance apart to be the number of edges in the path. The idea of a Λ-tree (Λ being an ordered abelian group) is obtained by considering Λ-valued metrics having certain properties in common with this metric. Some aspects of the theory of group actions on ordinary trees can be generalised to group actions (as isometries) on Λ-trees. However, there is at present no good analogue of the Bass-Serre structure theory for groups acting on ordinary trees. Much of my work has been motivated by the problem of developing a structure theory for groups acting on Λ-trees. My investigations on Λ-trees have continued. A recent theme has been generalisations of free groups, involving words indexed by ordered abelian groups instead of integers. This is ongoing work with, amomgst others, my colleague T.W. Müller at QMUL. It is inspired by several visits to the College by V.N. Remeslennikov. I have also been considering orderings of groups, following a visit by P.A. Linnell. I have investigated locally invariant orders on groups, an idea introduced by D. Promislow in the mid-eighties. They are relevant to recent work by T. Delzant and by S. Hair on unique product groups. Tree-free groups (those having an action as isometries on a Λ-tree which is free and without inversions) have a locally invariant order, and it follows that they are unique product groups.

The importance of Λ-trees was demonstrated by Morgan and Shalen, in a paper which uses R-trees to generalise the Thurston compactification of Teichmüller space. There has been considerable progress in the study of group actions, and spaces of group actions, on R-trees. The most notable result is obtained by combining further work of Morgan and Shalen with work of Rips and others. A finitely generated group has a free action on an R-tree if and only if it is a free product of free abelian groups and surface groups, with the exception of the groups of the non-orientable surfaces of genus less than or equal to 3.

Non-standard free groups act freely on Λ-trees, where Λ is a non-standard model of the integers, hence so do their subgroups. These subgoups can be characterised group-theoretically, as the "locally fully residually free" groups. The non-abelian subgroups of non-standard free groups are precisely the groups with the same universal theory as the free group of rank 2. These  results are  due to Gaglione and Spellman and to Remeslennikov. Recently, Myasnikov and Kharlampovich, using a powerful new technique (algebraic geometry over groups), have announced proofs of two long-standing conjectures of Tarski. The elementary theories of all the non-abelian free groups coincide, and the elementary theory of a free group is decidable. They have published several relevant papers, but not the final result. I understand that it will be published in 2006 in the AMS CRM series.The first conjecture has also been proved by Z. Sela in a series of six papers (available on his website), most of which have now appeared in print. He also proves other results, including a cnaracterisation of groups elementarily equivalent to the non-abelian free groups. In this area, I have written a paper with Remeslennikov which interprets old results on equations in free groups first in terms of algebraic geometry over groups, then in terms of non-standard free groups. The results are then proved by investigating cancellation in non-standard free groups.

There are also connections with model theory and the more general trees considered in two volumes of the AMS Memoirs, one by Adeleke and Neumann and one by Bowditch. Another theme of my work has been to show that various notions of generalised trees can be interpreted in terms of Λ-trees  (again Λ is a non-standard model of the integers). The latest of my papers on this is about the D-relations of Adelele and Neumann. A previous paper studied the connection between  Λ-trees and Dunwoody's notion of protree.