Finitely presented groups: where to go from here

This upcoming conference in group theory, to be held at the City College of New York Oct 2-3, 2009. The conference covers a range of topics in group theory, with speakers who are working at the forefront of research in group theory. The participants are drawn from not only the active community of researchers in the greater New York area, but also from institutions across the US and internationally. In 1865 Cayley dened a group as a combination of its symbols, making it possible to not only describe well-known groups in terms of a nite amount of data, termed a presenta- tion. It was not until 1882 that Walther von Dyck explicitly dened a presentation and in particular what is meant by a nitely presented group. The study of nitely presented groups became particularly important with the introduction of the fundamental group by Poincare in 1895, the discovery of knot groups by Wirtinger in 1905 and the proof by Tietze in 1908 that the fundamental group of a nite dimensional, compact, connected manifold is nitely presented. The notion of a presentation underlined the existence of presentations of a host of groups which had never been investigated. So it was nat- ural, but involved a great deal of insight, when Dehn, in 1912, raised three problems about nitely presented groups as a whole. The rst, and perhaps the most celebrated of these, is the word problem: given a nite presentation, is there an algorithm which decides whether or not any word in the generators is equal to 1 as a consequence of the dening relators? This problem, together with the conjugacy and isomorphism prob- lems have played a major role in the development of both combinatorial and geometric group theory. The theory of nitely presented groups impacts and is impacted by other branches of mathematics, especially algebraic topology and recursive function theory. Today the study of nitely presented groups oers as many challenges as it did in 1912 when Dehn raised his famous algorithmic problems, despite remarkable progress. In order to explain the rationale behind this proposed conference, it is necessary to brie y discuss some aspects of the theory at this time. First, in 1961, Graham Higman proved that a nitely generated group is a subgroup of a nitely presented group if and only if it can be dened in terms of nitely many generators linked by a recursively enumerable set of dening relations. This established a bond between recursive function theory and the subgroup structure of nitely presented groups and explained why almost all problems about nitely presented groups are algorithmically undecidable. Attempts to solve these algorithmic problems led to small cancellation theory, a theory which is implicit already in Dehn's earlier work on the nature of fundamental groups of surface groups and the work of Tartakovskii. The underlying geometric nature of these algebraic ideas was brought out in the work of Cannon, Thurston, Gromov and Rips and culminated in the theory of hyperbolic groups. Some of the underlying nature of the geodesics in the Cayley graph of these hyperbolic and related groups turned out to be governed by one of the facets of computer science, language theory in the guise of nite state automata. This was hinted at in some of the work of Bob Gilman and has given rise to the theory of automatic groups. The work of Makanin and Razborov has brought logic back into the study of groups via their elementary theories. This has culminated in the resolution by Sela, Kharlampovich and Miasnikov of an old problem of Tarski. Tarski asked, in particular, whether the elementary theory of a free group of rank two is the same as the elementary theory of a free group of rank three. This has turned out to involve what are now called, by some, limit groups. Somewhat surprisingly many years ago Baumslag proved that the fundamental groups of orientable surfaces are limit groups which brings one back to Dehn's original object of study. This discussion highlights and explains the objective of this proposed conference, namely for several leaders in the eld to sketch new directions for future development. Proposed speakers are William Thurston, Jim Cannon, Zlil Sela, Sasha Razborov, John Stallings, Mladen Bestvina, David Epstein, Mark Sapir. The main conference would be preceded by a day of expository lectures on related subjects designed, in the main, for graduate students. This aspect of the overall conference underlines one of the underlying aims, namely the participation of as many graduate students and postdocs as funding will permit,some under the auspices of CAISS, the Center for Algorithms and Interactive Scientic Software, a small research center of the City College of New York. CAISS has now taken over the running and continued maintenance of the New York Group Theory Seminar, perhaps the longest running research seminar in New York, more than 50 years. This seminar is a vehicle for the dissemination of cutting edge research in group theory and is one of the most important seminars in the subject in the world. Many researchers gravitate towards this seminar which plays an important role in the life of many of them in the greater New York area. So it is natural that a major conference in group theory be held in New York at the City College. The anticipated conference format is a two-day conference. The rst day's talks will be at a more introductory level, aimed at graduate students, postdocs and researchers in related elds. The aim of the rst day is to serve as an introduction and is aimed at a broad range of mathematical researchers. The second day's talks will cover the cutting-edge topics described above.