We are pleased to announce a two-day conference on finitely presented groups at The City College Of New York on October 1^st and 2^nd, 2009, funded by the National Science Foundation and City College, and sponsored by CAISS, the Center for Algorithms and Interactive Scientific Software. The conference will further the understanding of finitely presented groups, and point the way to new directions of study, research and investigation.

Today the study of finitely presented groups offers as many challenges as it did in 1911 when Dehn raised his famous algorithmic problems, despite remarkable progress. In order to explain the rationale behind this proposed conference, it is necessary to briefly discuss some aspects of the theory at this time. First, in 1961, Graham Higman proved that a finitely generated group is a subgroup of a finitely presented group if and only if it can be defined in terms of finitely many generators linked by a recursively enumerable set of defining relations. This established a bond between recursive function theory and the subgroup structure of finitely presented groups, and explained why almost all problems about finitely 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 finite state automata. This was hinted at in some of the work of Bob Gilman and has given rise to the solution 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.

Many problems still remain unsolved. In particular the very nature of the subgroup structure of finitely presented groups remains a mystery and the search for invariants has seemingly come to a halt. The objective of this proposed conference is to bring together some prominent mathematicians to discuss possible new directions for future development.

The first day of the conference will consist of expository lectures designed mainly for graduate students. This aspect of the conference will underline one of its principal aims, namely, the participation of as many graduate students and postdocs as funding will permit.  There will be two talks, one in the morning by Gilbert Baumslag and one in the afternoon by Alexei Miasnikov. The intervening time will be used for discussion. Mladen Bestvina, Jim Cannon, Peter Kropholler, Zlil Sela, and Bill Thurston have agreed to speak on the second day of the conference. Further information and a detailed outline of topics will be made available on our website in due course.

The conference is being organized by Gilbert Baumslag and Sean Cleary under the auspices of CAISS, the Center for Algorithms and Interactive Scientific Software, a small research center of The City College of New York, City University of New York, with the help of Vladimir Shpilrain, John Murray and Olga Mikhlina.

Conference Poster

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