Workshop on the Extended Family of R. Thompson Groups.

The proposed project is a workshop, with primary focus the theory of the extended family of R. Thompson groups.

The extended family of R. Thompson groups are an important family of (generally) finitely presented simple groups (or, nearly simple) which contain the first known examples of finitely presented infinite simple groups. The theory of this family intersects other areas of mathematics in often surprising and profound ways; providing examples and counterexamples to questions from areas such as logic (R. J. Thompson), solvable and unsolvable problems in group theory (R. J. Thompson and R. McKenzie), homotopy and category theory (P. Freyd and A. Heller), shape theory (J Dydak and H. Hastings), Teichmuller theory and mapping class groups (R. Penner) and various other areas as well.

The workshop will serve to both educate new and established researchers on the State of the Art in this dynamic area, as well as highlight some of the many connections between the theory of these groups and other areas of Mathematics. In particular, the workshop will host three mini-courses on the connections of the extended family of the R. Thompson groups to various outward-reaching areas of mathematics. The titles and presenters of these workshops are given below:

1) Semigroups, 'etale topological groupoids, C*-algebras, and Thompson groups; Mark V. Lawson.
2) Braids, logic and geometric presentations for Thompson's groups; Patrick Dehornoy.
3) Thurston's piecewise integral projective groups; Vladimir Sergiescu.

The workshop will also host a problem session focussing on these groups and the broader topics associated with them, and will publish summaries of the minicourses and the problem session discussion in a topical research journal.

The three main areas of impact are likely to be research impact, educational impact, and within a range of external applications.

The primary research impact of the proposed workshop is that it will encourage, by its main theme, new and established researchers to work on a broader base; instead of simply thinking of research in the extended family of R. Thompson groups for its own sake, also thinking of how this research reaches into other areas, encouraging a broader perspective which is, in turn, intended to generate more collaborative and cross-disciplinarian research. The theory of the R. Thompson groups provides a natural foundation for such broader research due to its ubiquitous nature; it tends naturally to interact with the theories of various fields of mathematics, physics, and computer science. In turn, this broader work, and meeting with overseas experts, should support the generation of new collaborations and a cross-pollination of ideas both across the UK and also between UK researchers and researchers overseas, and from diverse areas beyond pure group theory. One main point is to draw researchers' attentions to the various new fields of endeavour (related to the expanding theory of the extended family of R. Thompson groups) currently available, and perhaps supporting a partial shift of consciousness in some way from some of the traditional chestnuts of the area to these newer areas of broader applicability.

An example of the last point is, the workshop may indirectly generate broader UK research in some of the UK's more general current research agendas. E.g., the projects of 1) Gould, Lawson, and Ruskuc on automatic structures, 2) the continuing work on understanding the class CoCF of groups with context free co-word problem as established by Holt, Rees, Rover, and Thomas, and 3) the work on partial symmetries with connections between R. Thompson's group V and objects such as quasi-crystals and R. Penrose's irregular planar tilings. Each of these projects admit connections to the theory of the extended family of the R. Thompson groups, and may benefit by added attention from more researchers with backgrounds including expertise from the theory of the R. Thompson family of groups.

Perhaps supporting a partial argument in the other direction, as we will also be having talks summarising the State of the Art in various more traditional directions, there may also be research generated maintaining the main thrust of the traditional research in the theory of these groups; amenability, geometry, and the structure theory of these groups. Thus, the point is not to change the traditional direction of research in this family of groups, but to encourage a natural broadening of that research, reflective of the many types of new interactions that the theory of these groups has created in the last decade or so.

The primary educational impact will come from the transfer of knowledge from cross-disciplinary area experts to the new and established researchers in the main body of R. Thompson group research. Over time, this will also strengthen our national research profile while also helping us to establish an international reputation for, and also to maintain, the high level of expertise currently existant within the main body of the UK researchers in the area.

Finally, generated research may plausibly have impact in various applications, from data structures on trees (following Sleator, Tarjan, and Thurston), to circuit complexity (following Birget), to physics (following Dehornoy's work on braids and presentations), and to the areas mentioned previously when detailing various UK based on-going projects.