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Limit Groups over Partially Commutative Groups

Final Report Summary - LIMITGROUPS (Limit Groups over Partially Commutative Groups)

The research project lies in the field of geometric group theory and its interplay with geometry and model theory. A distinguished representative of the intersection of these branches of mathematics is the class of limit groups. Our project has centered at developing a theory of limit groups over partially commutative groups from algebraic, geometric, algorithmic and model theoretic viewpoints.

Partially commutative groups (or pc groups, for short, also commonly called right-angled Artin groups) are a class of groups widely studied on account of their simple definition, their intrinsically rich structure and their natural appearance in several branches of computer science and mathematics. They are defined by a finite presentation where relations are commutators of (some of the) generators and so they can be regarded as a class of groups that interpolates between free and free abelian groups.

One of the classical problems in model theory is the classification of objects from a logical point of view. Our first result is on the classification of partially commutative groups up to universal equivalence. More precisely, we characterise when two pc groups are universally equivalent in terms of strong embeddability. The importance of this characterisation is that it reduces a deep logical problem to an algebraic one and it equips us with new tools to attack it.

In view of this result, our research focused on the embeddability problem between pc groups. Our first results in this direction are counterexamples to the Extension Graph Conjecture and the Weakly Chordal Conjecture formulated by Kim and Koberda. This examples show unexpected subtleties and difficulties of the embeddability problem between pc groups.

We proceed in this direction and distinguish certain class of nice embeddings, namely the extension graph embeddings, and show that there is an algorithm to solve the embeddability problem for such embeddings. This is the first decidability result in this context and we believe that it can pilot other new algorithmic results.

The study of the embeddability problem brought us to a deeper understanding of the geometry of pc groups. In our work, we present a new connection between quasiisometric classification of pc groups and extension graph embeddability. Previous techniques used to study quasi-isometric classification of pc groups have had limited success and only apply to specific cases. The approach we suggest opens a way to attack the general problem of quasi-isometric classification - a very important problem in geometric group theory suggested and promoted by Gromov. We established yet another interesting relation between quasi-isometric rigidity of some classes of pc groups and the algebraic rigidity of their Q-completions. These results are in the spirit of classical Mostow rigidity for Lie lattices and call for further exploration.

The algebraic part of our project has centered in the study of the class of limit groups over pc groups. This study was begun in the article “Limit Groups over Partially Commutative Groups and Group Actions on Real Cubings” (Geometry and Topology, 118 pages, to appear) where we show that limit groups over pc groups have a natural hierarchical structure. This work has motivated a series of preprints on the analogue of Lyndon’s completion of a given pc group. One of the important consequences of this work is the finite presentability of the class of limit groups over coherent pc groups. This result generalises the well-known result for free groups proven by Khalampovich- Miasnikov and Sela and is key for studying algorithmic problems in this class of groups and establishing the decidability of the conjugacy and membership problems for limit groups over coherent pc groups.

The importance of this project lies in the results, the established interconnections between different disciplines and the new uncovered directions of research in each area.