Constructible groups and ramifications of JSJ theory

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The discovery by Poincaré of fundamental groups showed that groups could capture aspects of topological spaces. This phenomenon goes both ways: topological spaces can also describe groups. This perspective lead to group presentations, a description given by generators and relations, making a group into a set of strings of symbols. This formalism enabled Dehn in the 1920s to pose the three fundamental algorithmic problems of group theory: the word problem (are two elements equal?), the conjugacy problem (are two elements conjugate?), and the isomorphism problem (are two groups isomorphic?). In the 1950s, Novikov proved that the word problem in general is undecidable. As a consequence, a systematic study of all group considering only presentations is not feasible. However, further taking into account the metric on a group induced by a presentation, as well as the topology of presentation complexes, has been a spectacularly successful approach for certain classes of groups, yielding solutions to Dehn's problems. This is geometric group theory, the study of groups as geometric objects. My research program consists of several interconnected projects in geometric group theory aimed at describing algebraic structure and finding algorithms, unified by the themes of constructible groups and JSJ theory. Constructible groups are groups that can be obtained from the trivial group by successive gluings, or amalgamations, along 2-ended subgroups. JSJ theory (named after Jaco-Shalen and Johansson) describes the ways in which a group can be decomposed as an amalgam of groups. Constructible groups are relatively well-understood. This research program aims to describe quasi-isometry and commensurability classes of constructible groups. This description is one of the few remaining open questions about these groups. The proposed method to pursue this investigation is a novel application of JSJ theory to quasi-isometric rigidity. Until now, the techniques used to prove quasi-isometric rigidity results applied to different kinds of groups. Fast algorithms, that will be implemented on computers, will also be developed for constructible groups. I will also continue building on my work on the isomorphism problem for relatively hyperbolic groups to solve the conjugacy problem in Out(Fn),  an important open problem in my field that has withstood decades of attacks. An important component of the proposed approach is to exploit the fact that constructible groups actually play a key role in this problem. JSJ theory is powerful, but inaccessible. I will explore how the topic of the collapse of CAT(0) cube complexes can be used to unify and simplify many of the basic theorems of JSJ theory, as well as related results, while simultaneously  providing generalizations of JSJ theory. This research program will therefore provide radically new perspectives on well-established concepts of geometric group theory, yielding unexpected and far-reaching applications.