Actions on cube complexes and homomorphisms to families of groups

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AbstractAward: DMS 1507067, Principal Investigator: Daniel GrovesA group is an algebraic object which encodes the symmetries of a geometric object. This connection between geometry and algebra implies that the study of groups provides an algebraic language and a set of techniques for studying problems throughout geometry. In recent years, some of the most important problems in three-dimensional geometry (such as the Virtual Haken Conjecture on the structure of spaces that cover three-dimensional manifolds) have been solved by a combination of group theory and geometric properties of two-dimensional submanifolds of these spaces. Some of the names associated to this line of work are Agol, Wise, Kahn, Markovic, Haglund, Hsu, Bergeron, and Manning.A major focus of the first half of this project is to broaden the range of applications of the available theory, and to apply these new tools to many more problems in geometry and group theory. In the second half of this project, techniques will be developed to study problems about maps between families of groups. Again, problems from three-dimensional geometry will give key motivation. Due to the work of Agol and many others, the objects in this setting are now quite well understood. However, there are many fundamental questions about the maps between these objects, and this proposal will tackle some of these problems, as well as building a general toolkit for studying questions such as these. In the first half of the project, we study hyperbolic groups acting cocompactly on CAT(0) cube complexes. In the case that the cell stabilizers are finite, Agol's famous theorem implies that such a group is "virtually special" (an important concept introduced by Haglund and Wise) which implies many strong properties, such as linearity and strong residual finiteness properties for such a group. In the case of a one-dimensional complex (a tree), if the cell stabilizers are infinite but quasi-convex and virtually special, then Wise's Quasiconvex Hierarchy Theorem implies that the hyperbolic group is again virtually special. The main goal of the first half of this project is to prove a simultaneous generalization of these two theorems: If a hyperbolic group G acts cocompactly on a CAT(0) cube complex with quasi- convex and virtually special cell stabilizers, then G is virtually special. This will have many applications to relatively hyperbolic and hyperbolic groups acting on cube complexes. In the second half of this proposal, we will develop a quite general framework for studying the set of a maps from an arbitrary finitely generated group into natural families of groups. Examples of such families which will be studied include: Kleinian groups, arbitrary three-manifold groups, relatively hyperbolic and acylindrically hyperbolic groups, and others.