NSF Award
1105520
The suggested research plan centers around properties of self-similar groups acting on rooted trees. In most general terms, the theory of groups acting on rooted, spherically homogeneous, trees can be understood as study of residually finite groups by using the language, methods, ideas and the intuition from topology and geometry. The self-similarity of the object of action (the tree) and the presence of a fixed vertex (the root) in the theory of groups acting on rooted trees lead to a set of natural finiteness conditions, such as being a generated by a finite self-similar set, having rigid stabilizers of finite index, being defined by finitely many forbidden tree patterns, having finite nucleus, etc. Such conditions play a crucial role in the proposed problems and directions of study involving questions on presentations of self-similar groups, Bieri-Neumann-Strebel Sigma invariants, virtual endomorphisms and their applications, finitely constrained groups and other group shifts on trees, relations to Hausdorff dimension, and algorithmic problems, with special attention given to the conjugacy problem. <br/><br/>In many endeavors, in and outside of mathematics, understanding is achieved in two, often intertwined, phases. Namely, in the first phase one seeks understanding of some classes of objects and situations distinguished by their simplicity or regularity, and in the second understanding of the ways in which they fit together to build, or at least approximate, the more complex ones. Since, by its very nature, the notion of self-similarity concerns entities in which copies of the original can be found at various scales within the entity itself, the approach of building/understanding complex self-similar structures from simpler and more regular ones seems particularly well suited. The proposed research contributes to both natural phases in the understanding of self-similar group actions on rooted trees. For instance, all finite self-similar groups are being characterized, and they serve as the building blocks from which the finitely constrained groups, and more generally all self-similar groups, are put together. On the other hand, the decidability of algorithmic questions is explored in contexts in which the building blocks are already well understood and the solution of the problem in the simple instances could possibly be assembled into a solution in the composite structure.
