Ends of groups : a computational approach

Abstract

We develop ways in which we can find the number of ends of automatic groups and groups with solvable word problem. In chapter 1, we provide an introduction to ends, splittings, and computa- tion in groups. We also remark that the `JSJ problem' for finitely presented groups is not solvable. In chapter 2, we prove some geometrical properties of Cayley graphs that underpin later computational results. In chapter 3, we study coboundaries (sets of edges which disconnect the Cayley graph), and show how Stallings' theorem gives us finite objects from which we can calculate splittings. In chapter 4, we draw the results of previous chapters together to prove that we can detect zero, two, or infinitely many ends in groups with `good' automatic structures. We also prove that given an automatic group or a group with solvable word problem, if the group splits over a finite subgroup, we can detect this, and explicitly calculate a finite subgroup over which it splits. In chapter 5 we give an exposition of Gerasimov's result that one- endedness can be detected in hyperbolic groups. In chapter 6, we give an exposition of Epstein's boundary construction for graphs. We prove that a testable condition for automatic groups implies that this boundary is uniformly path-connected, and also prove that infinitely ended groups do not have uniformly path-connected boundary. As a result we are able to sometimes detect one endedness (and thus solve the problem of how many ends the group has)