Complexes of groups are higher‐dimensional analogues of graphs of groups. Triangles of groups are a special class of 2‐complexes of groups. Results of Bass‐Serre and Gersten‐Stallings are generalized to complexes of groups.

This paper considers extended finite automata over monoids, in the sense of Dassow and Mitrana. We show that the family of languages accepted by extended finite automata over a monoid K is controlled by the word problem of K in a precisely stated manner. We also point out a critical error in the proof of the main result in the paper by Dassow and Mitrana. However as one consequence of our approach, by analyzing a certain word problem, we obtain a complete proof of this result, namely that the family of languages accepted by extended finite automata over the free group of rank two is exactly the family of context-free languages. We further deduce that along with the free group of rank two, the only finitely generated groups with this property are precisely the groups that have a nonabelian free subgroup of finite index.

Groups called amalgamated sums that arise as inductive limits of systems of groups and injective homomorphisms are studied. The problem is to find conditions under which the groups in the system do not collapse in the limit. Such a condition is given by J. Tits when certain subsystems are associated to buildings. This condition can be phrased to apply to certain systems of abstract groups and injective homomorphisms. It is shown to imply that no collapse occurs in the limit in a strong sense; namely the natural homomorphism of the amalgamated sum of any subsystem into the amalgamated sum of the full system is injective. This answers a question of S. J. Pride.

We study the Dehn functions of amalgamations, introducing the notion of strongly undistorted subgroups. Using this, we give conditions under which taking an amalgamation does not increase the Dehn function, generalizing one aspect of the combination theorem of Bestvina and Feighn.To obtain examples of strongly undistorted subgroups, we define and study the relative Dehn function of pairs of groups. As a result we obtain a new method of constructing examples of pairs of groups that are relatively hyperbolic in the sense of Farb.

We generalize the idea of cofinite groups, due to B. Hartley, [2]. First we define cofinite spaces in general. Then, as a special situation, we study cofinite graphs and their uniform completions.The idea of constructing a cofinite graph starts with defining a uniform topological graph Γ, in an appropriate fashion. We endow abstract graphs with uniformities corresponding to separating filter bases of equivalence relations with finitely many equivalence classes over Γ. It is established that for any cofinite graph there exists a unique cofinite completion.