Jon M. Corson

EXTENDED FINITE AUTOMATA AND WORD PROBLEMS

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.

Amalgamated sums of groups

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.

A Quasi-Isometry Invariant Loop Shortening Property for Groups

We first introduce a loop shortening property for metric spaces, generalizing the property considered by M. Elder on Cayley graphs of finitely generated groups. Then using this metric property, we ...

A strong form of residual finiteness for groups

Abstract In general the extension of a residually finite group by a residually finite group may not be residually finite. We define a strong form of residual finiteness for groups and show that the property is closed under extensions. We then show how groups with this property can be constructed using amalgamated free products and HNN extensions. This leads to a multitude of examples.

ON DEHN FUNCTIONS OF AMALGAMATIONS AND STRONGLY UNDISTORTED SUBGROUPS

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.

Diagrammatically reducible complexes and Haken manifolds

We show that diagrammatically reducible two-complexes are characterized by the property: every finity subconmplex of the universal cover collapses to a one-complex. We use this to show that a compact orientable three-manifold with nonempty boundary is Haken if and only if it has a diagrammatically reducible spine. We also formulate an nanlogue of diagrammatic reducibility for higher dimensional complexes. Like Haken three-manifolds, we observe that if n ≥ 4 and M is compact connected n -dimensional manifold with a traingulation, or a spine, satisfying this property, then the interior of the universal cover of M is homeomorphic to Euclidean n -space.

ANNULAR DEHN FUNCTIONS OF GROUPS

For a finite presentation of a group, or more generally, a two-complex, we define a function analogous to the Dehn function that we call the annular Dehn function. This function measures the combinatorial area of maps of annuli into the complex as a function of the lengths of the boundary curves. A finitely presented group has solvable conjugacy problem iff its annular Dehn function is recursive. As with standard Dehn functions, the annular Dehn function may change with change of presentation. We prove that the type of function obtained is preserved by change of presentation. Further we obtain upper bounds for the annular Dehn functions of free products and, more generally, amalgamations or HNN extensions over finite subgroups.

ON PROFINITE GROUPS WHOSE POWER SUBGROUPS ARE CLOSED

ABSTRACT We say that a profinite group G has Property S if for each integer n, there is a bound k such that every element of the nth power subgroup is a product of k nth powers of elements of G. Finitely generated profinite groups with Property S are completely determined by their group structure in a way conjectured by Hartley to be true of all finitely generated profinite groups; namely, every subgroup of finite index is open. We show that the class of finitely generated profinite groups with Property S is closed under forming extensions of its members and under taking subgroups of finite index. As a consequence, we note that all profinite groups of finite rank have Property S.

COFINITE GRAPHS AND THEIR PROFINITE COMPLETIONS

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

Context-sensitive languages and G-automata

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