The present work shows that Cayley automatic groups are semiautomatic and exhibits some further constructions of semiautomatic groups and in particular shows that every finitely generated group of nilpotency class 3 is semiautomatic.

The girth of a finitely generated group G is defined to be the supremum of the girth of its Cayley graphs. Let G be a finitely generated subgroup of the mapping class group ModΣ, where Σ is an orientable closed surface with a finite number of punctures and with a finite number of components. We show that G is either a non-cyclic group with infinite girth or a virtually free-abelian group; these...

The notion of a group hyperbolic relative to a collection of subgroups was originally proposed by Gromov [15]. Since then many definitions of relatively hyperbolic groups were suggested by Bowditch [5], Farb [13], Drutu-Osin-Sapir [11, 33], Groves-Manning [17], Osin [29], Yaman [38] and others. All these definitions except for those suggested in [29] are equivalent and require the group to be f...

The class of rational subsets of a group G is the smallest class that contains all finite subsets of G and that is closed with respect to union, product and taking the monoid generated by a set. The rational subset membership problem for a finitely generated group G is the decision problem, where for a given rational subset of G and a group element g it is asked whether g ∈ G. This paper presen...

But before starting on this, let me try to explain again, in a different way, our approach. The one habit I’ve been trying to wean you of is the an over-reliance upon concrete examples to develop your understanding. The vector space R is a very concrete and familar example of a vector space over a field. To do calculations in this setting all you need to do is apply arithmetic (over and over an...

This paper deals with some results concerning finitely generated coreduced comultiplication modules over a commutative ring.

A subgroup Q of a group G is commensurated if the commensurator of Q in G is the entire group G. Our main result is that a finitely generated group G containing an infinite, finitely generated, commensurated subgroup H, of infinite index in G is 1-ended and semistable at ∞. If additionally, Q and G are finitely presented and either Q is 1-ended or the pair (G,Q) has 1 filtered end, then G is si...

Abstract A finitely generated group