The set of finitely generated subgroups of the group PL+(I) of orientation-preserving piecewiselinear homeomorphisms of the unit interval includes many important groups, most notably R. Thompson’s group F . In this paper we show that every finitely generated subgroup G < PL+(I) is either soluble, or contains an embedded copy of Brin’s group B, a finitely generated, non-soluble group, which veri...

1.1. Baumslag's theorem has another facet. It shows that in the variety 2l of metabelian groups, every finitely generated $I-group G occurs as a subgroup of some finitely presented 2I-group G. The analogous result for the variety of all groups is a celebrated theorem of G. Higman [8] which states that a finitely generated group G is the subgroup of a finitely presented group G if, and only if, ...

In this paper we study geometric versions of Burnside’s Problem and the von Neumann Conjecture. This is done by considering the notion of a translation-like action. Translation-like actions were introduced by Kevin Whyte as a geometric analogue of subgroup containment. Whyte proved a geometric version of the von Neumann Conjecture by showing that a finitely generated group is nonamenable if and...

We provide an example of a non-finitely generated group which admits nonempty strongly aperiodic subshift finite type. Furthermore, we completely characterize the groups with this property in terms their finitely subgroups and roots conjugacy classes.

This paper derives from a course in group theory which I gave at Berkeley in 1982. I wanted to prove the standard theorems on free groups, and discovered that, after a few preliminaries, the notion of "locally injective" map (or "immersion") of graphs was very useful. This enables one to see, in an effective, easy, algorithmic way just what happens with finitely generated free groups. One can u...