Narrow Results

We define a new invariant of a conjugacy class of subgroups which we call the breadth and prove that a quasiconvex subgroup of a negatively curved group has finite breadth in the ambient group. Utilizing the coset graph and the geodesic core of a subgroup we give an explicit algorithm for constructing a finite generating set for an intersection of a quasiconvex subgroup of a negatively curved group with its conjugate. Using that algorithm we construct algorithms for computing the breadth, the width, and the height of a quasiconvex subgroup of a negatively curved group. These algorithms decide if a quasiconvex subgroup of a negatively curved group is almost malnormal in the ambient group. We also explicitly compute a quasiconvexity constant of the intersection of two quasiconvex subgroups and give examples demonstrating that height, width, and breadth are different invariants of a subgroup.

We describe subgroups and overgroups of the generalized Thompson groups $V_n$ which arise via conjugation by rational homeomorphisms of Cantor space. We specifically consider conjugating $V_n$ by homeomorphisms induced by synchronizing transducers and their inverses. Our descriptions of the subgroups and overgroups use properties of the conjugating transducer to either restrict or augment the action of $V_n$ on Cantor space. We also consider a class $𝔓$ of transducers containing all invertible, initial transducers. We associate to every transducer $T$ in this class, a group ${\mathscr{H}}(T)$. We show that for a certain subclass of $𝔓$, the groups ${\mathscr{H}}(T)$ are simple.