Richard Thompson's Group $V$
Here is a useful website for an introduction to Richard Thompson's Groups.
Reading list on Thompson's groups - Matt Brin's Webpage
Wise words from Graham Higman:
Interview with Graham Higman
Finitely Presented Infinite simple groups - Graham Higman 1974
This book is devoted to the generalization, $V_{n,r}$ for $n>1$ and $r>0$, of a free algebra, $V_{2,1}$, in the variety of algebras with signature $\lt 2,1,1 \gt$ as described by B. Jonsson and A. Tarski in 1961. The automorphism groups of such algebras, $G_{n,r}$, are finitely presented infinite groups that are simple groups when $n$ is even (and contain a simple subgroup when $n$ is odd). The first of these groups, $G_{2,1}$, is isomorphic to Richard Thompson's group $V$.
Subgroups of Finitely Presented simple groups - Claas H.E.W. Roever 1999
This Thesis is devoted to the possible structure of subgroups of finitely presented infinite simple groups. A class of finitely presented simple groups are constructed that have subgroups isomorphic to Grigorchuk-Gupta-Sidki groups. This class of finitely presented simple groups contained the first groups that were finitely presented infinite simple groups that are not torsion locally finite.
Chapter One contains a detailed description "symbols" for elements of $G_{n,r}$ that were originally defined in Higman's book (above). Connections with the work and techniques of E. Scott, who described the method for constructing finitely presented groups (specifically ones containing the groups $GL(n,Z)$, $n\gt 2$, as subgroups), are described in Chapters One and Three.
Chapter Two is devoted to what is know about the subgroup structure of $G_{n,r}$. Specifically, the importance of the commutator subgroup of $G_{n,r}$ and that the class of subgroups of $G_{n,r}$ contains all countable locally finite groups and is closed under countable restricted direct products and finite extensions, contains the Houghton groups, non-abelian free subgroups of every countable rank. It is also shown that every free product of finitely many finite groups is embedded in $G_{n,r}$ for any choice of $n\gt 1$, $r\gt 0$.
The remaining Chapters answer algorithmic questions about the class of finitely presented infinite simple groups constructed.
My Research
Centralizers of elements of the Higman-Thompson group $G_{2,1}$.
We study centralizers of elements of the Higman-Thompson group using the original ideas of G. Higman from Finitely Presented Infinite Simple Groups. This work had previously been carried out using the dynamical techniques associated to Thompson's group $V$ by Bleak et al (see Centralizers in R. Thompson's group $V_{n}$ and Thompson's group $V$ from a dynamical viewpoint ).
Conjugacy problems for elements of the Higman-Thompson group $G_{2,1}$.
We look at the simultaneous and power conjugacy problems for the Higman-Thompson group $G_{2,1}$.
Studying the tower of Automorphisms of the free algebra $V_{2,1}$.
We consider the Automorphism group of the Automorphism group of the free object $V_{2,1}$ (that is the group Aut($G_{2,1}$)). By studying automorphisms of the variety structure $V_{2}$, we learn information about the elements of the group Out($G_{2,1}$) (the outer automorphism group).
Studying Automorphisms of the Higher Dimensional Thompson groups $nV$.
We consider the Automorphism group of the Automorphism group of the free Higman algebra $U(\{x_0\})$ (as defined by D. H. Kochloukova, C. Martinez-Perez and B. E. A. Nucinkis). This is joint work with Collin Bleak.
