Decision problems in groups of homeomorphisms of Cantor space

Group theory;Automorphisms;Infinite groups

Olukoya, Feyisayo ; Bleak, Collin Patrick ; Bleak, Collin Patrick

University:University of St Andrews

Department:Mathematics & Statistics (School of)

关键词: Group theory;    Automorphisms;    Infinite groups;   

Others  :  https://research-repository.st-andrews.ac.uk/bitstream/handle/10023/15885/FeyisayoOlukoyaPhDThesis.pdf?sequence=3&isAllowed=y

来源: DR-NTU

【 摘 要 】

The Thompson groups $F, T$ and $V$ are important groups in geometric group theory: $T$ and $V$ being the first discovered examples of finitely presented infinite simple groups. There are many generalisations of these groups including, for $n$and $r$ natural numbers and $1 < r < n$, the groups $F_{n}$, $T_{n,r}$ and$G_{n,r}$ ($T ≅ T_{2,1}$ and $V ≅ G_{2,1}$). Automorphisms of$F$ and $T$ were characterised in the seminal paper of Brin ([16]) and, later on, Brin and Guzman ([17]) investigateautomorphisms of $T_{n, n-1}$ and $F_{n}$ for $n>2$. However, their techniques give no information about automorphisms of $G_{n,r}$.The second chapter of this thesis is dedicated to characterising the automorphisms of $G_{n,r}$. Presenting results of the author's article [10], we show that automorphisms of $G_{n,r}$ arehomeomorphisms of Cantor space induced by transducers (finite state machines) which satisfy a strong synchronizing condition. In the rest of Chapter 2 and early sections of Chapter 3 we investigate thegroup $\out{G_{n,r}}$ of outer automorphisms of $G_{n,r}$. Presenting results of the forthcoming article [6] of the author's, we show that there is a subgroup $\hn{n}$ of $\out{G_{n,r}}$, independent of $r$, which is isomorphic to the groupof automorphisms of the one-sided shift dynamical system. Most of Chapter 3 is devoted to the order problem in $\hn{n}$ and is based on [44]. We give necessary and sufficient conditions for an element of $\hn{n}$ to have finite order, although these do not yield a decision procedure. Given an automorphism $\phi$ of a group $G$, two elements $f, g ∈ G$ are said to be $\phi$-twisted conjugate to one another if for some $h ∈ G$, $g = h⁻¹ f (h)\phi$. This defines an equivalence relation on $G$ and$G$ is said to have the $\rfty$ property if it has infinitely many $\phi$-twisted conjugacy classes for all automorphisms $\phi ∈ \aut{G}$.In the final chapter we show, using the description of $\aut{G_{n,r}}$,that for certain automorphisms, $G_{n,r}$ has infinitely many twisted conjugacy classes. We also show that for certain $\phi ∈ \aut{G_{2,1}}$ the problem of deciding when two elements of $G_{2,1}$ are $\phi$-twisted conjugate to one another is soluble.