The Higman-Thompson groups, their generalisations, and automorphisms of shift spaces

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The primary objective of the proposed research is to further explore a new and important connection between group theory, combinatorics and dynamical systems. Our research build upon techniques and methodology arising in several articles of the PI and collaborators some of which have been supported by EPSRC grant EP/R032866/1. We explain broadly what each of these areas are, and, then we say how our research relates to them.Group theory is an area of crucial importance in algebra and arguably arose from the quest to find solutions of polynomial equations of degree higher than 4. The defining characteristic of groups, the objects of study in group theory, is as a means of abstracting the inherent symmetry in a structure. For example, returning to the solutions of polynomial equations, groups arose via an understanding of the symmetry in the set of solutions to a given polynomial equation; the symmetries of a geometric object such as a square, or a circle also form a group. One might consider more complicated objects, for example fractal structures which have fine detail at infinitely small scales. More generally if one considers the collection of reversible transformations of an object which preserves some inherent structure one obtains a group. In our research we are interested in the Higman-Thompson groups which arise as symmetries of the Cantor space --- a fractal space.Group theory and combinatorics are intertwined areas of research. In our research, the connection to combinatorics is by transducers. These are finite state machines with fixed alphabet --- a given state of such a machine will read-in a symbol from the alphabet, possibly transition to a different state, and will output a symbol or string from the alphabet. One can imagine that the set of transducers which transform strings of a given alphabet in a reversible way gives rise to a group. A historical example of a transducer is the enigma machine --- a cipher device.Dynamics typically involves the study of long-term trends in the evolution of a system. Day-to-day examples of dynamical systems arise from the weather and the stock-market. Most dynamical systems can be studied in a symbolic way --- giving rise to the fundamental area of symbolic dynamics. This is our point of contact. We are interested in the shift dynamical system: this is an easily described symbolic dynamical system with numerous interesting features including chaos. One considers a fixed alphabet, and the collection of all bi-infinite (extending left and right) sequences over this alphabet. The shift map simply shifts all symbols of a given bi-infinite sequence one index to the left. Groups arise by considering reversible transformations of the space of infinite sequences which are invariant under the action of the shift map --- that is one cannot distinguish between the distinct processes of first applying the shift map and then such a transformation and applying the shift map first before applying the transformation. These are the so called groups of automorphisms of the shift dynamical system and are useful in understanding local dynamics of the system. Our research arises from the recent resolution, by the PI and collaborators, of the 20 year old problem of characterising the automorphism groups (group of symmetries) of the Higman-Thompson groups. The key insight was that these groups can be described by transducers which possess a synchronization property. This property means that under certain conditions automorphisms of the Higman-Thompson groups give rise to automorphisms of the shift dynamical system. One can go in the other direction -- automorphisms of the shift dynamical system give rise to automorphisms of the Higman Thompson groups. Our research aims, building on previous work, to further explore this new-found connection: expanding techniques and methodology across both fields, to shed further light on these areas of research.