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

这项研究的主要目的是进一步探索群论、组合学和动力系统之间的一种新的重要联系。我们的研究建立在PI和合作者的几篇文章中出现的技术和方法论上，其中一些得到了EPSRC赠款EP/R032866/1的支持。我们大致解释了这些领域中的每个领域，然后我们说我们的研究如何与它们相关。群论在代数中是一个至关重要的领域，可以说它起源于寻找高于4次的多项式方程的解。群的定义特征，群论的研究对象，是作为一种抽象结构中内在对称性的手段。例如，返回到多项式方程的解，群通过对给定多项式方程的解集合中的对称性的理解而产生；几何对象(如正方形或圆)的对称性也形成群。人们可以考虑更复杂的物体，例如在无限小尺度上具有精细细节的分形结构。更一般地，如果考虑保留某些固有结构的对象的可逆变换的集合，则得到一个群。在我们的研究中，我们感兴趣的是作为Cantor空间的对称性而产生的Higman-Thompson群-一个分形空间。群论和组合学是相互交织的研究领域。在我们的研究中，与组合学的联系是通过传感器来实现的。这些是具有固定字母表的有限状态机-这样的机器的给定状态将从字母表读入符号，可能转换到不同的状态，并将从字母表输出符号或字符串。人们可以想象，以可逆方式转换给定字母表的串的一组换能器会产生一组。换能器的一个历史例子是谜机器-一种密码设备。动力学通常涉及对系统演化的长期趋势的研究。动力系统的日常例子来自天气和股市。大多数动力系统都可以用符号的方式来研究-这就产生了符号动力学的基本领域。这是我们的联络点。我们感兴趣的是移位动力系统：这是一个容易描述的符号动力系统，具有包括混沌在内的许多有趣的特征。考虑一个固定的字母表，以及该字母表上所有双无限(向左和向右扩展)序列的集合。移位映射只是将给定的双无限序列的所有符号向左移位一个索引。群是通过考虑无限序列空间的可逆变换而产生的，这些变换在移位映射的作用下是不变的-也就是说，人们无法区分首先应用移位映射、然后应用这样的变换和在应用变换之前首先应用移位映射的不同过程。这些是移位动力系统的所谓自同构群，对于理解系统的局部动力学是有用的。我们的研究源于PI及其合作者最近解决了20年的Higman-Thompson群的自同构群(对称群)的刻画问题。关键的见解是，这些群可以用具有同步特性的换能器来描述。这一性质意味着在一定条件下，Higman-Thompson群的自同构导致移位动力系统的自同构。人们可以往另一个方向走--移位动力系统的自同构导致希格曼-汤普森群的自同构。我们的研究目的是在以前工作的基础上，进一步探索这种新发现的联系：在这两个领域扩展技术和方法论，以进一步阐明这些研究领域。

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.