Related papers: Topological Full Groups
The recent discovery of universal principles underlying many complex networks occurring across a wide range of length scales in the biological world has spurred physicists in trying to understand such features using techniques from…
Complex systems are characterized by specific time-dependent interactions among their many constituents. As a consequence they often manifest rich, non-trivial and unexpected behavior. Examples arise both in the physical and non-physical…
The topological fundamental group $\pi_{1}^{top}$ is a homotopy invariant finer than the usual fundamental group. It assigns to each space a quasitopological group and is discrete on spaces which admit universal covers. For an arbitrary…
We initiate a systematic study of the perfection of affine group schemes of finite type over fields of positive characteristic. The main result intrinsically characterises and classifies the perfections of reductive groups, and obtains a…
We associate to every action of a Polish group on a standard probability space a Polish group that we call the orbit full group. For discrete groups, we recover the well-known full groups of pmp equivalence relations equipped with the…
This is the first in a series of papers devoted to foundations of topological stacks. We begin developing a homotopy theory for topological stacks along the lines of classical homotopy theory of topological spaces. In this paper we go as…
The behavior of complex systems is determined not only by the topological organization of their interconnections but also by the dynamical processes taking place among their constituents. A faithful modeling of the dynamics is essential…
We generalise a fundamental graph-theoretical fact, stating that every element of the cycle space of a graph is a sum of edge-disjoint cycles, to arbitrary continua. To achieve this we replace graph cycles by topological circles, and…
We study topological groups of monotonic autohomeomorphisms on a generalized ordered space $L$. We find a condition that is necessary and sufficient for the set of all monotonic autohomeomorphisms on $L$ along with the function composition…
Large animal groups -- bird flocks, fish schools, insect swarms -- are often assumed to form by gradual aggregation of sparsely distributed individuals. Using a mathematically precise framework based on time-varying directed interaction…
The union of a directed family of topological groups can be equipped with two noteworthy topologies: the finest topology making each injection continuous, and the finest group topology making each injection continuous. This begs the…
We present the concept of the topological symmetry group as a way to analyze the symmetries of non-rigid molecules. Then we characterize all of the groups which can occur as the topological symmetry group of an embedding of the complete…
The notion of entropy appears in many fields and this paper is a survey about entropies in several branches of Mathematics. We are mainly concerned with the topological and the algebraic entropy in the context of continuous endomorphisms of…
This paper is a survey, with few proofs, of ideas and notions related to self-similarity of groups, semi-groups and their actions. It attempts to relate these concepts to more familiar ones, such as fractals, self-similar sets, and…
Different group structures which underline the integrable systems are considered. In some cases, the quantization of the integrable system can be provided with substituting groups by their quantum counterparts. However, some other group…
We give a description of the link between topological dynamical systems and their dimension groups. The focus is on minimal systems and, in particular, on substitution shifts. We describe in detail the various classes of systems including…
Different notions of entropy play a fundamental role in the classical theory of dynamical systems. Unlike many other concepts used to analyze autonomous dynamics, both measure-theoretic and topological entropy can be extended quite…
We develop the fundamental theory to study cubical isometry groups as totally disconnected, locally compact groups. We show how cubical isometries are determined by their local actions and how this can be applied in explicit constructions.…
In this paper, we investigate the topological full groups arising from the Cuntz and Cuntz-Toeplitz algebras and their crossed products with the Cartan subalgebras of Cuntz and Cuntz-Toeplitz algebras. We study the normal subgroups and…
We study topological full groups attached to groupoid models for left regular representations of Garside categories. Groups arising in this way include Thompson's group $V$ and many of its variations such as R\"over-Nekrashevych groups. Our…