Related papers: From telescopes to frames and simple groups

This note describes the first example of a group that is amenable, but cannot be obtained by subgroups, quotients, extensions and direct limits from the class of groups locally of subexponential growth. It has a balanced presentation…

Uhlenbeck proved that a set of simple elements generates the group of rational loops in GL(n,C) that satisfy the U(n)-reality condition. For an arbitrary complex reductive group, a choice of representation defines a notion of rationality…

This is the written version of the Bourbaki seminar given in January 2013 and published in 2014 (modulo an additional early reference added subsequently). It describes the first construction of infinite, finitely generated amenable simple…

In the current paper we attempt to transfer the notion of the projectional entropy, originally defined for multidimensional subshifts, to the case of actions of amenable groups. The main theorem states that if a system is strongly…

Let $K$ be a global function field of characteristic $p$, and let $\Gamma$ be a finite-index subgroup of an arithmetic group defined with respect to $K$ and such that any torsion element of $\Gamma$ is a $p$-torsion element. We define…

We show that the pair given by the power set and by the "Grassmannian"(set of all subgroups) of an arbitrary group behaves very much like the pair given by a projective space and its dual projective space. More precisely, we generalize…

Lenses have a rich history and have recently received a great deal of attention from applied category theorists. We generalize the notion of lens by defining a category $\mathsf{Lens}_F$ for any category $\mathcal{C}$ and functor $F\colon…

We generalize the classical definition of effectively closed subshift to finitely generated groups. We study classical stability properties of this class and then extend this notion by allowing the usage of an oracle to the word problem of…

We give a unified description of twisted forms of classical reductive groups schemes. Such group schemes are constructed from algebraic objects of finite rank, excluding some exceptions of small rank. These objects, augmented odd form…

The goal of this article is to study results and examples concerning finitely presented covers of finitely generated amenable groups. We collect examples of groups $G$ with the following properties: (i) $G$ is finitely generated, (ii) $G$…

The relation between manifold topology, observables and gauge group is clarified on the basis of the classification of the representations of the algebra of observables associated to positions and displacements on the manifold. The guiding,…

Working in the soft-element (classical) viewpoint, we introduce \emph{soft bitopological groups}: soft groups endowed with two soft topologies such that the induced topologies on the set of soft elements make the soft-element group into a…

We consider metric versions of the notions of local embeddability and LEF. We pay special attention to normally finitely generated groups with word metrics.

We show that amenability of a group acting by homeomorphisms can be deduced from a certain local property of the action and recurrency of the orbital Schreier graphs. This covers amenability of a wide class groups, the amenability of which…

A group, defined as set with associative multiplication and inverse, is a natural structure describing the symmetry of a space. The concept of group generalizes to group objects internal to other categories than sets. But there are yet more…

Network glasses are the physical prototype for many self-organized systems, ranging from proteins to computer science. Conventional theories of gases, liquids, and crystals do not account for the strongly material-selective character of the…

We consider (projectively) linearly sofic groups, i.e. groups which can be approximated using (projective) matrices over arbitrary fields, as a generalization of sofic groups. We generalize known results for sofic groups and groups which…

Arithmetic groups are groups of matrices with integral entries. We shall first discuss their origin in number theory (Gauss, Minkowski) and their role in the "reduction theory of quadratic forms". Then we shall describe these groups by…

Physical properties of matter are tightly related with the kind of symmetry of the medium. Group theory is a systematic tool, though not always easy to handle, to exploit symmetry properties, for instance to find the eigenvectors and…

This is Chapter 24 in the "AutoMathA" handbook. Finite automata have been used effectively in recent years to define infinite groups. The two main lines of research have as their most representative objects the class of automatic groups…