Residual properties of free products

Let C be a class of groups. We give sufficient conditions ensuring that a free product of residually C groups is again residually C, and analogous conditions are given for locally embeddable into C groups. As a corollary, we obtain that the class of residually amenable groups and the one of LEA groups (or initially subamenable groups in the terminology of Gromov) are closed under taking free products. Moreover, we consider the pro-C topology and we characterize special HNN extensions and amalgamated free products that are residually C, where C is a suitable class of groups. In this way, we describe special HNN extensions and amalgamated free products that are residually amenable.Comment: 17 pages, no figures. revised and expanded versio

Separating cyclic subgroups in graph products of groups

We prove that the property of being cyclic subgroup separable, that is having all cyclic subgroups closed in the profinite topology, is preserved under forming graph products. Furthermore, we develop the tools to study the analogous question in the pro-$p$ case. For a wide class of groups we show that the relevant cyclic subgroups - which are called $p$-isolated - are closed in the pro-$p$ topology of the graph product. In particular, we show that every $p$-isolated cyclic subgroup of a right-angled Artin group is closed in the pro-$p$ topology, and we fully characterise such subgroups.Comment: 37 pages, revised following referee's comments, to appear in Journal of Algebr

Scale function vs Topological entropy

In the realm of topological automorphisms of totally disconnected locally compact groups, the scale function introduced by Willis in \cite{Willis} is compared with the topological entropy. We prove that the logarithm of the scale function is always dominated by the topological entropy and we provide examples showing that this inequality can be strict. Moreover, we give a condition equivalent to the equality between these two invariants. Various properties of the scale function, inspired by those of the topological entropy, are presented.Comment: 21 page

New residually amenable groups, permanence properties, and metric approximations

Residuell mittelbare Gruppen treten als gemeinsame Verallgemenierung mittelbarer und residuell endlicher Gruppen auf. Diese beiden Klassen von Gruppen, die wenig miteinander gemeinsam haben, sind tief in der modernen Gruppentheorie verwurzelt und verbinden sie mit vielen anderen Zweigen der Mathematik. Vor kurzem erlangten residuell mittelbare Gruppen große Beachtung aufgrund ihrer Beziehung zur Eigenschaft "Soficity" (ein Begriff, der von Gromov eingeführt wurde, um Gottschalks Surjunctivity-Vermutung dynamischer Systeme zu behandeln). In dieser Arbeit untersuchen wir systematisch die Klasse der residuell mittelbaren Gruppen. Unser Ansatz konzentriert sich einerseits auf die strukturellen Eigenschaften dieser Klasse und andererseits auf eine quantitative Beschreibung der residuellen Mittelbarkeit. Wir beweisen, dass die Eigenschaft der residuellen Mittelbarkeit durch das Bilden freier Produkte, und allgemeiner durch Graph-Produkte, erhalten bleibt. Wir führen den Begriff einer pro-mittelbaren Topologie auf diskreten Gruppen ein und nutzen die Eigenschaften solcher Gruppen, um notwendige und hinreichende Bedingungen zu finden, die sicherstellen, dass bestimmte HNN-Erweiterungen und amalgamiert freie Produkte residuell mittelbarer Gruppen wieder residuell mittelbar sind. Im Allgemeinen gilt dies bei weitem nicht; durch diese Bedingungen konstruieren wir abzählbar viele paarweise nichtisomorphe endlich erzeugte Gruppen, die nicht residuell mittelbar sind. Wir untersuchen eine asymptotische Invariante für endlich erzeugte Gruppen, das residuell mittelbare Profil, das quantifiziert, wie sehr eine bestimmte Gruppe residuell mittelbar ist. Dies verallgemeinert den bekannten Begriff von Folner-Funktionen für mittelbare Gruppen auf die Klasse der residuell mittelbaren Gruppen. Wir analysieren das Verhalten des residuell mittelbaren Profiles im Bezug auf gruppentheoretische Operationen, die die Klasse der residuell mittelbaren Gruppe erhalten, wie direkte Produkte, freie Produkte und Erweiterungen mit mittelbarem Quotienten. Schließlich untersuchen wir die Klasse von Sofic-Gruppen und bringen einen neuen, unabängigen Beweis für die Tatsache, dass jede Gruppenerweiterung mit einem Sofic-Kern und mittelbarem Quotient wiederum eine Sofic-Gruppe ist. Dieser Beweis zeigt, wie auch der ursprüngliche von Elek und Szabó, wie die Voraussetzung der Mittelbarkeit die bestmoglichste für Resultate in diese Richtung sein könnte. Wir schlagen einen verschärften Begriff für Soficity vor, nämlich Konjugations-Soficity, und untersuchen die Eigenschaften dieses Begriffes. In diesem neuen Kontext können (im Gegensatz zur Soficity) die Soficity-Approximationen nicht mehr Homomorphismen auf endliche symmetrische Gruppen sein, sondern vielmehr Karten, die keine Homomorphismen sind.Residually amenable groups arise as a common generalisation of amenable and residually finite groups. These two classes of groups, that share little in common, are deeply rooted in modern group theory and connect it with many other branches of mathematics. Recently, residually amenable groups attracted considerable attention for their relation to soficity, a notion introduced by Gromov to tackle Gottschalk's surjunctivity conjecture in dynamical systems. In this thesis we systematically study the class of residually amenable groups. Our approach focuses on one side on the structural properties of the class, and on the other on a quantitative description of residual amenability. We prove that residual amenability is preserved by taking free products and, more generally, by graph products. We introduce the notion of proamenable topology on a discrete group, and we take advantage of its properties to provide necessary and sufficient conditions, for certain HNN extensions and amalgamated free products of residually amenable groups, to be again residually amenable. This is far from being true in general, and exploiting these obtained conditions we construct countably many pairwise non-isomorphic finitely presented groups that are not residually amenable. We study an asymptotic invariant for finitely generated groups, the residually amenable profile, which quantifies how much a given group is residually amenable. This generalises the well-known notion of Folner functions for amenable groups to the residually amenable setting. We analyse the behavior of the residually amenable profile with respect to group-theoretic operations that preserve the class of residually amenable groups, as for direct products, free products, and extensions with amenable quotient. Lastly, we investigate the class of sofic groups, producing a new, independent, proof of the soficity of any group extension with sofic kernel and amenable quotient. This proof, as well as the original one of Elek and Szabó, underlines how the amenability assumption might be the optimal one for results in this direction. We then propose a strengthened notion for soficity, that is, conjugacy soficity, and we study its properties.In this new context, and in contrast to soficity, the sofic approximations cannot anymore be homomorphisms onto finite symmetric groups, but are forced to be maps that are not homomorphisms