## Abstracts

#### Mini-courses

**Helge Glöckner** (Universität Paderborn)**Endomorphisms of totally disconnected groups**

Abstract: Several subgroups can be associated to an automorphism (or endomorphism) f of a locally compact group G, like the contraction group con(f) of all group elements whose forward orbit under f converges to the neutral element e, or the group par(f) of all group elements whose forward orbit is relatively compact.

For totally disconnected G, the study of such subgroups was started in [1] and [5], and connections were established there to notions from the structure theory of totally disconnected groups (like tidy subgroups and the scale, which is also related to topological entropy, [3]).

In the talks, I'll give an introduction to this area of research, including some recent results both in the general case and for the special case of endomorphisms of Lie groups over local fields (as in [2] and [4]).

Bibliography:

[1] U. Baumgartner and G. Willis, Contraction groups and scales of automorphisms of totally disconnected locally compact groups, Isr. J. Math. 142 (2004), 221-248.

[2] T.P. Bywaters, H. Glockner, and S. Tornier, Contraction groups and passage to subgroups and quotients for endomorphisms of totally disconnected locally compact groups, to appear in Israel J. Math. (cf. arXiv:1612.06958)

[3] A. Giordano Bruno and S. Virili, Topological entropy in totally disconnected locally compact groups, Ergod. Th. & Dynam. Sys. 37 (2017), 2163-2186.

[4] H. Glockner, Endomorphisms of Lie groups over local fields, to appear in: The 2016 MATRIX annals, Springer-Verlag (see www.matrix-inst.org.au/2016-matrix-annals/)

[5] G.A. Willis, The scale and tidy subgroups for endomorphisms of totally disconnected locally compact groups. Math. Ann. 361 (2015), 403-442.

**Andrea Sambusetti** (Sapienza Università di Roma)**The amenability conjecture for Gromov-hyperbolic groups**

Abstract: We will present a classical conjecture on amenability for groups related to growth. The original result, which is due to Cohen and, independently, Grigorchuk, dates back to the 80's: a group Q finitely presented as F/N, where F is a free group, is amenable if and only if the exponential growth rates (or entropies) \omega(N) and \omega(F) of N and F coincide.

This is related to Kesten's criterion for amenability, and has been declined in many different ways and different contexts during the years. The most accredited version of the Amenability Problem in the last decade was: given a group G acting properly on a CAT(-1) space X and a normal subgroup N of G, does the equality \omega(N) = \omega(G) imply that the quotient group Q = G/N is amenable?

We prove a general version of the amenability conjecture for Gromov word-hyperbolic groups or a cocompact groups of isometries of a CAT(-1)-space.

For this, we prove a quantified version of an amenability criterion due to Stadlbauer (generalizing Kesten's Criterion) for group extensions of a topologically transitive subshift of finite type, in terms of the spectral radii of the classical Ruelle transfer operator and its corresponding extension.

As a consequence, we are able to show that, in our enlarged context, there is a gap between the entropy of a group with Kazhdan’s property (T) and the entropies of all of its infinite index subgroups. This also generalizes a well-known theorem of Corlette for lattices of the quaternionic hyperbolic space or the Cayley hyperbolic plane.

We will try to sketch the overall strategy and, time permitting, to give an idea of all the background material needed in the proof (dynamics on hyperbolic groups, Gromov's coding, transfer operator for subshifts of finite type, spectral amenability criterions for group extensions etc). The seminar is based on the joint work with F. Dal'Bo and R. Coulon arxiv.org/pdf/1709.07287.pdf

#### Plenary talks

**Michael Megrelishvili** (Bar-Ilan University)**Generalized amenability of topological groups and Banach representations**

Abstract: We study topological groups and continuous actions which can be represented on Rosenthal Banach spaces (i.e., Banach spaces which contain no copy of l_{1}). Such actions are tame. For example this happens for actions on circularly ordered systems and dendrons. By a dendron we mean a compact space where every pair of distinct points can be separated by a third point. We are interested in topological groups G for which the universal minimal G-system M(G) or/and the universal irreducible affine G-system IA(G) are tame (dynamically small). This leads to generalized (extremal) amenability of topological groups. We discuss some examples and open problems. It is a joint work with Eli Glasner.

**Luigi Salce** (Università degli Studi di Padova)**Intrinsic valuation entropy**

Abstract: The intrinsic entropy for endomorphisms of modules over a non-discrete archimedean valuation domain R is presented, which extends the analogous notion for endomorphisms of Abelian groups, using the natural non-discrete length function introduced by Northcott and Reufel for such a category of modules. With the aid of new techniques suitable for the non-discrete setting, we prove that this notion of entropy is a length function for the category of R[X]-modules, it satisfies a suitably adapted version of the Yuzvinski Formula, and it is essentially the unique invariant for Mod(R[X]) with these properties. (Joint work with Simone Virili)

**Manuel Sanchis** (Universitat Jaume I de Castelló)

The Ellis semigroup of a nonautonomous discrete dynamical system

Abstract: We introduce the Ellis semigroup of a nonautonomous discrete dynamical system (X,f_{1,∞}) when X is a metric compact space. The underlying set of this semigroup is the pointwise closure of {f_{1}^{n}: n∈N} in the space X^{X}. By using the convergence of a sequence of points with respect to an ultrafilter it is possible to give a precise description of the semigroup and its operation. This notion extends the classical Ellis semigroup of a discrete dynamical system. We show several properties that connect this semigroup and the topological properties of the nonautonomous discrete dynamical system. (Joint paper with Salvador García-Ferreira)

**Klaus Schmidt** (Universität Wien)**Algebraic actions of the discrete Heisenberg group**

Abstract: Algebraic actions of finitely generated abelian groups are by now pretty well understood. Very little is known, however, about algebraic actions of countable nonabelian groups. In this talk I’ll describe the state of knowledge (or lack of knowledge) about algebraic actions of one of the simplest nonabelian groups, the discrete Heisenberg group.

**Pablo Spiga** (Università di Milano-Bicocca)**Milnor-Wolf Theorem for the growth of endomorphisms of locally virtually soluble groups**

Abstract: Here we are interested in the growth of group endomorphisms. This notion is well studied and great deal is known for endomorphisms of abelian groups. However, our current understanding on the growth of group endomorphisms for arbitrary groups is not as clear. In this talk, we show that some results valid for abelian groups still hold for endomorphisms of locally virtually soluble groups. Indeed, we discuss an analogue of the Milnor-Wolf’s Theorem for the growth of finitely generated soluble groups: if G is a locally virtually soluble group and if f:G→G is an endomorphism, then f has either polynomial or exponential growth.

**Luchezar Stoyanov** (University of Western Australia)**Topological and metric entropy for group and semigroup actions**

Abstract: It is well-known that the classical definition of topological entropy for group and semigroup actions is frequently zero in some rather interesting situations, e.g. smooth actions of Z_{+}^{k} (k>1) on manifolds.

A different definition was proposed in 1995 by K.H. Hofmann and the speaker which produces topological entropy not trivially zero for such smooth actions. In the present talk we will discuss this particular approach, and also some of the main properties of the topological entropy defined in this way, its advantages and disadvantages compared with the classical definition. We will also discuss some recent results (obtained jointly with Dikran Dikranjan and Anna Giordano Bruno) of a similar definition of metric entropy, i.e. entropy with respect to an invariant measure for a group or a semigroup action, and some of its properties.

#### Contributed talks

**Taras Banakh** (Ivan Franko National University of Lviv, Jan Kochanowski University in Kielce)**Closedness and completeness of topological groups and semigroups**

Abstract: Given a class C of topologized semigroup, we shall discuss the problem of recognizing topologized (semi)groups X which are closed in each topologized semigroup Y in C that contains X as a subsemigroup.

**Andrzej Biś** (University of Lodz)**Topological conditional entropy of a solenoid**

Abstract: Misiurewicz developed a theory of topological conditional entropy of a single map. We investigate topological conditional entropy of a solenoid. A solenoid can be presented either in an abstract way, as an inverse limit of maps, or in a geometric way, as nested intersections of solid tori.

**Serhii Bardyla** (Ivan Franko National University of Lviv)**On complete semitopological semilattices**

Abstract: We find (completeness type) conditions on topological semilattices X, Y guaranteeing that each continuous homomorphism h:X→Y has closed image h(X) in Y. More precisely we investigate semitopological semilattices with compact-like maximal chains and introduce a notion of Comfort topology on semitopological semilattices.

**Jung Kyu Canci** (University of Basel)**Scarcity of periodic points for rational functions over a number field**

Abstract: I will present a recent joint work with S. Vishkautsan where we provide an explicit bound on the number of periodic points of a rational function of degree at least 2 defined over a number field. The bound depends only on the number of primes of bad reduction and the degree of the function, and is linear in the degree. We show that under stronger assumptions (but not so strong, in terms of ramification) the dependence on the degree of the map in the bounds can be removed.

**Jose S. Canovas** (Universidad Politécnica de Cartagena)**On topological entropy on non compact spaces**

Abstract: Several non conjugate notions of topological entropy for non compact spaces are analyzed and compared.

**Ulderico Dardano** (Università degli Studi di Napoli Federico II)**Inertial methods in group theory**

Abstract: I will consider the concept of inertial subgroup as a generalization on that of normal subgroup (in the theory of abstract groups) and discuss results which are obtained using tools from the theory of entropy in abelian group theory.

**Michal Doucha** (Czech Academy of Sciences, Prague)**Kazhdan groups and generic unitary representations**

Abstract: We show that countable discrete groups with property (T) and with the property that their unitary dual has densely-many finite-dimensional representations have the following characterization: they are the only countable discrete groups that admit a generic unitary representation, i.e. a unitary representation whose class under unitary equivalence is dense G_\delta in the Polish space of all unitary representations. We also show that these generic representations are Koopman representations of some measure preserving actions on a standard probability space; this addresses a certain question of Kechris. It is joint work with Maciej Malicki and Alain Valette.

**Stefano Ferri** (Universidad de los Andes, Bogotà)**Banach space actions, admissible algebras and group uniformities**

Abstract: In this talk we shall present some conditions which give a relation between the richness of certain algebras of functions defined on a topological group, the fact that the group acts isometrically on certain Banach spaces and certain properties of the group uniformity. Namely, we shall prove the following: Let G be a topological group and suppose that the coefficients of representations of G as isometries of Banach spaces which belong to some class F, preserved under l_{2} sum generate the topology of G, then G acts as isometry group on some space B in F and this happens if and only if the uniformity of G belongs to F (in a sense which will be made clear during the talk).

The talk include a short introduction to the relation between Ellis enveloping semigroup and the results mentioned above and is based on several paper joint works with Itaï Ben Yaacov, Alexander Berenstein, Jorge Galindo, Camilo Gòmez y Matthias Neufang.

**Carmelo Antonio Finocchiaro** (Università degli Studi di Padova)**On topology and flatness**

Abstract: As it is well-known, multiplication by an ideal I of a ring usually does not commute with the intersection of a family of ideals. But, when the ideal I is fat and the family is finite, then multiplication commutes with intersection. In this talk we will provide a generalization of the previous result by proving that finite families of ideals can be replaced by compact subspaces of a natural topological space. Several consequences of this fact will be presented. This talk is based on a paper in collaboration with Dario Spirito.

**Antongiulio Fornasiero** (Università degli Studi di Firenze)**Algebraic entropy for actions of amenable groups**

Abstract: I will give an axiomatic approach to some entropies of actions of amenable groups (including ent and h_{alg} for actions on Abelian groups), with the view of proving the corresponding addition theorems.

**Marzieh Forough** (Institute for Research in Fundamental Sciences, Tehran)**An equivariant Choi-Effros Lifting Theorem and applications**

Abstract: We shall discuss an equivariant version of the Choi-Effros lifting theorem, an important result from the theory of C*-algebras, for second-countable locally compact groups. We then show some applications to C*-dynamical systems. We explain all the necessary background. This is joint work with Eusebio Gardella and Klaus Thomsen.

**Oleg Gutik** (Ivan Franko National University of Lviv)**Feebly compact semitopological symmetric inverse semigroups of a bounded finite rank**

Abstract: For an arbitrary positive integer n, we discuss feebly compact shift-continuous T_{1}-topologies on the symmetric inverse semigroup I_{l}^{n} of finite transformations of rank less or equal n.

**Peter Kropholler** (University of Southampton)**Amenability, cellular automata, and group graded rings**

Abstract: We'll look at a generalization of results of Bartholdi to group graded rings showing an interconnection between amenability of a group G and the behaviour or cellular automata structure on objects that are G-graded.

**Bibgbing Liang** (Max-Planck Institute for Mathematics in Bonn)**Sofic mean rank**

Abstract: We define an invariant, called sofic mean rank, for G-modules of a sofic group G. This invariant generalizes the case when G is amenable. It is shown that the sofic mean rank of every G-module coincides with the mean dimension of the associated topological dynamical system. This is a joint work with Hanfeng Li.

**Markus Oliver Steenbock** (Ecole Normale Supérieure Paris, CNRS, PSL Research University)**Product set growth in groups and hyperbolic geometry**

Abstract: We discuss product theorems in groups acting on hyperbolic spaces: for every hyperbolic group there exists a constant a>0 such that for every finite subset U that is not contained in a virtually cyclic subgroup, |U^{3}|>(a|U|)^{2}. We also discuss the growth of |U^{n}| and conclude that the entropy of U (the limit of 1/n log|U^{n}| as n goes to infinity) exceeds 1/2 log(a|U|). This generalizes results of Razborov and Safin, and answers a question of Button. We discuss similar estimates for groups acting acylindrically on trees or hyperbolic spaces. This talk is on a joint work with T. Delzant.

**Francesco Veneziano** (Centro di Ricerca Matematica Ennio De Giorgi - SNS Pisa)**An effective criterion for periodicity of l-adic continued fractions**

Abstract: The theory of continued fractions has been generalized to l-adic numbers by several authors and presents many differences with respect to the real case. For example, in the l-adic case, rational numbers may have a periodic non-terminating expansion; moreover, for quadratic irrational numbers, no analogue of Lagrange’s theorem holds, and the problem of deciding whether the continued fraction expansion is periodic seems to be still open. In our paper (joint work with Laura Capuano and Umberto Zannier) we investigate the l-adic continued fraction expansions of rationals and quadratic irrationals using the definition introduced by Ruban. We give general explicit criteria to establish the possible periodicity of the expansion in both the rational and the quadratic case.

**Federico Vigolo** (University of Oxford)**Warped cones: a coarse geometric invariant of actions**

Abstract: A warped cone is a metric space that can be constructed given any action of a finitely generated group on a compact metric space. Warped cones have been successfully used to construct families of expanders, superexpanders and new classes of counterexamples to the coarse Baum-Connes conjecture.

In this talk I will show that the coarse geometry of warped cones encodes information about both the metric space and the dynamic of the action. Time permitting, I will explain why this allows us to produce a wealth of genuinely new examples of not coarsely equivalent families of expanders.