Abstract: I will describe the construction of groups of intermediate word growth, starting with Slava Grigorchuk's original proof and showing how it can be optimized to give a sharp upper bound on his group's growth. This method is combinatorial, but has a geometric avatar, which gives growth estimates of the type enα for some finitely generated simple groups, including a simple group containing Grigorchuk's original example. This last part is joint work with Tianyi Zheng and Volodymyr Nekrashevych.

Abstract: A finitely generated, residually finite group is profinitely rigid if its set of finite quotients distinguishes it from all other finitely generated, residually finite groups.  This is the first of two talks (the second one by Martin Bridson) on profinite rigidity. This talk will survey some of our work, for example on proving certain lattices in PSL(2,R) and PSL(2,C) are profinitely rigid, and setting up the second talk in which Martin will talk about some of our recent work on profinite rigidity and finiteness properties.

Abstract: Dimension groups are complete invariants for orbit equivalence of minimal Cantor dynamics. This was shown in 1995 by Thierry Giordano, Ian Putnam and Christian Skau. There is no doubt that this is one of the best results in topological dynamics and ergodic theory of the 90's and probably even more. In this talk we will survey many properties of these abelian groups and its counterpart for dynamical systems. We will discuss, for example, the invariant measures, the entropy, the dynamical eigenvalues, mixing properties, ... This will be illustrated by numerous examples such as substitution, S-adic, dendric, Ferenczi, Toeplitz subshifts.

Abstract: The Rokhlin lemma is a finite approximation property that underpins a great many constructions in classical ergodic theory, including most spectacularly those involving entropy in the Ornstein isomorphism theory for Bernoulli shifts. In the 1970s Ornstein and Weiss showed amenability to be the natural setting for finite approximation in dynamics by establishing a general form of the Rokhlin lemma in this setting, and this led, among other things, to a much broader recasting of the Ornstein isomorphism theory. Over the last couple of decades a growing interest in the interplay between dynamics and the geometric and analytic structure of groups has set the stage for a resurgence of applications of the Ornstein-Weiss Rokhlin lemma, once again confirming its utility as a powerful and versatile tool. I will sketch some of the recent applications to entropy and orbit equivalence.

Abstract: In this talk I will show some examples of Schreier graphs of automaton groups focusing on the problem of their (topological and isometric) classification at the boundary. In the second part I will present some new results concerning the Schreier graphs of the p-Basilica groups.

Abstract: Most of the classical limit theorems of probability theory for sums of independent random variables (the law of large numbers, the central limit theorem, etc.) admit analogues for observables over finite state spaces Markov chains. This together with some general ideas from geometry and dynamical systems allows to get fine counting estimates on languages defined by automata, hence in particular in hyperbolic groups. In the present talk, I will present such counting estimates in the particular case of non abelian finitely generated free groups. This is a joint work with Rostislav Grigorchuk.

Abstract: The Whitehead algorithm for a free group Fr decides whether, given two elements w,v ∈ Fr , there exists an automorphism φ of Fr such that φ(w)=v.

Whitehead produced an algorithm for solving this problem in a classic 1936 paper. Despite much study since then, the worst-case complexity of Whitehead’s algorithm is still not well understood, and for r ≥ 3 the only known worst case bound is exponential time in max{|w|,|v|}. 

We study the generic-case complexity of Whitehead’s algorithm for inputs generated by several natural types of random walks on Fr , including group-based and graph-based random walks. We prove that for “random” elements wn ∈ Fr generated by such walks Whitehead’s algorithm works in at most quadratic time on wn , v for any v ∈ Fr.

The machinery of geodesic currents on free groups serves as the main tool for the proof. A key step is establishing that there is a geodesic current μ on Fn  with nice properties such that along a random trajectory w1 , w2 , w3 ,… of the walk the conjugacy classes [wn] converge projectively to the current μ.

Abstract: There is a rich interplay between group theory and theoretical computer science, and more precisely the theory of automata and formal languages.

In this talk, we will give an introduction to automata and formal languages, explaining their connections with groups of automorphisms of rooted trees. Then, we present examples, recent developments, and open problems.

Abstract: Shannon entropy of simple random walks can be thought of as a probabilistic version of (logarithm of) growth. Slava Grigorchuk's arguments to estimate growth, as described in Laurent Bartholdi's talk, can be adapted to obtain sublinear estimates on entropy and on speed of simple random walks. In the opposite direction, the existence of (non simple) random walks with linear entropy on the first Grigorchuk group is central in the sharp lower bounds estimates obtained by Anna Erschler and Tianyi Zheng.

Abstract: The Thompson’s group F is one of the most famous and very important groups related to many areas of mathematics.  The open question about amenability of this group can be expressed in terms of spectra of Markov operators of random walks on F. Moreover, amenability of F would imply amenability of all Schreier graphs of F. This motivates studying spectral properties of these graphs.

In this talk I will present results on spectral properties of the family of Schreier graphs associated to the action of the Thompson group F on the interval [0,1]. In particular, I will describe spectra of Laplace type operators associated to these Schreier graphs and calculate certain spectral measures associated to them.  As a byproduct we will obtain the asymptotics of the return probabilities of the simple random walk on the orbit of 1/2. The talk is based on a joint work with Rostislav Grigorchuk.

Abstract: A few years ago, McReynolds, Reid, Spitler and I proved that certain arithmetic lattices in PSL(2,R) and PSL(2,C) are profinitely rigid; Alan Reid will discuss this in his talk.

By taking central extensions of the Fuchsian examples, one obtains infinite families of 3-manifold groups that are profinitely rigid. Certain of these 3-manifold groups G have the property that no other *finitely presented*, residually finite group has the same finite quotients as G×G, but there are infinitely many non-isomorphic *finitely generated* subgroups H<G×G that have the same finite quotients as G×G. (This is joint work with Alan Reid and Ryan Spitler.)

In this talk I will explain why different finiteness properties are important in the context of  profinite rigidity.

Abstract: In symbolic dynamics, the Garden of Eden Theorem (originally proved by E.F. Moore and J. Myhill for cellular automata over Zd) yields a new characterization of group amenability.

After introducing the notions of cellular automata, I'll briefly sketch the main ideas and tools (such us entropy) used to prove these results.

Then, after introducing the notion of a subshift (together with its irreducibility and finiteness properties) I'll concentrate on a version of the GOE theorem for cellular automata on subshifts over Z and discuss some interesting graph theoretical constructions involved.

Abstract: Branch groups, which were first formally introduced by Grigorchuk in 1997, are a special kind of groups of automorphisms of rooted trees. They have attracted a fair amount of scrutiny due to their propensity for strange or unusual behaviour, possessing properties which are often difficult or impossible to find in more classical groups.

The normal subgroup structure of a branch group is, in a weak sense, similar to the rooted tree on which it acts. This suggests that one could use this action to extract algebraic information about general subgroups of branch groups. In this talk, we will survey the work that has been done in this direction and present new results.

Abstract: Let n ≥ 2 be an integer and let Gn  denote the space of n-generated marked groups. We show that there exist 2ℵ0 non-atomic Aut(Fn )-invariant probability measures on Gn  such that the action of Aut(Fn ) on Gn is ergodic. Our proof makes use of acylindrical hyperbolicity of Aut(Fn ). On the other hand, we obtain some restrictions on the supports of Aut(Fn )-invariant ergodic probability measures on Gn. In particular, we show that the support of such a measure cannot contain the perfect core of Gn.

Abstract: In this talk I will give an overview of the work done in the last 10 years to characterise solutions to equations in groups in terms of formal languages, starting with free and hyperbolic groups.  I will then describe the most recent progress in the area, which has been made for groups without negative curvature, such as virtually abelian, the integral Heisenberg group, or the soluble Baumslag-Solitar groups, where the approaches to describing the solutions are different from the negative curvature setup.

Abstract: I will introduce the Gromov density model and other models of random groups and will explain why a random group with density d<1/2 has linear isoperimetric inequality and is therefore hyperbolic (Gromov, Ollivier's result).  I will discuss our results about equations for d<1/16, that all their solutions are coming with overwhelming probability from solutions in free groups, and discuss other properties of random groups.

Abstract: We show that both the branch completion and the rigid kernel of a regular branch group G are finitely constrained groups (group tree shifts defined by finitely many forbidden patterns). If, in addition, G is self-replicating, then its closure is also a finitely constrained group. The last property says that the closure of a self-replicating regular branch group G is branching over a level stabilizer even when G itself does not have this property (as in the case of Hanoi Towers group). We introduce several examples during the talk, including groups with infinite rigid kernels, groups with rigid kernels of various exponents and, time permitting, a group with a nonabelian rigid kernel. We note that some of these examples, in the spirit of the conference title, have dynamical origin (as iterated monodromy groups of rational maps on the Riemann sphere). Most of the presented work is joint with Alejandra Garrido and/or Jone Uria Albizuri.

Abstract: In this talk, we describe a process whereby we can embed any hyperbolic group into a finitely presented infinite simple group.  This gives a proof of what is, in some sense, the ​"typical" case of the Boone-Higman Conjecture of 1973 (that a finitely generated group G has a solvable word problem if and only if G can be embedded into a finitely presented simple group).

The talk will proceed in three roughly independent parts: A short history of the Boone-Higman Conjecture,  a discussion of Hyperbolic groups and an embedding of these into the Rational group of Grigorchuk, Nekrashevych, and Suschanskii, and finally, a discussion of why the topological full group over this rational group is finitely presented and simple, or, at least provides a route to a further embedding in a group that is finitely presented and simple.  

Joint with James Belk, Francesco Matucci, and Matthew Zaremsky.

Abstract: We will discuss the spectrum of the Laplacian on Schreier graphs of some self-similar groups, like the Grigorchuk group, Hanoi group, and the Basilica group. Classical Schur renormalization transformations act on suitable spectral parameters as rational maps in two variables.  We will show that the spectra in question can be interpreted as the asymptotic distributions of sliced pullbacks of certain algebraic curves under the iterates of these rational maps. In the Grigorchuk case the density of states turns out to be absolutely continuous, in the Hanoi case it is singular and supported on a Cantor set of zero length, yet in the Basilica case it is singular but supported on a Cantor set of positive length. We will give a self-contained non-technical introduction to the interplay between spectral theory and dynamics that helps to address such problems.

Based upon a joint work with E. Bedford, N.-B. Dang, and R. Grigorchuk.

Abstract: We suggest a new definition of higher-dimensional automata motivated by cocompact quotients of buildings. We construct infinite series of such automata and produce very explicit constructions of Ramanujan higher-dimensional graphs. The talk is based on joint results with Ievgen Bondarenko, Rostislav Grigorchuk and Jakob Stix.