I will discuss recent results establishing hyperfiniteness of the equivalence relations induced by actions on the Gromov boundaries of various hyperbolic spaces. This includes boundary actions of hyperbolic groups (joint work with T. Marquis) and actions of the mapping class group on the boundaries of the arc graph and the curve graph (joint work with P. Przytycki).
I will discuss about quantitative versions of Measure Equivalence for locally compact compactly generated groups, a notion introduced by Tessera, Le Maître, Delabie and Koivisto on the finitely generated case. Moreover, they introduced as well quantitative asymmetric versions of it, called $L^p$-measured subgroups, and in particular they proved that coarse embeddings between amenable groups imply the existence of a $L^\infty$-measured coupling. In this talk, I will prove the same statement on the locally compact case, which will gives us an obstruction to coarse embeddings for locally compact compactly generated groups.
Given a topological dynamical system (A group G acting by homeomorphisms on a compact space X) a measures on X is said to be characteristic if it is invariant to the automorphism group of the system. A system is called minimal if it has no closed G invariant subsystems. In this talk I will give a brief introduction to characteristic measures before explaining the main result: a minimal dynamical system without characteristic measures. This is joint work with Brandon Seward and Andy Zucker
The theory of IRS (invariant random subgroups) has proven to be very useful to the study of lattices and their asymptotic invariants. However, restricting to invariant measures (on the space of subgroups) is a big compromise (since our groups are non-amenable) and limits the scope of problems that one can investigate. I will explain a fundamental inequality concerning the discreteness function, proved in a joint work with G.A. Margulis and A. Levit, which allows extending various results about IRS to SRS (stationary random subgroups). The same inequality gives control on certain random walks on the space ofdiscrete subgroups. Finally I will outline the proof obtained jointly with M. Fraczyk of the following conjecture of Margulis: Let G be a higher rank simple Lie group and D a discrete subgroup. If D is confined, i.e. if there is a compact set in G\{1} which meets every conjugate of D, then D is a lattice in G. This result gives a far reaching generalisation of the celebrated normal subgroup theorem of Margulis.
In a recent work, I introduced the notion of allosteric actions: a minimal action of a countable group on a compact space, with an ergodic invariant measure, is allosteric if it is topologically free but not essentially free. In the first part of my talk I will explain some properties of allosteric actions, and their links with Invariant Random Subgroups (IRS). In the second part, I will explain a recent result of mine: the fundamental group of a closed hyperbolic surface admits allosteric actions.
In the context of large (non-locally compact) topological groups, one frequently witnesses an extreme form of amenability: extreme amenability. A topological group G is called extremely amenable if everycontinuous action of G on a non-void compact Hausdorff space admits afixed point. Most of the currently known manifestations of this phenomenon have been exhibited using either structural Ramsey theory, or concentration of measure. The talk will be focused on the latter method. Among other things, I will discuss a new concentration result for convolution products of invariant means, based on a suitable adaptation of Azuma's inequality. Furthermore, I will show how this result can beused to prove extreme amenability of certain topological groups arising from von Neumann's continuous geometries.
In the first half, we report on joint work with Mehrdad Kalantar in which we completely characterize C*-simplicity of quasi-regular representations associated to stabilizers of boundary actions in terms of amenability of the isotropy groups of the groupoid of germs of the action. For quasi-regular representations associated to "open" stabilizers, a complete characterization of C*-simplicity is still missing, and we illustrate this fact with an ad hoc proof that, for Thompson's group F < T, the quasi-regular representation of T associated to [F,F] properly weakly contains the one associated to F (a year ago Kalantar spoke at this seminar and I will emphasize the new results and examples obtained since then).In the second half, we show that the topological full group of a minimal action on the Cantor set is C*-simple if and only if the alternating full group is non-amenable. We use this to conclude that, e.g., for free actions of groups of subexponential growth, non-amenability of the topological full group is equivalent to C*-simplicity, but in general this equivalence is an open problem.
Automata groups are a family of groups acting on rooted trees that have a simple definition yet exhibit a very rich behavior. Automaton groups include many interesting examples such as Grigorchuk groups, the Basilica group, Hanoi tower groups and lamplighter groups.
The activity of an automaton group, introduced by Sidki, can be viewed as a measure of complexity that can grow either polynomially (with some degree) or exponentially. Sidki proved that polynomial activity automata groups do not contain free subgroups, which prompted him to ask “Are all polynomial activity automata groups amenable?”
This was answered positively for degree 0 (“bounded”) by Bartholdi-Kaimanovich-Nekrashevych and for degree 1 (“linear”) by Amir-Angel-Virag.
Juschenko, Nekrashevych and de la Salle gave a general approach allowing to deduce the amenability of groups from recurrence of the orbital Schreier graphs of group actions satisfying some conditions. This allowed, among other things, to reprove the amenability of automata groups of degree 0 and 1, and to prove the conditional result that if the "natural" action of a quadratic activity (d=2) automata group is recurrent then it is amenable.
In recent work with Omer Angel and Balint Virag, we prove that the natural Schreier graphs of the quadratic activity mother groups, a special family into which all quadratic activity automata groups can be embedded, is recurrent. This allows us to conclude the amenability of all quadratic activity automata groups.The proof relies on bounding the electrical resistance between vertices in the Schreier graphs, which in turn relies on a "combinatorial" analysis of the graph structure together with new Nash-Williams type lower bound on resistances.
After surveying some background on automata groups, mother groups and electrical resistance, and some previous amenability results on automata groups, we will focus on the new analysis giving the resistance lower bounds. No previous knowledge on random walks, automata groups or electrical resistance will be assumed. This talk is based on joint work with O. Angel and B. Virag.
Random groups are one way to study "typical" behavior of groups. In the Gromov density model, we often find a threshold density above which a property is satisfied with probability 1, and below which it is satisfied with probability 0. Two properties of random groups that have studied are cubulation and Property (T). In this setting these are mutually exclusive, but the threshold densities are not known. In this talk I'll present a method to demonstrate cubulation on groups with density less than 3/14, and discuss how this might be extended to demonstrate cubulation for densities up to 1/4. In particular, I will describe a construction of walls in the Cayley complex X which give rise to a non-trivial action by isometries on a CAT(0) cube complex.
This extends results of Ollivier-Wise and Mackay-Przytycki at densities less than 1/5 and 5/24, respectively.
Magnus' Freiheitssatz states that if a group is defined by a presentation with m generators and a single cyclically reduced relator, and this relator contains the last generating letter, then the first m-1 letters freely generate a free subgroup. We study an analogue of this theorem in the Gromov density model of random groups, showing a phase transition phenomenon at density d_r = min{1/2, 1-log_{2m-1}(2r-1)} with 0<r<m: we prove that for a random group with m generators at density d, if d<d_r then the first r letters freely generate a free subgroup; whereas if d>d_r then the first r letters generate the whole group.
A random group in the triangular binomial model Gamma(n,p) is given by the presentation < S|R >, where S is a set of n generators and R is a random set of cyclically reduced relators of length 3 over S, with each relator included in R independently with probability p. When n tends to infinity, the asymptotic properties of groups in Gamma(n,p) vary widely with the choice of p=p(n). By Antoniuk-Łuczak-Świątkowski and Żuk, there existconstants C, C', such that a random triangular group is asymptotically almost surely (a.a.s.) free if p<Cn^{-2} and a.a.s.infinite, hyperbolic, but not free, if p in (C'n^{-2}, n^{-3/2-\varepsilon}). We generalize the second statement by findinga constant c such that if p\in(cn^{-2}, n^{-3/2-\varepsilon}), then a random triangular group is a.a.s. not left-orderable. We prove this by linking left-orderability of Gamma in Gamma(n,p) to thesatisfiability of the random propositional formula, constructed from the presentation of Gamma. The left-orderability of quotients will be also discussed.
Button observed that finitely generated matrix groups containing no nontrivial unipotent matrices behave much like groups admitting proper actions by semisimple isometries on complete CAT(0) spaces. It turns out that any finitely generated matrix group possesses an action on such a space whose restrictions to unipotent-free subgroups are in some sense tame. We discuss this phenomenon and some of its implications for the representation theory of certain 3-manifold groups.
Buildings of type Ã_2 are commonly associated to groups such as G=SL_3(k), where k is a non-archimedean local field. Lattices in such a group G have strong rigidity properties (for example, they satisfy Margulis' superrgidity). But there are also buildings for which the automorphism group is smaller, and much less understood - but in some cases still cocompact. In this talk I will explain how these other "exotic" lattices are still very rigid, and raise some open questions.
Given a two measure-preserving ergodic action of countable groups on a standard probability space, Dye's reconstruction theorem asserts that any isomorphism between the associated full groups must come from an isomorphism of the space which sends the first partition of the space into orbits to the second. It is thus natural to ask what happens more generally for homomorphisms between full groups. I will present a joint work with Alessandro Carderi and Alice Giraud where we show that any such homomorphism comes from a measure-preserving action of the equivalence relation or of one of its symmetric powers. Such a result is very similar in spirit to Matte Bon's striking classification of actions by homeomorphisms of topological full groups, but we will see that the proof is much simpler modulo the Thomas-Tucker-Drob classification of invariant random subgroups of the dyadic symmetric group. As an application, we characterize Kazhdan's property (T) of a measure-preserving equivalence relation in terms of its full group: the equivalence relation has (T) if and only if all non-free ergodic Boolean actions of its full group are strongly ergodic.
We will present a new induction technique based on ideas of Gromov and Shalom. Given two finitely generated groups H and G and a Lipschitz injective map from H to G, we construct a topological coupling space between them. If H is amenable, then this enables us to view H as a ``measured subgroup" of G. Using this formalism, we manage to prove that the Folner function of G grows faster than the Folner function of H.
An application of this result is the following (new) theorem: an amenable group coarsely embeds into a hyperbolic group if and only it is virtually nilpotent.
After an introduction to the topic of highly transitive groups, I will present a joint work with F. Le Maître, S. Moon and Y. Stalder in which we characterize groups acting on trees which are highly transitive.
A JSJ decomposition is a graph of groups decomposition that allows one to classify all splittings of a group over certain subgroups. I will discuss a JSJ decomposition for relatively hyperbolic groups splitting over elementary subgroups that depends only on the topology of its boundary. This decomposition could potentially be of use for understanding groups that have homeomorphic boundaries, but are not necessarily quasi-isometric. (Joint work with Matt Haulmark.)
After a general introduction to lamplighter groups and their asymptotic geometry, I will describe a complete quasi-isometric classification of lamplighters over one-ended finitely presented groups. The proof will be briefly overviewed, and the rest of the talk will be dedicated to the central tool of the argument: an embedding theorem proved thanks to (quasi-)median geometry.
This second talk will be dedicated to the asymmetry between amenable and non-amenable groups in the quasi-isometric classification previously described. In particular, I will explain why lamplighters over non-amenable groups are more often quasi-isometric than lamplighters over amenable groups. Also, I will show how the distance from a quasi-isometry between amenable groups to a bijection can be quantified, introducing quasi-k-to-one quasi-isometries for an arbitrary real k>0, and explain how this notion is fundamental in the understanding of the asymptotic geometry of lamplighters over amenable groups.
A recurring question in the theory of random walks on hyperbolic spaces asks whether the hitting (harmonic) measures can coincide with measures of geometric origin, such as the Lebesgue measure. This is also related to the inequality between entropy and drift.
For finitely-supported random walks on cocompact Fuchsian groups with symmetric fundamental domain, we prove that the hitting measure is singular with respect to Lebesgue measure; moreover, its Hausdorff dimension is strictly less than 1.
Along the way, we prove a purely geometric inequality for geodesic lengths, strongly reminiscent of the Anderson-Canary-Culler-Shalen inequality for free Kleinian groups.
Joint with P. Kosenko.
Consider a random walk on a nonelementary hyperbolic space (proper or not, but one may just think of a free group for simplicity). It is known that the walk is converging almost surely towards a point at a boundary, and that the rate of escape is positive. We will discuss quantitative versions of these statements: when can one show that these facts hold with an exponentially small probability for exceptions? While there are several such results in the literature, the originality of our approach is that it does not require any moment condition on the random walk. We will discuss the main technical new idea in the case of the free group.
In 80s Vadim Kaimanovich presented a construction of a non-degenerate measure on the standard lamplighter group which has trivial right random walk boundary and non-trivial left random walk boundary. I will show that examples of such kind are possible exactly for amenable groups with non-trivial ICC factors.
It is easy to see that there are Riemannian manifolds homeomorphic to R3 with infinite asymptotic dimension. In contrast to this we showed with K. Fujiwara that the asymptotic dimension of Riemannian planes (and planar graphs) is bounded by 3. This was improved to 2 by Jorgensen-Lang and Bonamy-Bousquet-Esperet-Groenland-Pirot-Scott.
There is a class of random walks on some countable discrete groups that capture the asymptotic behaviour of certain random walks on totally disconnected locally compact second countable (t.d.l.c.) groups which are completions of the discrete group. We will see that the Poisson boundary of the t.d.l.c. group is always a factor of the Poisson boundary of the discrete group, when equipped with these random walks. All this is done by means of a so-called Hecke subgroup.
In particular, if the two Poisson boundaries are isomorphic then this Hecke subgroup is forced to be amenable. The reverse direction holds whenever there is a uniquely stationary compact model for the Poisson boundary of the discrete group. Furthermore, we will deduce some applications to concrete examples, like the lamplighter group over Z and solvable Baumslag-Solitar groups and show that they are prime, i.e. there are random walks such that the Poisson boundary and the one-point-space are the only boundaries.
This is a joint work with Michael Björklund (Chalmers, Sweden) and Yair Hartman (Ben Gurion University, Israel).
Let G be a discrete amenable group. We study interrelations between topological and measure-theoretic actions of G. For a given continuous representation of G on a compact metric space X we consider the set of all ergodic invariant measures on X. For any such measure we associate the corresponding measure-theoretic dynamical system. The general wild question is what the family M of these systems could be up to measure-theoretic isomorphisms.
The topological system for which M coincides with a given class S of ergodic actions is called universal. B.Weiss’s question regards the existence of a universal system for the class of all zero-entropy actions. For the case of Z, the negative answer was given by J. Serafin.
Our main result establishes the non-existence of a universal zero-entropy system for any non-periodic amenable group. The condition of non-periodicity is crucial in our arguments so the question is still open for general torsion amenable groups.
Our proof bases on the slow entropy type invariant called scaling entropy introduced by A. Vershik. This invariant characterizes the intermediate growth of the entropy in a sense on the verge of topological and measure-preserving dynamics. I will present a brief survey of scaling entropy and show how this invariant applies to the non-existence theorem.
De Bruijn graphs represent word overlaps in symbolic dynamical systems. They naturally occur in dynamical systems and combinatorics, as well as in computer science and bioinformatics. We will show that de Bruijn graphs converge to a Cayley graph of the Lamplighter group and and will also compute their spetra. We will then discuss some generalizations of them as for examples Spider web graphs or Rauzy graphs.
Based on a joint work with R. Grigorchuk and T. Nagnibeda.
We are motivated by looking for traces of hyperbolicity in a space or group which is not Gromov-hyperbolic. One previous approach in this direction is the notion of Morse quasigeodesics, which describes "negatively-curved" directions in the spaces; another previous approach is "higher rank hyperbolicity" with one example being that though triangles in products of two hyperbolic planes are not thin, tetrahedrons made of minimal surfaces are "thin". We introduce the notion of Morse quasiflats, which unifies these two seemingly different approaches and applies to a wider range of objects. In the talk, we will provide motivations and examples for Morse quasiflats, as well as a number of equivalent definitions and quasi-isometric invariance (under mild assumptions). We will also show that Morse quasiflats are asymptotically conical, and comment on potential applications. Based on joint work with B. Kleiner and S. Stadler.
Event structures, trace automata, and Petri nets are fundamental models in concurrency theory. There exist nice interpretations of these structures as combinatorial and geometric objects and both conjectures can be reformulated in this framework. Namely, from a graph theoretical point of view, the domains of prime event structures correspond exactly to median graphs; from a geometric point of view, these domains are in bijection with CAT(0) cube complexes.
Thiagarajan conjectured that regular event structures correspond exactly to event structures obtained as unfoldings of finite 1-safe Petri nets. Using the bijections between event structures, median graphs and CAT(0) cube complexes, we disproved this conjecture. Our counterexample is derived from an example by Wise of a nonpositively curved square complex whose universal cover is a CAT(0) square complex containing a particular plane with an aperiodic tiling.
On the positive side, we show that event structures obtained as unfoldings of finite 1-safe Petri nets correspond to the finite special cube complexes introduced by Haglund and Wise.
A "Complete Square Complex" is a 2-complex X whose universal cover is the product of two trees. Obvious examples are when X is itself the product of two graphs but there are many other examples. I will give a quick survey of complete square complexes with an aim towards describing some problems about them and describing some small examples that are "irreducible" in the sense that they do not have a finite cover that is a product.
Let G be a finitely generated group with generating set S. We study the cogrowth sequence {an(G,S)}, which counts the number of words of length n over the alphabet S that are equal to 1 in G. I will survey rеcent asymptotic and analytic results on the cogrowth sequence, motivated by both combinatorial and algebraic applications. I will then present our recent work with Kassabov on spectral radii of Cayley graphs, which are also governed by the asymptotics of cogrowth sequences.
This talk concerns the situations when the Poisson boundaries of different random walks on the same group coincide. In some special cases, Furstenberg and Willis addressed this question. However, the scopes of their constructions are limited. I will show how randomized stopping times can construct measures that preserve Poisson boundaries and discuss their applications regarding the Poisson boundary identification problem. This talk is based on joint work with Kaimanovich.
A continuous action of a group G on a compact space X is said to be a boundary action if the weak*-closure of the orbit of every Borel probability on X under G-action contains all point measures on X. Given a boundary action of a discrete countable group, we prove that at any continuity point of the stabilizer map, the quasi-regular representation of the stabilizer subgroup is weakly equivalent to every representation that it weakly contains. We also completely characterize when these quasi-regular representations weakly contain the GNS representation of a character on the group. This is joint work with Eduardo Scarparo.
In this talk, we construct Hölder maps to the Heisenberg group H, answering a question of Gromov. Pansu and Gromov observed that any surface embedded in H has Hausdorff dimension at least 3, so there is no α-Hölder embedding of a surface into H when α > 2/3. Züst improved this result to show that when α > 2/3, any α-Hölder map from a simply-connected Riemannian manifold to H factors through a metric tree. We use new techniques for constructing self-similar extensions to show that any continuous map to H can be approximated by a (2/3 - ε)-Hölder map. This is joint work with Stefan Wenger.
Sol, one of the eight Thurston geometries, is a solvable three-dimensional Lie group equipped with a canonical left invariant metric. Sol has sectional curvature of both signs and is not rotationally symmetric, which complicates the study of its Riemannian geometry.
Our main result is a characterization of the cut locus of Sol, which implies as a corollary that the metric spheres in Sol are topological spheres. This is joint work with Richard Schwartz.
This is a sequel talk, following Matei Coiculescu’s talk about our joint work characterizing the cut locus of the identity in Sol. In this talk, I will explain my result that the area of a metric sphere of radius r in Sol is at most Cer for some uniform constant C. That is, up to constants, the sphere of radius r in Sol has the same area as the hyperbolic disk of radius r.
It is well-known that the quasi-convex hull of finitely many points in a hyperbolic space is quasi-isometric to a tree. I will discuss an analogous fact in the context of hierarchically hyperbolic spaces, a large class of spaces and groups including mapping class groups, Teichmueller space, right-angled Artin and Coxeter groups, and many others. In this context, the approximating tree is replaced by a CAT(0) cube complex. I will also briefly discuss applications, including how this can be used to construct bicombings. Based on joint works with Behrstock-Hagen and Durham-Minsky.
For any hierarchical hyperbolic group, and in particular any mapping class group, we define a new metric that satisfies a coarse Helly property. This enables us to deduce that the group is semihyperbolic, i.e. that it admits a bounded quasigeodesic bicombing, and also that it has finitely many conjugacy classes of finite subgroups. This has several other consequences for the group. This is a joint work with Nima Hoda and Harry Petyt.
From its hyperplanes, one can always characterise a CAT(0) cube complex as the subset of some (often infinite) cube consisting of the solutions to a system of "consistency" conditions. Analogously, a hierarchically hyperbolic space (HHS) can be coarsely characterised as a subset of a product of Gromov-hyperbolic spaces consisting of the "solutions" to a system of coarse consistency conditions.
HHSes are a common generalisation of hyperbolic spaces, mapping class groups, Teichmuller space, and right-angled Artin/Coxeter groups. The original motivation for defining HHSes was to provide a unified framework for studying the large-scale properties of examples like these.
So, it is natural to ask about the structure of asymptotic cones of hierarchically hyperbolic spaces.
Motivated by the above characterisation of a CAT(0) cube complex, we introduce the notion of an R-cubing. This is a space that can be obtained from a product of R-trees, with the ℓ1 metric, as a solution set of a similar set of consistency conditions. R-cubings are therefore a common generalisation of R-trees and (finite-dimensional) CAT(0) cube complexes. R-cubings are median spaces with extra structure, in much the same way that HHSes are coarse median spaces with extra structure.
The main result in this talk says that every asymptotic cone of a hierarchically hyperbolic space is bilipschitz equivalent to an R-cubing. This strengthens a theorem of Behrstock-Drutu-Sapir about asymptotic cones of mapping class groups. Time permitting, I will talk about an application of this result which is still in progress, namely uniqueness of asymptotic cones of various hierarchically hyperbolic groups, including mapping class groups and right-angled Artin groups. This is joint work with Montse Casals-Ruiz and Ilya Kazachkov.
A group G is called a Q-group if for any natural number n and any element g from G there exists a unique nth root of g in G. These groups were introduced by G. Baumslag in the sixties under the name of D-groups. The free Q-group on X can be constructed from the free group on X by applying an infinite number of amalgamations over cyclic subgroups. In this talk I will explain how to show that free Q-groups are residually torsion-free nilpotent. This solves a problem raised by G. Baumslag. A key ingredient of our argument is the proof of one instance of the Lueck approximation in characteristic p corresponding to an embedding of a finitely generated group into a free pro-p group. For more details see http://matematicas.uam.es/~andrei.jaikin/preprints/baumslag.pdf.
Slides for this talk are available here.
(joint work with Jacob Fox and Huy Tuan Pham.)
We develop novel techniques which allow us to prove a diverse range of results relating to subset sums and complete sequences of positive integers, including solutions to several longstanding open problems. These include: solutions to the three problems of Burr and Erdos on Ramsey complete sequences, for which Erdos later offered a combined total of $350; analogous results for the new notion of density complete sequences; the solution to a conjecture of Alon and Erdos on the minimum number of colors needed to color the positive integers less than n so that n cannot be written as a monochromatic sum; the exact determination of an extremal function introduced by Erdos and Graham on sets of integers avoiding a given subset sum; and, answering a question of Tran, Vu and Wood, a strengthening of a seminal result of Szemeredi and Vu on long arithmetic progressions in subset sums.