Kaimanovich asked whether every unimodular random rooted graph admits a growth exponent. Timar answered this question on the negative, producing a one-ended unimodular random rooted tree with different upper and lower growth. Such trees are always hyperfinite, in particular, when the degrees are bounded by d, their upper growth is bounded by (d-1)^1/2. We show that once the upper growth supersedes this bound, the growth will exist. The proof uses invariant percolation theory and a new result on the existence of random walk sampling exponent for subrelations of countable p.m.p. relations. We conjecture that more is true and every unimodular random rooted tree of bounded degree and infinitely many ends will admit a growth exponent. This is joint work with Mikolaj Fraczyk and Ben Hayes. This initial work opens up the possibility of a Patterson-Sullivan theory for unimodular random rooted trees and Riemannian surfaces. I will discuss possible directions and some initial results with various research groups.

How long does it take for a random walk to cover all the vertices of a graph? And what is the structure of the uncovered set (the set of points not yet visited by the walk) close to the cover time? We completely characterize the vertex-transitive graphs of bounded degree for which the cover time is in the Gumbel universality class. Surprisingly our characterization is in terms of a simple global geometric condition; this is furthermore equivalent to the decorrelation of the uncovered set at a time (in the sense that it is close to a product measure). To prove this result we rely on recent breakthroughs in geometric group theory which give a quantitative form of Gromov’s theorem on groups of polynomial growth. We also prove along the way optimal quantitative estimates giving an exponential approximation for hitting time of sets of vertices (irrespective of their geometry). Joint work with Jonathan Hermon (UBC) and Lucas Teyssier (UBC).

I will recall basic facts about the Rapid Decay property, and explain how we prove it for polynomially distorted graphs of groups, which describe 3-manifold groups without sol components. This is joint work with François Gautero.

The study of groups from the geometric, respectively from the dynamical/probabilistic viewpoint, have several points of encounter. Two among them are the investigations of actions of groups on their various types of boundaries, and on their asymptotic cones. In this setting, several notions encoding non-positive curvature (e.g. combings, globalizations of local properties) are relevant, as well as various coarse invariants (such as divergence). In this talk, I shall overview these topics and explain some recent advances. Part of the talk is based on joint work with Davide Spriano and Stefanie Zbinden.

Measure equivalence was introduced by Gromov as a measure-theoretic analogue of quasi-isometry. In this talk we will study the behaviour of graph products under measure equivalence and its ergodic counterpart: orbit equivalence. We will also highlight the links with right-angled Artin groups and geometric group theory. This is joint work with Camille Horbez

Every countable discrete group G admits a canonical topological dynamics on a 0-dimensional space, namely the action by conjugation on its space Sub(G) of subgroups. The process of successively removing the isolated points produces the perfect kernel K(G) in Sub(G) and the Cantor-Bendixson rank rk(G). I will survey some results ranging from computations of the perfect kernels and Cantor-Bendixson ranks to topological transitivity properties of the action of G, illustrated with such examples as the Baumslag-Solitar groups, hyperbolic-like groups, right-angled-Artin groups including direct product of free groups. These are joint works with P. Azuelos, S. Bontemps, A. Carderi, F. Le Maître and Y. Stalder.

The metric associated with the Liouville quantum gravity (LQG) surface has been constructed through a series of recent works and several proper- ties of its associated geodesics have been studied. In this talk we will discuss a proof of the folklore conjecture that the Euclidean Hausdorff dimension of LQG geodesics is strictly greater than 1 for all values of the so-called Liouville first passage percolation (LFPP) parameter ξ. This is deduced from an adaptation of a general criterion due to Aizenman and Burchard which in our case amounts to near-geometric bounds on the probabilities of certain crossing events for LQG geodesics in the number of crossings. We obtain such bounds using the axiomatic characterization of the LQG metric after proving a special regularity property for the Gaussian free field (GFF). If time permits, we will also discuss an analogous result for the LFPP geodesics.

In my talk I will try to connect seemingly unrelated topics from my title.

We consider inclusions of discrete, probability measure-preserving orbit equivalence relations. In previous work with Abért-Fraçzyk, we established the pointwise almost sure existence of the cospectral radius relative to the subrelation of a random walk on the equivalence classes in the bigger relation. In this paper, we investigate the connections of this cospectral radius to the coamenability of the inclusion, analogous to the group situation. We also undertake a systematic study of coamenability for inclusions of relations, establishing several equivalence formulations of this notion. Time permitting, I will discuss connections to problems in percolation theory. Previous knowledge of cospectral radius and coamenability for relations will not be assumed.

Ol'shanskii, Osin and Sapir constructed a lacunary hyperbolic group which is amenable using some kind of 'non-commutative lamplighter'. We will show that the same basic idea allows to construct all kinds of interesting groups.

It is well-established that Patterson-Sullivan measures on the boundary of a hyperbolic space, along with the associated Bowen-Margulis-Sullivan measure, provide valuable insights into the action of a group of isometries on the space’s boundary through analysis of the geodesic flow. Given that paths of a random walk on a hyperbolic groups lie close to the group’s quasi-geodesics, it is natural to ask whether similar behaviour can also be seen in the flow along bi-infinite random walk paths. In this talk, I will show how studying bi-infinite random walks on a discrete hyperbolic groups G leads to an analogue of the Patterson-Sullivan measure on ∂²G. This measure can be constructed in multiple measure-equivalent ways, each giving distinct perspectives on its intrinsic structure. Moreover, as in the classical case, the action of G on ∂²G is ergodic with respect to this measure. This talk is based on joint work in progress with Mahan Mj and Chiranjib Mukherjee.

Given a discrete set of points in a metric space, called nuclei, one associates to each such nucleus its Voronoi cell, which consists of all points closer to it than to other nuclei. In Euclidean space, one commonly uses a homogeneous Poisson point process to assign the locations of the nuclei. As the intensity of the point process tends to 0, the nuclei spread out and disappear in the limit, with each pair of space points eventually belonging to the same cell. Surprisingly, this does not happen in other settings such as hyperbolic space. We will describe properties of such a limiting tessellation, as well as analogous behavior on Cayley graphs of finitely generated groups. In particular, we discuss various ergodic properties. We will illustrate results with many pictures and videos. The talk is based on work of Sandeep Bhupatiraju and joint work with Matteo d’Achille, Nicolas Curien, Nathanael Enriquez, and Meltem Ünel.

The Poisson Voronoi Tessellation (PVT) is a natural model from stochastic geometry. There has been recent interest in its behaviour on nonamenable spaces as the intensity of the Poisson process goes to zero. It turns out that there is a limiting tessellation (the IPVT). The existence of this object can be used to prove fixed price one for some groups, resolving some conjectures of Gaboriau and Abert-Nikolov-Gelander. In this talk, I will discuss the IPVT in detail for the special case of H² × H², and sketch the proof of the key property responsible for fixed price one – in this tessellation, every pair of cells has a metrically unbounded intersection.

We study Ising models and Gibbs states on general hyperbolic groups. For large values of the inverse temperature, Series and Sinai (1990) constructed an uncountable number of mutually singular Gibbs states for Fuchsian groups, i.e. lattices in SL(2,R). We extend this result to all non-elementary hyperbolic groups. This is joint work with Ritwik Chakraborty.

Many dynamical systems admit a natural symbolic representation as subshifts over a finite alphabet. Gérard Rauzy suggested to study their complexity by associating to the subshift an infinite family of finite graphs that describe the local structure of its orbits; such graphs are known as Rauzy graphs. In this talk we will be interested in convergence properties of these sequences of graphs and of their spectra.

In this model introduced by Ráth and Rokob, we have an infinite Cayley graph Γ, a probability measure ν on rooted finite connected subsets, called lattice animals, and we place iid Poisson(λ) independent copies of them at each vertex. If the expected volume of the animals wrt ν is infinite, then the whole Γ is covered for any λ > 0. If the second moment of the volume is finite, then it is easy to see that for small enough λ the union of the animals has only finite clusters, while for λ large enough there are also infinite clusters. In joint work with Sándor Rokob, we show that: 1. If Γ is a free product, then for ANY ν with an infinite second but finite first moment and any λ > 0, there will be infinite clusters, despite having arbitrarily low density. 2. The same result holds for ANY nonamenable Γ, when the lattice animals are worms: random walk trajectories of random length. Then, how does a low density infinite cluster in a factor of iid percolation on a nonamenable Cayley graph look like? In joint work with Endre Csóka and Péter Mester, we show that, on trees, they must be just barely nonamenable, and use this to prove a quantitative strengthening for free groups of the Chifan-Ioana indistinguishability theorem. Does this strong indistinguishability theorem hold for every nonamenable group? The Poisson Zoo results suggest that at least our strategy (maybe also the result) might break down for Gromov monsters.

Given a locally finite, connected graph, a bond percolation is a random subgraph obtained by keeping or deleting edges. It is a fundamental question how geometric features of the graph are reflected in the behavior of bond percolation models and vice versa. In this talk, we will relate several large-scale geometric properties to the connectivity decay of bond percolation models with large marginals. In the setting of Cayley graphs of finitely generated groups, we will describe characterizations of the Haagerup property and Kazhdan’s property (T) through invariant bond percolations, making progress on a program suggested by Russell Lyons. We will also provide a characterization of the non-equivariant L¹-compression exponent of graphs in the possible absence of a non-trivial group action. We will highlight a common thread, which is a new construction of bond percolation models using Poisson point processes on so-called spaces with measured walls, which yields explicit bounds on the two-point function. This talk is based on joint work with Chiranjib Mukherjee.

The talk will survey recent developments regarding the (near-)critical behaviour of certain percolation models in dimensions larger than 2. Unlike the classical “long-range percolation” model obtained by adding bonds to arbitrary distances, these models involve different probabilistic objects, such as random walks and the Gaussian free field. One specific model in this class allows the rigorous derivation of various critical exponents below the upper-critical dimension. These results are proved for a large class of transient base graphs G but depend only on a small number of key parameters associated to G, a manifestation of universality.

The Poisson boundary of a random walk on a countable group is a probability space that encodes the asymptotic behavior of sample paths. Given a group G endowed with a probability measure μ, it is a natural problem to identify an explicit model of the associated Poisson boundary, described in terms of the geometry of G. In this talk I will speak about the identifica- tion problem for wreath products, the main example being the lamplighter groups Z/2Z ≀ Z^d, for d ≥ 3. I will explain results that describe the Poisson boundary for probability measures with finite entropy and that satisfy a stabilization condition that naturally arises in this context. This is joint work with Joshua Frisch.

We show that there are 2^ℵ0 non-isomorphic universal sofic groups. This proves a conjecture of Simon Thomas. This is joint work with Vadim Alekseev.

Several recent methods for proving pointwise ergodic theorems for pmp actions of free groups critically use weak mixing properties of Markov measures on the boundary of a free group of finite rank. However, it was not known exactly which Markov measures are weak mixing. In joint work with Jenna Zomback, we give a complete characterization of such measures. It turns out that, under mild non-degeneracy assumptions, they are exactly the Markov measures arising from strictly irreducible transition matrices – a condition introduced by Bufetov in 2000 for a different purpose. The proof of this characterization goes through proving equivalences with a new combinatorial condition on the action that we call chaining, which is interesting in its own right.

We show that all non-amenable non-unimodular Baumslag-Solitar groups BS(r,s), 2 ≤ |r| < s, are measure equivalent to each other, thereby completing the measure equivalence classification of Baumslag-Solitar groups. Consequently, each BS(r,s) belongs to one of three measure equivalence classes according to whether it is amenable (|r| = 1 or |s| = 1), virtually isomorphic to F_n × Z (2 ≤ |r| = |s|), or non-amenable and non-unimodular (2 ≤ |r| < s). This is joint work with Damien Gaboriau, Antoine Poulin, Anush Tserunyan, and Konrad Wrobel.

A length function ℓ on a group G is a function from G to the nonnegative real numbers satisfying the following for all group elements x and y: ℓ(x) = 0 if and only if x=1_G, ℓ(x^-1) = ℓ(x), and ℓ(xy) ≤ ℓ(x)+l(y). In this talk, we will investigate the asymptotic behavior of compatible length functions on Polish groups, and in particular, the extent to which a sphere of large radius with respect to one length function looks spherical with respect to another. This is joint work with Christian Rosendal.