BGU Probability and Ergodic Theory (PET) seminar Ical Atom

I will talk about the coarse geometry of lattices in real semisimple Lie groups. One great result from the 1990’s is the quasi-isometric rigidity of these lattices: any group that is quasi-isometric to such a lattice must be, up to some minor adjustments, isomorphic to lattice in the same Lie group. In this talk I present a partial generalization of this result to the setting of Sublinear Bilipschitz Equivalences (SBE): these are maps that generalize quasi-isometries in some `sublinear’ fashion.

Subshifts of finite type are the central objects studied in symbolic dynamics.

In the one dimensional case, (e.g. subshifts of finite type when the acting group is Z, the group of integers), although there are difficult standing unsolved problems (in particular, the isomorphism problem), there is a reasonable and fairly developed structure theory:

• Any Z-subshift of finite type “decomposes” into irreducible components and wandering points, where any irreducible SFT becomes topologically mixinig after passing to some power of the shift.

• Krieger’s embedding theorem provides “essentially checkable” necessary and sufficient conditions for an arbitrary subshift to embed in a given topologically mixing SFT.

• Boyle’s factor theorems give “essentially checkable” conditions for factoring between mixing SFTs.

The situation for multidimensional subshifts is far less structured and far more mysterious.

By now it is well-known that multidimensional subshifts of finite type can exhibit a wild variety of ``pathological behavior’’.

One is soon faced with undecidability issues, and there seems to be little hope to obtain a tractable structure theory in complete generality.

Over the years various properties of multidimensional subshifts have been introduced and studied, in an attempt to recover and generalize some structural aspects of the one-dimensional theory for a natural class.

Among these properties: “square mixing”, “block gluing”, “strong irreducibility”, “topological strong spatial mixing”, “the finite extension property” and more…

In this talk I will introduce a natural class of multidimensional subshifts of finite type for which I have obtained extensions of the fundamental theorems from the one dimensional case.

This new class of subshifts has various equivalent characterizations. The first characterization is the map extension property of subshifts.

The map extension property has been introduced implicitly by Mike Boyle in the early 1980’s for Z-subshifts.

In a suitable natural formulation, in the context of subshifts, it turns out to coincide with the notion of an absolute retract, introduced by Borsuk in the 1930’s.

The map extension property is a stronger property than strong irreducibly, but it still holds for a variety of ``reasonable’’ subshifts such as any subshift with a safe symbol or proper colorings of (the standard Cayley graph of) Z^2 with 5 or more colors.

A Z-subshift has the map extension property if and only if it is a mixing subshift of finite type.

The map extension property allows a meaningful complete multidimensional generalization of both Kreiger’s embedding theorem and of Boyle’s lower entropy factor theorem (partial generalization have been obtained in previous work for other classes).

In this talk I will discuss a necessary and sufficient condition for denseness of horopherical orbits in the non-wandering set of a higher-rank homogeneous space $G / \Gamma$, for a Zariski dense discrete subgroup $\Gamma < G$, possibly of infinite covolume. In rank one this condition (established in this setting by Eberlein and Dal’bo) implies in particular that the horospherical subgroup acts minimally on the non-wandering set if and only if the discrete group $\Gamma$ is convex co-compact. In contrast, we show that Schottky groups in higher-rank can support non-minimal horospherical actions. This distinction between rank-one and higher-rank is due to the role that Benoist’s limit cone plays in the analysis. Based on joint work with Hee Oh.

Horofunction boundaries are a nice way to approach questions about the behavior of metric spaces at infinity and learn about their geodesics. In the case of Cayley graphs of finitely generated groups, they are also fruitful when studying group actions, algebraic properties and geometric properties (such as the growth rate of the group).

The basic construction is the embedding of the group G in a space of 1-Lipschitz functions on it, by the map sending x to the function b_x(y)=d(x,y)-d(x,1_G). This gives a compactification of G and a compact boundary. The elements in the boundary are called horofunctions. Some of the horofunctions (and in some cases, all of them) are realized as limits of geodesic rays in G, and these are called Busemann points.

The boundary depends on the metric on G, so different Cayley graphs can give rise to different (non-homeomorphic) boundaries. Thus, we are interested in finding out which properties of the boundary are invariants of the group, and we are mainly focused on the cardinality in a broad sense (i.e. finite, countable or uncountable boundary) and the existence of a finite orbit under the group action on the boundary.

In this talk we will review quickly the main definitions and examples and then focus on groups with finitely many Busemann points. We will hopefully go through the main steps of proving that a group with finitely many Busemann points in every Cayley graph horofunction boundary are virtually-cyclic, and in that case every horofunction is a Busemann point.

Joint work with Ariel Yadin

A classical Theorem of Delorme-Guichardet states that a group G has property (T) if and only if every continuous affine isometric action of G on a Hilbert space has a fixed point.

There was a conjecture (attributed to Margulis) that for simple higher rank algebraic groups, this result has the following far reaching generalization: For a simple higher rank algebraic group with a finite center G, every continuous affine isometric action of G on a uniformly convex space has a fixed point.

This conjecture was recently settled by the joint works of V. Lafforgue, Liao for the non-Archimedean case, and myself, and de Laat and de la Salle in the real case.

In my lecture, I will discuss the history of the conjecture mentioned above and a further generalization of its solution beyond algebraic groups (namely, for higher rank universal lattices and Steinberg groups).

Let G be an algebraic group over a local field. We will show that the image of G under an arbitrary continuous homomorphism into any (Hausdorff) topological group is closed if and only if the center of G is compact. We will show how mixing properties for unitary representations follow from this topological property.

A Grassmannian complex is a family of linear subspaces of a given linear space, closed under inclusion. In the talk we will explore the properties of Grassmannian complexes over a finite field and define boundary maps that give rise to notions of connectivity and high dimensional expansion. In contrast to the simplex, where all the homology groups are trivial, the complete Grassmannian (consisting of all subspaces of a given linear space) may have a non-trivial homology, and other exciting phenomena.

We will show analogues to the theorems of Linial, Meshulam and Wallach on the expansion of the complete Grassmannian, and to the phase transition of the connectivity of a random complex.

If time permits, we will discuss related extremal problems and topological overlap.

Based on joint work with Ran Tessler.

(A joint work with Waltraud Lederle) In order to avoid technicalities I will focus on one specific example for a higher $\mathbb{Q}$-rank lattice: the group $\Gamma = \mathrm{SL}_3(\mathbb{Z})$. This group exhibits strong rigidity properties, some of which are naturally expressed in topological terms. For example, one of the earliest rigidity results, the congruence subgroup property which was established independently by Mennicke and Bass-Milnor-Serre, can be expressed as an equality between two group topologies on $\Gamma$: The profinite and the congruence topologies. Margulis’ celebrated normal subgroup theorem can be thought of as the statement that even the normal topology coincides with these two. Here the normal topology is defined by taking all infinite normal subgroups as a basis of identity neighborhoods for a topology on $\Gamma$. Together with Waltraud Lederle we introduce an a-priori much finer topology on $\Gamma$ called the boomerang topology and show that in fact even this topology coincides with the congruence topology. As a result we obtain a generalization of a rigidity theorem for probability measure preserving actions due to Nevo-Stuck-Zimmer.

During the 60,s and the 70,s Furstenberg developed two parallel theories regarding boundaries of groups of different flavours. One is topological, and the other is measurable and relates to random walks. The research of these two theories and their connections with rigidity theory and operator algebra theory is still very active, yet many questions are open. In an attempt to understand better the connections between them, I’ll show that they share the same driving force. We סwill develop one machinery to produce them both at the same time. Two Boundary Theories for the price of one.

The size of the smallest $k$-regular graph of girth $g$ is denoted by the well studied function $n(k,g)$. We suggest generalizing this function to $n(H,g)$, defined as the smallest size girth $g$ graph covering the, possibly non-regular, graph $H$. We prove that the two main combinatorial bounds on $n(k,g)$, the Moore lower bound and the Erdos-Sachs upper bound, carry over to the new setting of lifts, even in their non-asymptotic form.

We also consider two other generalizations of $n(k,g)$: i) The smallest size girth $g$ graph sharing a universal cover with $H$. We prove that it is the same as $n(H,g)$ up to a multiplicative constant. ii) The smallest size girth $g$ graph with a prescribed degree distribution. We discuss this known generalization and argue that the new suggested definitions are superior.

We conclude with experimental results for a specific base graph and with some conjectures and open problems.

https://arxiv.org/abs/2401.01238

Let $T$ be a bounded linear operator on a Banach space $X$ satisfying $\lVert T^n\rVert/n\rightarrow 0$. We prove that $T$ is uniformly ergodic if and only if the one-sided ergodic Hilbert transform $H(T)x:=\lim_{n\rightarrow \infty}\sum_{k=1}^nk^{-1}T^kx$ converges for every $x\in \overline{(I-T)X}$. When $T$ is a power-bounded (or more generally $(C,\alpha)$ bounded for some $0<\alpha<1$), then $T$ us uniformly ergodic if and only if the domain of $H$ equals $(I-T)X$.

To shake things up a little we’ll talk about the Ising model. I will explain a phenomenon in thermodynamics called the “infrared bound”, and what it is usually good for. The only known way to prove this bound on a graph is using a property called “reflection positivity”. But this basically limits the graph in question to Z^d, the Euclidean lattice.

Recently with Tom Meyerovitch we have been thinking of a new method of proving the infrared bound on other (transitive) graphs. I will present a necessary and sufficient condition for something called “Gaussian domination” which in turn implies the infrared bound. The main idea of the talk is to present the different ideas that arise in these kinds of thermodynamic models.

No background is assumed.

Let $(\lambda_{1},…,\lambda_{d})=\lambda\in(0,1)^{d}$ be with $\lambda_{1}>…>\lambda_{d}$ and let $\mu_{\lambda}$ be the distribution of the random vector $\sum_{n\ge0}\pm (\lambda_{1}^{n},…,\lambda_{d}^{n})$, where the $\pm$ are independent fair coin-tosses. Assuming $P(\lambda_{j})\ne 0$ for all $1\le j\le d$ and nonzero polynomials with coefficients $\pm1,0$, we show that $\operatorname{dim}\mu_{\lambda}=\min \big(d,\dim_{L}\mu_{\lambda} \big)$, where $\dim_{L}\mu_{\lambda}$ is the Lyapunov dimension. This extends to higher dimensions a result of Varjú from 2018 regarding the dimension of Bernoulli convolutions on the real line. Joint work with Haojie Ren.