# Seminars & Events for Minerva mini-course

##### An overview of Benjamini-Schramm convergence in group theory and dynamics

When studying an infinite geometric object or graph it is natural to want a "good" finite or bounded model for the sake of computations. But what does "good" mean here? This notion is formalized by Benjamini-Schramm convergence: "good" means that locally the finite object looks like the infinite one, except for a small density of singularities. While this notion appears rather weak, it is surprisingly useful. Indeed, there are many problems in transitive graphs, countable groups and stationary processes (for example) that can be solved only by viewing the underlying structure as a limit of finite models. A small sampling includes: classification of Bernoulli shifts over sofic groups, direct finiteness for group rings, computation of L2 betti numbers and other spectral invariants.

##### $L^2$ invariants and Benjamini-Schramm convergence

Does there exist a sequence of free subgroups $F_k$ of the isometry group of hyperbolic $n$-space such that the Cheeger constant of the quotient space $H^n/F_k$ tends to zero as $k$ tends to infinity? I will explain how to answer this (and related questions) when $n$ is even using a curious result of $G$.

##### Classification of Bernoulli shifts

Bernoulli shifts over amenable groups are classified by entropy (this is due to Kolmogorov and Ornstein for $Z$ and Ornstein-Weiss in general). A fundamental property is that entropy never increases under a factor map. This property is violated for nonamenable groups. In spite of this, sofic entropy theory makes sense even for nonamenable groups and Bernoulli shifts are classified by sofic entropy. Time permitting, I'll also discuss recent results of B. Seward showing how Rohlin entropy can be used to extend this classification to all countable groups (conditional on a natural conjecture).

##### When does injectivity imply surjectivity?

Any injective map from a finite set to itself is surjective. Ax's Theorem extends this to algebraic varieties and regular maps. Gromov invented sofic groups as a way to extend to this result to cellular automata and other settings. We'll re-prove his results via sofic entropy theory. Similarly we use sofic mean dimension to prove non-embeddability results in topological dynamics and $l^p$ dimension to prove non-embeddability results in representation theory.

##### Ergodic theorems beyond amenable groups

Let G be a locally compact group acting by measure-preserving transformations on a probability space (X,mu). To every probability measure on G there is an associated averaging operator on L^p(X,mu). Ergodic theorems describe the pointwise and norm limits of sequences of such operators. In joint work with Amos Nevo, we develop a new general approach based on reducing the problem to the amenable case. From this we obtain ergodic theorems for sector and spherical averages when G is a rank 1 Lie group or a countable Gromov hyperbolic group.