Activities This Week

In a seminal 1973 paper, Williams recast conjugacy and eventual conjugacy for subshifts of finite type purely in terms of equivalence relations between adjacency matrices of the directed graphs. Williams expected these two notions to be the same, but after around 20 years the last hope for a positive answer, even under the most restrictive conditions, was extinguished by Kim and Roush.

In this talk, we will discuss operator algebras associated with adjacency matrices / directed graphs, which are naturally $\mathbb{Z}$-graded algebras. These operator algebras were first introduced by Cuntz and Krieger in tandem with early attacks on Williams’ problem, and manifest several natural properties of subshifts through their classification up to various kinds of isomorphisms.

The works on Cuntz-Krieger algebras later inspired a systematic study of purely algebraic versions called Leavitt path algebras, promoting new interactions between pure algebra and analysis. A well-known conjecture of Hazrat claims that two Leavitt path algebras are graded isomorphic if and only if their unital graded Grothendieck K0 groups are isomorphic. The topological version of this problem asks for a characterization of graded (stable) isomorphisms between Cuntz-Krieger algebras in terms of equivariant K-theory.

A solution to these problems has been sought after by many, and although substantial progress has been made, a proof is still missing in general. In joint work with Carlsen and Eilers we were able to discover subtle obstructions to certain algebraic methods of proof for the latter conjecture, by building on the counterexamples of Kim and Roush

It is well known that if a finite set of integers A tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization A+B=Z_M of a finite cyclic group. Coven and Meyerowitz (1998) proved that when the tiling period M has at most two distinct prime factors, each of the sets A and B can be replaced by a highly ordered “standard” tiling complement. It is not known whether this behaviour persists for all tilings with no restrictions on the number of prime factors of M. In joint work with Izabella Laba (UBC), we proved that this is true when M=(pqr)^2. In my talk I will discuss this problem and introduce some ingredients from the proof.

Given a countable group G acting on a Cantor set X by transformations preserving a probability measure, the action is essentially free if the set of points with trivial stabilizers has a full measure. In this talk, we consider actions where no point has a trivial stabilizer, and investigate the properties of the points with non-trivial holonomy. We introduce the notion of a locally non-degenerate action, and show that if an action is locally non-degenerate, then the set of points with trivial holonomy has full measure in X. We discuss applications of this work to the study of invariant random subgroups, induced by actions of countable groups. This is joint work with Maik Gröger.