Experimental Mathematics Colloquium

Speaker: Mike Boyle (Maryland)

Title: Decidability of equivalence of poset blocked matrices over a finite group ring, for Cuntz-Krieger  algebras and flow equivalence

Abstract:

This is part of work in progress with Ben Steinberg.

Let M be a class of matrices and C a class of invertible matrices. For certain M and C, we address the decidability of the problem: given matrices A and B from M, do there exist matrices U,V from C such that UAV = B?

Let ZG be the integral group ring of a finite group G. The problem is decidable in the following cases:

• I. M is the set of square matrices over ZG; C = GL(n,ZG), SL(n,ZG) or El(n,ZG) .
• II. The poset-blocked analogues of I.
• III. A more complicated version of El(n,ZG) poset-blocked matrix equivalence.

For ZG=Z, Case II (applying B-Huang) shows decidability of flow equivalence of SFTs and (applying Restorff) stable isomorphism of Cuntz-Krieger algebras satisfying the Cuntz Condition II.

Along with a stabilization result, Case III (applying B-Carlsen-Eilers) shows decidability of G-flow equivalence of G-SFTs.

All of this appeals to the 1980 Annals of Math paper of Grunwald and Segal, of which the results for ZG=Z are easy corollaries. For C=El(nZG), more is needed.