The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Not all rationals between 0 and 1 occur as commuting probabilities. In fact Keith Joseph conjectured in 1977 that all limit points of the set of commuting probabilities are rational, and moreover that these limit points can only be approached from above. In this talk we'll discuss a structure theorem for commuting probabilities which roughly asserts that commuting probabilities are nearly Egyptian fractions of bounded complexity. Joseph's conjectures are corollaries.

# Past Algebra Seminar

We will introduce both the class of right-angled Artin groups (RAAG) and

the Nielsen realisation problem. Then we will discuss some recent progress

towards solving the problem.

Recently several conjectures about l2-invariants of

CW-complexes have been disproved. At the heart of the counterexamples

is a method of computing the spectral measure of an element of the

complex group ring. We show that the same method can be used to

compute the finite field analog of the l2-Betti numbers, the homology

gradient. As an application we point out that (i) the homology

gradient over any field of characteristic different than 2 can be an

irrational number, and (ii) there exists a CW-complex whose homology

gradients over different fields have infinitely many different values.

String algebras are tame - their finite-dimensional representations have been classified - and the Auslander-Reiten quiver of such an algebra shows some of the morphisms between them. But not all. To see the morphisms which pass between components of the Auslander-Reiten quiver, and so obtain a more complete picture of the category of representations, we should look at certain infinite-dimensional representations and use ideas and techniques from the model theory of modules.

This is joint work with Rosie Laking and Gena Puninski:

G. Puninski and M. Prest, Ringel's conjecture for domestic string algebras, arXiv:1407.7470;

R. Laking, M. Prest and G. Puninski, Krull-Gabriel dimension of domestic string algebras, in preparation.