A regular map is a highly symmetric embedding of a finite graph into a closed surface. I will describe a programme to study such embeddings for a rather large class of graphs: namely, the class of orbital graphs of finite simple groups.

# Past Algebra Seminar

Groups which act on rooted trees, and branch groups in particular, have provided examples of groups with exotic properties for the last three decades. This and their links to other areas of mathematics such as dynamical systems has made them the object of intense research.

One of their more useful properties is that of having a "tree-like" subgroup structure, in several senses.

I shall explain what this means in the talk and give some applications.

the only permitted defining relators are commutators of the generators. These groups and their subgroups play an important role in Geometric Group Theory, especially in view of the recent groundbreaking results of Haglund, Wise, Agol, and others, showing that many groups possess finite index subgroups that embed into RAAGs.

In their recent work on limit groups over right angled Artin groups, Casals-Ruiz and Kazachkov asked whether for every natural number n there exists a single "universal" RAAG, A_n, containing all n-generated subgroups of RAAGs. Motivated by this question, I will discuss several results showing that "universal" (in various contexts) RAAGs generally do not exist. I will also mention some positive results about universal groups for finitely presented n-generated subgroups of direct products of free and limit groups.

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.

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.