Constructing irreducible representations of finitely presented algebras

By combining well-known techniques from both noncommutative algebra and computational commutative algebra, we observe that an algorithmic approach can be applied to the study of irreducible representations of finitely presented algebras. In slightly more detail: Assume that $n$ is a positive integer, that $k$ is a computable field, that $\bar{k}$ denotes the algebraic closure of $k$, and that $M_n(\bar{k})$ denotes the algebra of $n \times n$ matrices with entries in $\bar{k}$. Let $R$ be a finitely presented $k$-algebra. Calculating over $k$, the procedure will (a) decide whether an irreducible representation $R \to M_n(\bar{k})$ exists, and (b) explicitly construct an irreducible representation $R \to M_n(\bar{k})$ if at least one exists. (For (b), it is necessary to assume that $k[x]$ is equipped with a factoring algorithm.) An elementary example is worked through.Comment: 9 pages. Final version. To appear in J. Symbolic Computatio

The virtual Haken conjecture: Experiments and examples

A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture says that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. Here, we discuss two interrelated topics concerning this conjecture. First, we describe computer experiments which give strong evidence that the Virtual Haken Conjecture is true for hyperbolic 3-manifolds. We took the complete Hodgson-Weeks census of 10,986 small-volume closed hyperbolic 3-manifolds, and for each of them found finite covers which are Haken. There are interesting and unexplained patterns in the data which may lead to a better understanding of this problem. Second, we discuss a method for transferring the virtual Haken property under Dehn filling. In particular, we show that if a 3-manifold with torus boundary has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of the Dehn filled manifolds are virtually Haken. We use this to show that every non-trivial Dehn surgery on the figure-8 knot is virtually Haken.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper12.abs.htm

One class of wild but brick-tame matrix problems

We present a class of wild matrix problems (representations of boxes), which are "brick-tame," i.e. only have one-parameter families of \emph{bricks} (representations with trivial endomorphism algebra). This class includes several boxes that arise in study of simple vector bundles on degenerations of elliptic curves, as well as those arising from the coadjoint action of some linear groups.Comment: 19 page