Associated to a complex admissible representation of a p-adic group is an invariant known is the "canonical dimension". It is closely related to the more well-studied invariant called the "wavefront set". The advantage of the canonical dimension over the wavefront set is that it allows for a completely different approach in computing it compared to the known computational methods for the wavefront set. In this talk we illustrate this point by finding a lower bound for the canonical dimension of any depth-zero supercuspidal representation, which depends only on the group and so is independent of the representation itself. To compute this lower bound, we consider the geometry of the associated Bruhat-Tits building.

If \(F\) is a free group and \(F/N\) is a presentation of a group \(G\), there is a natural way to turn the abelianisation of \(N\) into a \(\mathbb ZG\)-module, known as the relation module of the presentation. The images of normal generators for \(N\) yield \(\mathbb ZG\)-module generators of the relation module, but 'lifting' \(\mathbb ZG\)-generators to normal generators cannot always be done by a result of Dunwoody. Nevertheless, it is an open problem, known as the relation gap problem, whether the relation module can have strictly fewer \(\mathbb ZG\)-module generators than \(N\) can have normal generators when \(G\) is finitely presented. In this talk I will survey what is known and what is not known about this problem and its variations and discuss some recent progress for groups with a cyclic relation module.