Mathematics

Let H be a cyclically-presented group on n generators with a single defining relator. Attempts have been made to classify such groups by their order, their status as a 3-manifold group, and the asphericity status of their presentations. For groups with a defining relator of length 3 these classifications are nearly complete, with only two groups, H(9,4) and H(9,7), representing outstanding cases in each classification. We complete the asphericity classification for the presentations of these two groups and show that H(9,4) is not a 3-manifold group.

A translation surface $(X,\omega)$ is a collection of polygons in $\mathbb{R}^2$ together with a set of side-pairing identifications such that identified sides are parallel, equal length and of opposite orientation. There is a natural action of $\text{GL}_2\mathbb{R}$ on the set of all translation surfaces. The subgroup of orientation-preserving elements of the stabilizer of $(X,\omega)$ under this action is called the Veech group, denoted $\text{SL}(X,\omega)$.