Non-affine horocycle orbit closures on strata of translation surfaces: new examples

We investigate specific examples of locally-defined real vector-fields on strata of translation surfaces. Integrating SL(2,R)-loci of Veech surfaces along these vector-fields yield interesting new examples of horocyle-invariant ergodic measures. These measures are supported on closed immersed manifolds with boundary that approach the boundary of strata in a novel way.Comment: 48 pages, 27 figures. Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosoph

Enumeration of nilpotent loops up to isotopy

We modify tools introduced by Daniel Daly and Petr Vojtechovsky in order to count, for any odd prime q, the number of nilpotent loops of order 2q up to isotopy, instead of isomorphy.Comment: 20 pages; important typos corrected (section 6

The Space of Geometric Limits of One-generator Closed Subgroups of PSL2(R)

We give a complete description of the closure of the space of one-generator closed subgroups of PSL2(R) for the Chabauty topology, by computing explicitly the matrices associated with elements of Aut(D) = PSL2(R), and finding quantities parametrizing the limit cases. Along the way, we investigate under what conditions sequences of maps transform convergent sequences of closed subsets of the domain into convergent sequences of closed subsets of the range. In particular, this allows us to compute certain geometric limits of PSL2(R) only by looking at the Hausdorff limit of some closed subsets of C.Comment: 28 pages, 6 figure