arXiv subjects
Publications and source records attributed to Slavyana Geninska.
For a translation surface, we define the systole to be the length of the shortest saddle connection. We give a characterization of the maxima of the systole function on a stratum, and give a family of examples providing local but nonglobal maxima on each stratum of genus at least three. We further study the relation between (locally) maximal values of the systole function and the number of shortest saddle connections.
In this paper we prove that the systole fonction on a connected component of area one translation surfaces admits a local maximum that is not a global maximum if and only if the connected component is not hyperelliptic.
In this paper we give a description of the possible limit sets of finitely generated subgroups of irreducible lattices in $PSL(2,R)^r$.
We provide examples of finitely generated infinite covolume subgroups of $PSL(2,R)^r$ with a "big" limit set, e.g. that contains an open subset of the geometric boundary. They are given by the so called semi-arithmetic Fuchsian groups admitting modular embeddings.
While lattices in semi-simple Lie groups are studied very well, only little is known about discrete subgroups of infinite covolume. The main class of examples are Schottky groups. Here we investigate some new examples. We consider subgroups $Γ$ of arithmetic groups in $PSL(2,C)^q \times PSL(2,R)^r$ with $q+r>1$ and their limit set. We prove that the projective limit set of a nonelementary finitely generated $Γ$ consists of exactly one point if and only if one and hence all projections of $Γ$ to the simple factors of $PSL(2,C)^q \times PSL(2,R)^r$ are subgroups of arithmetic Fuchsian or Kleinian groups. Furthermore, we study the topology of the whole limit set of $Γ$. In particular, we give a necessary and sufficient condition for the limit set to be homeomorphic to a circle.