James W Anderson | Searcharxiv

arXiv2013

Gaps in the space of skeletal signatures

Skeletal signatures were introduced in [J W Anderson and A Wootton, A Lower Bound for the Number of Group Actions on a Compact Riemann Surface, Algebr. Geom. Topol. 12 (2012) 19--35.] as a tool to describe the space of all signatures with which a group can act on a surface of genus $σ\geq 2$. In the present paper we provide a complete description of the gaps that appear in the space of skeletal signatures, together with proofs of the conjectures posed in our earlier work.

math.GT↗

arXiv2013

Limit set intersection theorems for Kleinian groups and a conjecture of Susskind

We continue here the investigation of the relationship between the intersection of a pair of subgroups of a Kleinian group, and in particular the limit set of that intersection, and the intersection of the limit sets of the subgroups. Of specific interest is the extent to which the intersection of the limit sets being non-empty implies that the intersection of the subgroups is non-trivial. We present examples to show that a conjecture of Susskind, stating that the intersection of the sets of conical limit points of subgroups $Φ$ and $Θ$ of a Kleinian group $Γ$ is contained in the limit set of $Φ\cap Θ$, is as sharp as can reasonably be expected. We further show that Susskind's conjecture holds most of the time.

math.GT↗