Shadows of Teichm\"uller discs in the curve graph
Abstract
We consider several natural sets of curves associated to a given Teichm\"uller disc, such as the systole set or cylinder set, and study their coarse geometry inside the curve graph. We prove that these sets are quasiconvex and agree up to uniformly bounded Hausdorff distance. Furthermore, we describe two operations on curves and show that they approximate nearest point projections to their respective targets. Our techniques can be used to prove a bounded geodesic image theorem for a natural map from the curve graph to the filling multi-arc graph associated to a Teichm\"uller disc.
Keywords
Cite
@article{arxiv.1510.04259,
title = {Shadows of Teichm\"uller discs in the curve graph},
author = {Robert Tang and Richard C. H. Webb},
journal= {arXiv preprint arXiv:1510.04259},
year = {2015}
}
Comments
25 pages, 4 figures; Added Section 3.3 which includes a bounded geodesic image theorem (Theorem 1.4); Distance bound is now log(2) for both parts of Proposition 7.1; Minor revisions in Section 7.2