In the presence of certain topological conditions, we provide lower bounds for the infimum of the length function associated to a collection of curves on Teichmüller space that depend on the dual cube complex associated to the collection, a concept due to Sageev. As an application of our bounds, we obtain estimates for the “longest” curve with self-intersections, complementing work of Basmajian [J. Topol. 6 (2013) 513–524].
"Infima of length functions and dual cube complexes." Algebr. Geom. Topol. 17 (2) 1041 - 1057, 2017. https://doi.org/10.2140/agt.2017.17.1041