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http://dx.doi.org/10.4134/JKMS.j190407

GEOMETRIC SIMPLICIAL EMBEDDINGS OF ARC-TYPE GRAPHS  

Parlier, Hugo (Department of Mathematics University of Luxembourg)
Weber, Ashley (Department of Mathematics Brown University)
Publication Information
Journal of the Korean Mathematical Society / v.57, no.5, 2020 , pp. 1103-1118 More about this Journal
Abstract
In this paper, we investigate a family of graphs associated to collections of arcs on surfaces. These multiarc graphs naturally interpolate between arc graphs and flip graphs, both well studied objects in low dimensional geometry and topology. We show a number of rigidity results, namely showing that, under certain complexity conditions, that simplicial maps between them only arise in the "obvious way". We also observe that, again under necessary complexity conditions, subsurface strata are convex. Put together, these results imply that certain simplicial maps always give rise to convex images.
Keywords
Arc graphs; flip graphs; mapping class groups;
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