2012/11/13 by Lucas Sabalka, Sabalka, Lucas, Dmytro Savchuk +1 · 1 citation
Computer Science · Mathematics · #20F28 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.GR #math.GT #msc:20F28
paper · pdf · doi:10.48550/arxiv.1211.3111
30 pages, 12 figures. This is a preliminary version; comments and suggestions are welcome
arxiv created 2012/11/13 · openalex publication_date 2012/11/13 · arxiv updated 2012/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
One of the most useful tools for studying the geometry of the mapping class group has been the subsurface projections of Masur and Minsky. Here we propose an analogue for the study of the geometry of Out(Fn) called submanifold projection. We use the doubled handlebody Mn = #n S2 × S1 as a geometric model of Fn, and consider essential embedded 2-spheres in Mn, isotopy classes of which can be identified with free splittings of the free group. We interpret submanifold projection in the context of the sphere complex (also known as the splitting complex). We prove that submanifold projection satisfies a number of desirable properties, including a Behrstock inequality and a Bounded Geodesic Image theorem. Our proof of the latter relies on a method of canonically visualizing one sphere `with respect to' another given sphere, which we call a sphere tree. Sphere trees are related to Hatcher normal form for spheres, and coincide with an interpretation of certain slices of a Guirardel core.