9,453 research outputs found
Products of Farey graphs are totally geodesic in the pants graph
We show that for a surface S, the subgraph of the pants graph determined by
fixing a collection of curves that cut S into pairs of pants, once-punctured
tori, and four-times-punctured spheres is totally geodesic. The main theorem
resolves a special case of a conjecture made by Aramayona, Parlier, and
Shackleton and has the implication that an embedded product of Farey graphs in
any pants graph is totally geodesic. In addition, we show that a pants graph
contains a convex n-flat if and only if it contains an n-quasi-flat.Comment: v2: 25 pages, 16 figures. Completely rewritten, several figures added
for clarit
Coarse and synthetic Weil-Petersson geometry: quasi-flats, geodesics, and relative hyperbolicity
We analyze the coarse geometry of the Weil-Petersson metric on Teichm\"uller
space, focusing on applications to its synthetic geometry (in particular the
behavior of geodesics). We settle the question of the strong relative
hyperbolicity of the Weil-Petersson metric via consideration of its coarse
quasi-isometric model, the "pants graph." We show that in dimension~3 the pants
graph is strongly relatively hyperbolic with respect to naturally defined
product regions and show any quasi-flat lies a bounded distance from a single
product. For all higher dimensions there is no non-trivial collection of
subsets with respect to which it strongly relatively hyperbolic; this extends a
theorem of [BDM] in dimension 6 and higher into the intermediate range (it is
hyperbolic if and only if the dimension is 1 or 2 [BF]). Stability and relative
stability of quasi-geodesics in dimensions up through 3 provide for a strong
understanding of the behavior of geodesics and a complete description of the
CAT(0)-boundary of the Weil-Petersson metric via curve-hierarchies and their
associated "boundary laminations."Comment: References added apropos of equivalence of the notion of
asymptotically tree-graded and strong relative-hyperbolicity in the sense of
Drutu-Sapir. We thank Jason Behrstock for bringing this to our attention.
Proof of thickness in higher dimension streamlined, some comments, questions
and references adde
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