Ideal triangulations of pseudo-Anosov mapping tori

“…The critical point moves in a 3-cycle 0 c c 2 + c (2) . The optimal elastic graph 1 The black edges have the indicated lengths, which come from looking at the external rays landing at the α fixed point of f 1 . Give the colored edges an equal and long elastic length (say, 100).…”

Section: Example 21mentioning

confidence: 99%

“…Consider a quadratic differential on 2 with a very large proportion of its area in 1 . Then most U i (as weighted by Area q (U i )) must have most of their q-area in U i ∩ 1 .…”

Section: Sf[ φ]mentioning

confidence: 99%

“…Proof sketch For C a curve on 1 In contrast with the case for surfaces (Sect. 6.2), it is easy to see how the stretch factor for graphs behaves under covers.…”

Section: Remark 619mentioning

confidence: 99%

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From rubber bands to rational maps: a research report

Thurston

2016

Res Math Sci

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This research report outlines work, partially joint with Jeremy Kahn and Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal surfaces with boundary. On one hand, this lets us tell when one rubber band network is looser than another and, on the other hand, tell when one conformal surface embeds in another. We apply this to give a new characterization of hyperbolic critically finite rational maps among branched self-coverings of the sphere, by a positive criterion: a branched covering is equivalent to a hyperbolic rational map if and only if there is an elastic graph with a particular "self-embedding" property. This complements the earlier negative criterion of W. Thurston.

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Ideal triangulations of 3‐manifolds up to decorated transit equivalences

Benedetti¹

2020

Mathematische Nachrichten

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We consider 3‐dimensional pseudo‐manifolds trueM̂ with a given set of marked point V such that M̂∖V is the interior of a compact 3‐manifold with boundary M. An ideal triangulation T of (M̂,V) has V as set of vertices. A branching (T,b) enhances T to a Δ‐complex. Branched triangulations of (M̂,V) are considered up to the b‐transit equivalence generated by isotopy and ideal branched moves which keep V pointwise fixed. We extend a well known connectivity result for “naked” ideal triangulations by showing that branched ideal triangulations of (M̂,V) are equivalent to each other. A pre‐branching (T,ω) is a system of transverse orientations at the 2‐facets of T verifying a certain global constraint; pre‐branchings are considered up to a natural pb‐transit equivalence. If M is oriented, every branching (T,b) induces a pre‐branching (T,ωb) and every b‐transit induces a pb‐transit. The quotient set of pre‐branchings up to transit equivalence is far to be trivial; we get some information about it and we characterize the pre‐branchings of the type ωb. Pre‐branched and branched moves are naturally organized in subfamilies which give rise to restricted transit equivalences. In the branching setting we revisit, with some complement, early results about the sliding transit equivalence and outline a (partially conjectural) conceptually different approach to the branched ideal connectivity and eventually also to the naked one. The basic idea is to point out some structures of differential topological nature on M which are carried by every branched ideal triangulation (T,b) of trueM̂, are preserved by the sliding transits and can be modified by the full branched transits. The non ambiguous transit equivalence already widely studied on pre‐branchings lifts to a specialization of the sliding equivalence on branched triangulations; we point out a few specific insights, again in terms of carried structures preserved by the non ambiguous and which can be modified by the whole sliding transits.

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Ideal triangulations of pseudo-Anosov mapping tori

Cited by 60 publications

References 22 publications

Fibered faces, veering triangulations, and the arc complex

Fibered faces, veering triangulations, and the arc complex

From rubber bands to rational maps: a research report

Ideal triangulations of 3‐manifolds up to decorated transit equivalences

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