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From: Shadows of Teichmüller Discs in the Curve Graph

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1 From: Shadows of Teichmüller Discs in the Curve Graph
Fig. 1. Top: a monogon and bigon; Bottom: a figure–8 and barbell. From: Shadows of Teichmüller Discs in the Curve Graph Int Math Res Notices. Published online February 04, doi: /imrn/rnw318 Int Math Res Notices | © The Author(s) Published by Oxford University Press.This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

2 From: Shadows of Teichmüller Discs in the Curve Graph
Fig. 2. Splitting a large branch; and sliding a mixed branch. From: Shadows of Teichmüller Discs in the Curve Graph Int Math Res Notices. Published online February 04, doi: /imrn/rnw318 Int Math Res Notices | © The Author(s) Published by Oxford University Press.This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

3 From: Shadows of Teichmüller Discs in the Curve Graph
Fig. 3. Smoothing $\alpha^q$ at a singularity $x$ to locally produce a train track. From: Shadows of Teichmüller Discs in the Curve Graph Int Math Res Notices. Published online February 04, doi: /imrn/rnw318 Int Math Res Notices | © The Author(s) Published by Oxford University Press.This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

4 From: Shadows of Teichmüller Discs in the Curve Graph
Fig. 5. The auxiliary polygons for $\alpha$, corresponding to the half-translation structures $q$, $\rho_\theta \cdot q$, $p$, and $\rho_{\frac{\pi}{4}}\cdot p$ respectively, are nested between pairs of ellipses. The grid lines indicate the pair of transverse slopes corresponding to $\mathcal{G}$. From: Shadows of Teichmüller Discs in the Curve Graph Int Math Res Notices. Published online February 04, doi: /imrn/rnw318 Int Math Res Notices | © The Author(s) Published by Oxford University Press.This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

5 From: Shadows of Teichmüller Discs in the Curve Graph
Proof. Applying $A \in SL(2,\mathbb{R})$ to the polygon $P$ to make it “round”. From: Shadows of Teichmüller Discs in the Curve Graph Int Math Res Notices. Published online February 04, doi: /imrn/rnw318 Int Math Res Notices | © The Author(s) Published by Oxford University Press.This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.

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