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In Umemoto's thesis 1 on Dirichlet fundamental domains for Fuchsian groups, Theorem 24 involves assumed angles for subdivided hexagons in the proof (Fig.18 on page 35). However, the rationale behind these angles is unclear.

For instance, I thought, as its reflection $\theta_6$ of bottom left corner of the top triangle should be the top left corner of the internal triangle.

Could someone explain why these angles are chosen and their role in the proof?

Fig.18

The author only goes as follows;

we can find that the angles of the hexagon is like a figure in Fig. 18. enter image description here

[1] Umemoto, Yuriko. "On Dirichlet fundamental domains for Fuchsian groups." Diss. Master Thesis, Graduate School of Science, Osaka City University, 2011.

Rowing0914
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    Not an answer (I think one would have to read much of the context in the thesis to provide one) but this paper might be relevant. https://www.cs.umb.edu/~eb/dirichlet/RecognizingDTs.pdf https://link.springer.com/article/10.1007/BF00181470 – Ethan Bolker Mar 13 '24 at 23:58
  • @EthanBolker Thank you for your suggestion of further reading! Will take a look at them! – Rowing0914 Mar 14 '24 at 01:17
  • Just realised but reflection isn't conformal. So, my guess about $\theta_6$ in question is naturally wrong. And I think this would change my understanding of angles! – Rowing0914 Mar 14 '24 at 01:19
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    Let me suggest that you look at our guidelines for formatting and writing your post, where you will read this: Don't force someone to click on an external link just to see or understand your question. As currently written, your post consists of an diagram and a question about that diagram, but the question makes no sense without clicking on the linked paper to understand what that link says about how the angles were chosen and their role in the proof (and... what proof?). – Lee Mosher Mar 17 '24 at 13:26

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