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Wikipedia states that the boundary of any fundamental domain should have some restrictions, such as smoothness of polyhedrality. Further, this thesis on Page 8, imposes the condition that the boundary should be piecewise smooth.

My question is why are these "nice" conditions necessary in the definition of the fundamental domain, especially in the case of the modular group?

Adam Karlson
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  • What does "necessary" even mean in this context? Definitions are written to describe useful properties of certain objects. – Lee Mosher Mar 05 '22 at 05:17
  • Exactly. What is the usefulness which is imparted by imposing the "smoothness" condition? If my fundamental domain does not have a smooth boundary, what do I lose? – Adam Karlson Mar 05 '22 at 05:22
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    Not an answer but maybe it’s a good start: in the case of the modular group, some properties (such as the $k/12$-lemma or the Eichler-Shimura isomorphism) use integration on the fundamental domain and Stokes’ theorem to relate this to some contour integrals. This approach usually requires some smoothness assumptions on the boundary. – Aphelli Mar 05 '22 at 09:10
  • @Mindlack Thanks! That helps. – Adam Karlson Mar 05 '22 at 10:38

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