Modular Forms-Fundamental Domain

[binbagsss]

Apologies if this is a stupid question, but is the fundamental domain unique?

And what exactly is the definition of a cusp- a quick google tells me it is 'where two curves meet', so looking at the fundamental domain,I would say $\omega=\exp^{\frac{2\pi i}{ 3}}$ and $\omega*$ are?

thanks

 

[.Scott]

Here is a link:
https://en.wikipedia.org/wiki/Cusp_(singularity)

For modular forms, the cusps would be all those points along the X axis.

 

[mathwonk]

no it is not unique. any image of one fundamental domain by any element of the modular group is another fundmental domain. in the illustration posted above in post #2, every curvilinear polygon shown is a fundamental domain. the infinite shaded one is the usual choice simply because it is more symmetric, but the other infinite ones on either side of it are also fundamental domains obtained from it by translation. the bounded ones nearer the x - axis look a different shape but that is because they are images of the standard one by group elements that change shape, i.e. elements like z--> -1/z.