Is there any homeomorphism between fundamental domains?

Question

It is known that for the same fuchsean group if I take two different points I will get two different fundamental domains. Is there any homeomorphism between these fundamental domains?

Answer 1

Of course the quotient of the group action is well-defined independent of the choice of fundamental domain. Therefore the quotient of one fundamental domain is homeomorphic to the quotient of any other fundamental domain, both of them being homeomorphic to the quotient of the group action. But you didn't ask about quotients, you asked about the fundamental domains themselves.

If the group action is cocompact then the fundamental domain is homeomorphic to the closed disc $D^2$. So yes, two fundamental domains of this type will be homeomorphic to $D^2$ and therefore homeomorphic to each other.

If the group action is not cocompact then the fundamental domain will be homeomorphic to a disc with some finite number of points removed from the boundary. In this case, the number of points removed determines the topology of the fundamental domain and so two such fundamental domains are homeomorphic if and only if they have the same number of removed points. But it's not hard to construct counterexamples where two fundamental domains for the same Fuchsian group action have different number of points removed from their boundary, and therefore those two fundamental domains are not homeomorphic.

For example, in the case that the quotient surface is homeomorphic to a once-punctured torus, there will be a fundamental domain which gives the usual square gluing diagram with gluing word $a b a^{-1} b^{-1}$ and with all four corners removed, and so that fundamental domain homeomorphic is to a disc with four boundary points removed. But there will also be a fundamental domain which gives a hexagonal gluing diagram with gluing word $a b a^{-1} b^{-1} c c^{-1}$, and with only the vertex between the $c$ and $c^{-1}$ edge removed, so that fundamental domain is homeomorphic to a disc with one boundary point removed.