Vertex of Fundamental Domains & Elliptic Points

[Fangyang Tian]

Dear Folks:
Suppose [itex]\Gamma[/itex] is a discrete subgroup of SL₂(R), which acts on the upper half complex plane as Mobius transformation. F is its fundamental domain. If z is a vertex of F which does not lie on the extended real line ( that is R[itex]\bigcup[/itex][itex]\infty[/itex] ) ,then must x be an elliptic point?? Many thanks!
For example, if [itex]\Gamma[/itex] is SL₂(Z), then x is either e^{i[itex]\frac{2}{3}[/itex][itex]\pi[/itex]} or e^{i[itex]\frac{4}{3}[/itex][itex]\pi[/itex]}, both of them are elliptic points.

 

[pat_connell]

Yes, if z is a vertex of the fundamental domain F then x must be an elliptic point. This is because all Mobius transformations that are part of \Gamma leave the set of elliptic points invariant. Therefore, any point that is a vertex of the fundamental domain must also be an elliptic point.