What is a thesis statement about friendship and betrayal in Memoirs of a Geisha?

The themes of friendship and betrayal are both present in this novel, so I think that you have a good starting point for a possible thesis statement. Friendship and betrayal are quite opposite, but they do come hand-in-hand at times within this novel. For that reason, I recommend a thesis statement that contains a point followed by a counter-point. It needs to be a single sentence, so make the sentence a complex sentence beginning with the word "although." This will allow you to have a dependent clause that introduces either friendship or betrayal. That is followed by the independent clause that introduces the second option. For example:

Although geisha school is all about teaching various forms of friendship and companionship, the competition that exists within the structure leads to inevitable betrayals.

For textual evidence to support that thesis, I would look to the relationship that exists between Chiyo/Sayuri and Pumpkin. There are times when Pumpkin is or appears to be Sayuri's only friend; however, Pumpkin's final betrayal of Sayuri really shows readers just how shallow that friendship likely was.