When Turgoff uses the word "anarchist" he means "nihilist".
Turgoff: When the police come, Turgoff go -- always -- without exception.
Turgoff: I am an anarchist. All I believe in is nothing.
Holmes: Well, why not give them the vote. They can't do any worse than we have.
Holmes: You are without honor in your own country.
Holmes refers to Doreen as a prophet in saying that one day the women will have the vote by alluding to "A prophet is not without honor, save in his own country" (Matthew 13:57)
Women gained full voting rights in Great Britain in 1928.