The town of arkham was invented by H.P.Lovecraft for his horror stories. They are now largely public domain. The town name of Arkham, therefore, is public domain. Yet WB Games have launched a continuing series of Arkham: Something games for their Batman license. This would suggest that they are using a public domain name for a running line of flagship game titles for a (I assume) flagship genre franchise. That seems odd. Just as Atlantis cannot be a trademarked name, neither shoud Arkham. Of course, they can use it, but is anything keeping others from making Arkham: Pony Rider or Arkham Flyfishing, since the name is public domain??
You are confusing public domain (a concept related to copyright) with the use of a word as a trademark.
Lovecraft never had any copyright over Arkham; a single word or short phrase is never subject to copyright. Similarly batman is not subject to copyright.
However, both of those terms are subject to active trademarks as they have been and continue to be actively used as product identification (they are probably also registered but this is not strictly necessary). Now trademarks are industry specific so there is no problem using Arkham Mining for a mining company or Batman Exterminators for vermin eradication providing you took care that it was clear there was no association between you and the trade mark holders. Use them in a computer game, novel or film? Expect to be successfully sued.