It depends if the America trademark covers America and the rest of the world. A trademark has two parts, the log and text. If they trademark the domain, that is one part of it. Most Trademarks have to have a class or many classes. Each class describes a category of good like apparel and what is within that apparel minus what is not include - mittens, gloves, woollen hats... minus surgical gloves, surgical hats etc. Basically, it depends on the details of the Trademark and whether it has included Greece as a country where it has been registered and approved.
Companies tend to do that, I mean they try to shut you down. Usually they send you a very harsh 'cease and desist' letter where they demand you give up your domain or they take you to the court. Some people are getting scared of that and naturally give up their domain names. But you shouldn't. If the domain was registered before the trademark, or that particular trademark covers some other countries but not yours, it's okay, they can't really do anything about it