Is it wrong to ask her to cut off ties with her ex?

I've been with my gf for almost 2 years. At the start she told me she was still friends with her ex (who lives overseas) and I was cool with it as long as i knew when they spoke. recently I found out that earlier last year she had spoke to him and also around his birthday. The first time- from the conversation i seen, she had told him about an argument we had been having (which i'm not happy about) and... the second time- was for his birthday where all she said was "i miss you"- She claims the friendship- is just that...a friendship- and that she 'misses' the whole group. I guess we've both agreed it's inappropriate.. and she's agreed not to do it anymore.. as in tell him she misses him and talk about our arguments with him. Now, I want her to cut off all ties...She doesn't want to and is claiming that i knew they were friends when we started dating- so i shouldn't have a problem. Do I have the right to ask that? should she? Whats going on? is it okay to private msg your ex to say you miss them?... i'm confused.. and when ever we bring it up- we get into an argument that doesn't fix because we aren't seeing things the same. Please help!