“but there is no man appointed by the king to listen to you.” It seems that for Absalom to make this claim it was likely true. One possibility for that may have been that earlier, when the United Kingdom was less organized and more tribal, David may have been able to hear the cases that came to Jerusalem himself, but as the United Kingdom became more organized and grew in its centralized power that became more difficult. Or, David may have been able to hear many of the cases but Absalom quietly intervened. There are some other possibilities as well.
We have to wonder why David did not seem know what Absalom was doing. It may be that, especially as Absalom gained popularity and strength, no one wanted to get involved and tell king David. If someone had informed about Absalom and David did nothing about it, Absalom would have been a formidable enemy. It is also possible that David did hear that Absalom was judging some cases, but he did not see any danger in it. David had a weakness for not seeing trouble when it came to his sons.
It is also worth asking how Absalom could tell someone their grievance was good and right without having witnesses and representatives of both sides present. It is possible that 2 Samuel 15:3 is just giving a kind of summary and there were times when Absalom adjudicated over complicated cases. Another possibility is that Absalom may have only really gotten involved with certain cases—those involving the state—and being able to subtlety speak about the failures of the state would have undermined David while increasing his own popularity. The United Kingdom of Israel was brand new, and no doubt there were a number of unpopular moves being made on the state level: state incursions into what had formerly been tribal matters, issues with taxes, problems with recruiting manpower for state projects, conflicts about who got appointed to which positions in the kingdom, and much more. David’s attention had to be focused on building and defending the kingdom, changes that were necessary but that would have offended many people. The various changes affected the whole kingdom and in part explains how Absalom could have followers throughout all the tribes of Israel (2 Sam. 15:10). In contrast to David, Absalom did not actually have to make any of the hard choices and changes in the kingdom. He could simply propagandize and tell people that if he were king then things would be different; and over the four years he quietly undermined David he stole the hearts of the people of Israel (2 Sam. 15:6-7).