Factual error: A jury in Hamburg is asked if Mr Troppoff is guilty. There are no juries at all in the German court system.
Factual error: The entire reason Sherlock takes up the case is because he is intrigued how a man managed to disappear from a tube carriage in between stations - it appears to be impossible to do. However, any Londoner will tell you that it is perfectly simple to do: all tube carriages have doors between them linking them. So if the man wanted to leave the carriage between stations, he'd just use the door at the end of the carriage. The train employee would not be puzzled by this, nor would Sherlock consider the case worthy of his time.
Suggested correction: The footage we see doesn't show much of the interior of the last car, so if they get from that that nobody is in the car, they must have checked the interior somehow, which isn't shown or mentioned, and likely they checked the entire train. So the reasoning is: one man gets on the train at point A, the train is proven empty at point C (not shown), but at point B, the only possible exit in between, nobody got out.
