Possible meanings are 1) the word "their" refers to the Jews of that town. AT: "the synogogue" or 2) the word "their" refers to the Pharisees that Jesus had just spoken to, and this was the synagogue that they and other Jews in that town attended. The word "their" does not mean that the Pharisees owned the synagogue. AT: "the synagogue that they attended"
They did not just want to accuse Jesus in front of the people. The Pharisees wanted Jesus to give an answer that contradicted the law of Moses so they could take him before a judge and legally charge him with breaking the law. (See: [[rc://en/ta/man/translate/figs-explicit]])