From 1986 to 1990, the number of Jews authorized to emigrate from the Soviet Union grew from 914 to 186,815. However, there was still a group of 11,000 “refuseniks,” or people who had been denied.
Natan Sharansky was a refusenik in the mid 1970s, and he became the de facto spokesperson for them. He brought attention to the desire to leave and human rights abuses happening in the USSR. In 1978, he was accused of treason and sentenced to 13 years of forced labor.
You can read more about the refuseniks at the link.