Thirty-five years after she broke out of a New Jersey prison, Joanne Chesimard is at the center of the debate over President Obama's move to reestablish relations with Cuba.

Chesimard, who was convicted of fatally shooting a state trooper following a traffic stop on the New Jersey Turnpike in 1973, turned up in Cuba in the 1980s after escaping from a women's correctional facility where she was serving a life sentence.

A Black Liberation Army member who now is known as Assata Shakur and is said to be the step-aunt and godmother of the slain rapper Tupac Shakur, Chesimard, now 67, is thought to be living in Cuba, according to the FBI, which put her on its Most Wanted Terrorists list in 2013.

An agency spokeswoman on Tuesday described Chesimard as a leader of the BLA, "known to law enforcement as one of the most violent militant black organizations of the 1970s."

Chesimard has said she is not guilty in the slaying of Werner Foerster, who was shot twice in the head while providing backup for another trooper who pulled over the car Chesimard was riding in for a faulty taillight.

Police said Chesimard and two others in the car were armed. Chesimard fired the first shot, police said, wounding the other trooper.

She continued to fire, police said, and a gun battle ensued. Police said Foerster was killed with his own gun on the side of the road; Chesimard's gun was found by his side.

Lennox Hinds, a Rutgers University criminal justice professor who has represented Chesimard, could not be reached Tuesday.

Law enforcement agencies over the years have ratcheted up the reward for information leading to Chesimard's capture - currently $1 million from the FBI and $1 million from the New Jersey State Police.

The state police have a detective assigned to the case.

"We're never going to stop seeking her return to New Jersey to finish out the sentence she was convicted of," Sgt. Jeff Flynn said Tuesday. He would not comment on what authority police had to act on tips about her whereabouts in Cuba.

While state officials have in the past advocated for Chesimard's return, their calls made new headlines last week following Obama's announcement of plans to normalize relations with Cuba.

Several state and federal lawmakers, including Sen. Robert Menendez (D., N.J.), demanded that Chesimard be sent back. Chesimard "needs to be brought back to justice in New Jersey," Menendez, a critic of Obama's decision to reestablish ties, said last week. Menendez's parents emigrated from Cuba.

On Sunday, Gov. Christie's office circulated a letter he sent Obama urging Chesimard's return and criticizing the president for not making it a requirement of the deal with Cuba.

"What I'm going to do is do what I need to do as governor of New Jersey, which is, one of our state troopers was murdered in cold blood, his killer was convicted, and these thugs in Cuba have given her political asylum for 30 years," Christie said in an interview Monday night on NJTV. "It's unacceptable, and I'm going to continue to speak out."

A Cuban Foreign Ministry official on Monday told the Associated Press that the country had "sovereign and legitimate rights to grant political asylum to people it considers to have been persecuted."

Cuba has previously ignored calls to return Chesimard. After she was put on a U.S. terrorist watch list in 2005, Fidel Castro said in a television address that officials "wanted to portray her as a terrorist, something that was an injustice, a brutality, an infamous lie."

While a bilateral extradition treaty with Cuba remains in force, it has not been invoked for more than 50 years, according to a State Department spokesperson.

Lawyers familiar with international law said it was unlikely that the treaty could be used to extradite Chesimard.

"The fact that it was negotiated by the prior regime suggests that if any extradition is sought, it will likely have to be via a newly negotiated treaty," said Daniel Arshack, a New York criminal defense lawyer.

But even under that treaty, Arshack and another lawyer said, Chesimard would not be eligible for extradition if she has been granted political asylum, as reported. The agreement says Cuba "shall not" extradite a fugitive criminal "if the offense in respect of which his surrender is demanded be of a political character."

Given those provisions, Christie's demand of Obama is "more political rhetoric, frankly," said Douglas McNabb, an international criminal lawyer based in Washington and Houston who specializes in extradition.

There are 75 FBI fugitives in Cuba, the agency said.

Bernadette Meehan, a National Security Council spokesperson, said, "We will continue to press in our engagement with the Cuban government for the return of U.S. fugitives in Cuba to pursue justice for the victims of their crimes."

A State Department spokesperson could not answer questions Tuesday about whether the department was seeking extradition of any fugitives in Cuba.

Extradition may be discussed as the administration considers Cuba's designation as a state sponsor of terrorism, said Cynthia Arnson, director of the Latin American program at the Woodrow Wilson Center in Washington. But she said the issue was tricky.

"There are any number of American Cuban exiles who have been alleged to have participated in acts of terrorism against Cuba, including the downing of a civilian airliner in the 1970s and various assassination attempts" against Castro, Arnson said. "I suspect there are many more people who would be of interest by Cuba" than by the United States, she said.