Two gardaí travelled to Jordan to liaise with the authorities following the arrest of a man there whom they want to speak to in connection with the murder of a 42-year-old woman in Co Kerry earlier this month.
Jamey Carney, the mother of a teenage girl, was found dead at her home in Killarney on 7 July.
A 28-year-old Jordanian man left the country before her body was discovered.
He took a bus to Dublin in the early hours of that morning, a taxi from the city centre to the airport and flew to Turkey before making his way to Jordan.
The ministries of interior and justice said he was arrested and detained in Jordan after the security services received a report from the Irish authorities.
Two garda detective inspectors travelled to the capital Amman to meet with the Jordanian authorities, but they did not interview the man.
They have since returned home.
Garda Commissioner Justin Kelly today thanked the Jordanian authorities for their co-operation and described it as a complex case which gardai were working through at the moment.
The Department of Foreign Affairs and the Irish Embassy in Jordan were also involved.