The arrest came after an international search involving Interpol, New Zealand police and Swiss authorities.
Swiss police allege the man fled the country after killing the woman in her apartment.
At today's hearing Crown lawyer Simon Bar outlined Switzerland's case against the man.
No oral evidence was presented, rather Mr Barr gave a summary of written depositions provided by police and witnesses, including the victims friends and family, in Switzerland.
He detailed the victim's relationship with the accused. Some of those details were suppressed but the Herald can report that they had been in a relationship but broke up several months before her death.
She died as a result of blood loss after being stabbed three times in the neck, probably from behind while the perpetrator had her in a choke hold.
The same day, it is alleged the accused contacted his employer and said he "had a problem" and would not be returning to his job.
Mr Barr said forensic evidence linked the accused to the dead woman's address. A glass with his fingerprints on it was located on a coffee table, suggesting he had been there recently before her death.
At his apartment investigators also found a plastic bag in a rubbish bin that was inside out and appeared to have been washed. It contained traces of human blood and DNA belonging to the victim.
A partial shoe print found at his apartment matched a second partial print found at the crime scene next to a cupboard where the victim kept her plastic bags.
A knife with the victim's DNA on it was found wrapped in material in the glove box of the accused's car.
"It is very strong evidence indeed," Mr Barr said.
Mr Barr said the accused man omitted spending time in Switzerland when he came to New Zealand. He argued that the omission could be inferred at a trial as evidence that the accused was "fleeing".
Mr Barr said the accused's lawyer had "alternative theories" for the evidence put forward by Swiss police.
"That's not a matter that this court needs to be concerned with at this time," he said.
"Competing theories or explanations are a matter for the trial court not for this court."
The accused man's lawyer Grant Illingworth QC said the prosecution evidence was all circumstantial and not sufficient to justify a trial. Thus, the man should not be extradited.
He said the accused had lived an "exemplary life" since he arrived in New Zealand.
"He hasn't got the slightest hint of any difficulty," he said.
"Extradition is one of the most drastic things that can happen to a person - in some ways it's worse than being imprisoned because the person is plucked out of their country of residence and taken, in this case, to a country on the other side of the world and into a foreign jurisdiction."
He said it was "crucial" that that test for extradition, which carried "extreme and serious consequences", was properly met.
He argued that the forensic evidence did not at all show that his client was the woman's killer.
"It is unsurprising that there would be evidence he, the former boyfriend, had been in the apartment. It cannot tell the court when those traces were left in the apartment," he said.
Mr Illingworth said there was also no evidence the knife found in the accused's car was used in the murder.
He said the Crown had "overstated seriously" the reference to it in its evidence.
"It's got... No link at all with the death scene. There is no forensic link between the knife and the injuries to the deceased."
Mr Illingworth told the court that the plastic bag evidence presented by the Crown was easily explained.
"It is common ground that The two had been in a relationship. It is perfectly permissible the bag had been used to carry items that could have left traces of DNA.
Any other theory about the bag amounted to "guess work and speculation".
"A jury could not properly conclude the evidence established guilt beyond reasonable doubt," he said.
"The evidence does no more than create suspicion. suspicion plus suspicion equals suspicion - It doesn't equal proof.
"In our submission the evidence falls short of the mark... (The accused) cannot be plucked away from his established life in New Zealand just because he is a suspect."
Judge Eddie Paul has reserved his decision until Friday 22 May.
