The prosecution is seeking life imprisonment for 42-year-old Fabrice Anthamatten, who admits the killing but denies it was premeditated, according to news reports.
The country was shocked by the brutal killing of the female therapist, Adeline Morel, in September 2013. The 34-year-old was the mother of an eight-month-old child.
A two-time convicted rapist, Anthamatten was serving a 20-year sentence at the Geneva la Pâquerette detention centre when he was allowed out on day release with Morel.
Her body was found in the woods near an equestrian centre where she had been escorting Anthamatten for a therapy session.
The French-Swiss inmate who fled the crime scene was arrested in Poland and extradited to Switzerland.
The prosecution argues that Anthamatten meticulously planned the crime over several months, exploiting failings at the detention centre to obtain day release, choose where to spend it, and acquire the knife he used to kill his victim.
The defendant denies premeditation.
Anthamatten, who is also charged with sexual assault and deprivation of liberty, gave testimony on the first day of his trial, scheduled to last two weeks.
He described his difficult childhood, with a mother who neglected him and a father who was an alcoholic.
Four psychiatric experts – two French and two Swiss – will also be called to give crucial evidence on whether the suspect can be cured.
If they successfully argue he cannot be, Anthamatten could face life in prison if found guilty.
The case has sparked a national debate about whether prisoners are being treated too leniently and the Geneva authorities promised changes to tighten regulations.
The authorities acknowledged that Anthamatten should have never been released alone in Morel's custody.