Chengdu police said suspects have been detained and admitted carrying out the killings.
There are no indications so far that the murder was driven by sectarian or political factors, said Robbie Barnett, a Tibet expert at New York's Columbia University. "He was very judicious in political matters and was careful not to be assertive on such matters, so did not attract serious hostility from either side," Barnett said.
Britain's Foreign Office confirmed the death of a British citizen in Chengdu and said a consular team had traveled to the city to "liaise with Chinese authorities about this case."
The monk's philanthropic work in China won him respect and admiration among Tibetans, who referred to him using the honorary title of Rinpoche.
"Akong Rinpoche was very committed to the Tibetan community. He tried to come to China almost every year and helped build many schools for Tibetans," Woeser said.
Born in 1939, Tarap Shetrup Akong was groomed to become an abbot in a Tibetan monastery, but fled to India after the 1959 Tibetan uprising against Communist rule. He settled in England in 1963 and later co-founded the Samye Ling Monastery in Scotland in 1967, the first Tibetan Buddhist center to be established in the West.
Decades later, he helped set up schools, clinics and medical colleges in Tibet, according to a website affiliated with the monastery.
Tarap Shetrup Akong maintained political ties with Beijing and met with Jia Qinglin, then chairman of China's top political advisory body, when Jia traveled to Britain in 2006 to explain Beijing's policies in Tibet, according to China's official Xinhua News Agency.
"Though we have settled abroad for a long time, we are always concerned about the development of our motherland, in particular the development of Tibet," Xinhua said Tarap Shetrup Akong told Jia.
The death came days after tensions flared as Chinese security forces fired into a crowd of Tibetan residents who were demanding the release of a fellow villager detained for protesting orders to display the national flag. Overseas rights groups said about 60 Tibetans were injured in the unusual shooting, a sign that Beijing is tightening its control in the Himalayan region following a wave of self-immolations protesting Beijing's rule.
The Chinese Communist government's rule over Tibet has been turbulent. Beijing says it has made vast investments to boost the region's economy and improve the quality of life for Tibetans, but many Tibetans say Beijing's economic policies there have mainly benefited ethnic Chinese migrants. They also resent the government's strict limits on Buddhism and Tibetan culture, while their spiritual leader Dalai Lama remains in exile.
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Associated Press journalists Isolda Morillo in Beijing and Sylvia Hui in London contributed to this report.