* Rights groups say evidence in the case is fabricated
* Case at dead end on eve of murder's anniversary
By Alissa de Carbonnel
MOSCOW, July 14 (Reuters) - A Russian investigation into the murder two years ago of human rights activist Natalia Estemirova is mired in false evidence and has ground to a halt, activists said on Thursday.
The killing of Estemirova, 50, who was shot dead and dumped in nearby Ingushetia after being kidnapped in her native Chechnya, has increased scrutiny of Chechen leader Ramzan Kadyrov.
"The investigators are digging exclusively down one path, for which they do not have a scrap of serious evidence," the head of Russian rights group Memorial, where Estemirova worked, told reporters.
A coalition of Russian rights groups dismissed claims by President Dmitry Medvedev last year that Estemirova's killer -- Islamist militant Alkhazur Bashayev -- had been identified.
Russia says Bashayev was killed in a counter-terrorism operation four months after Estemirova's murder but that his brother, Anzor, now living in France, is also a suspect. France has turned down repeated requests for extradition.
Of the likelihood of Bashayev's involvement, Orlov said: "This is completely nonsense." He added that Bashayev did not have a plausible motive for murder and the official investigation was chasing a dead end.
The group said it conducted its own DNA test using samples from Bashayev's brother which disproved his involvement.
The group presented its own report, which accuses investigators of failing to probe suspicions implicating local security forces, whom Estemirova had accused only a week before her death of being involved in an extrajudicial killing.
"What we are seeing is the fabrication of an investigation, not an actual investigation," said Svetlana Gannushkina, also from Memorial.
Among the doubtful evidence of the case is a silencer gun that does not match the weapon found at the scene of the crime, and the alleged car used to kidnap her which was found without any traces of blood or a struggle inside, the group said.
A lawyer for Estemirova's family said they had appealed to the influential France-based European Court of Human Rights for access to the official investigation material.
The longer the probe flounders "down the wrong path", the slimmer the chances of finding the true culprits behind the crime, the lawyer Ramzan Karpinsky said.
Global rights groups Amnesty International, the Norwegian Helsinki Committee and Human Rights Watch (HRW) issued a joint statement on Thursday criticising Russia's slow progress and questioning the Kremlin's will to solve the case.
There is no direct evidence to implicate Kadyrov himself, but activists accuse him of ruling in a climate of fear and impunity in Chechnya, site of two separatist wars since 1994.
Kadyrov --- a one-time rebel the Kremlin relies on to keep a lid on insurgency violence in the Muslim region where it fought two wars -- has denied any involvement in both her death and that of her friend and reporter, Anna Politkovskaya in 2006.
The Chechen leader has described activists employed by Memorial as "enemies of the people" and has won two defamation cases against the group. But he lost a libel case last month against Orlov, who had said Kadyrov was responsible for Estemirova's murder.