They say Alkhanov's statement indicates neither a move toward reconciliation, nor an accurate reflection of the situation on the ground.
On 6 June, President Alkhanov said the number of kidnappings by Russian troops in the republic is on the decline.
"Today...out of all kidnappings, those carried out by federal forces make up [only] about 5 to 10 percent," Alkhanov said.
Several years ago, federal troops were accused of being behind many more such kidnappings. Moscow has said the situation in the breakaway republic is "stabilizing."
Meanwhile, Alkhanov said 58 people had been reported as missing in the republic during the first five months of 2005, but that only 23 people are still officially listed as missing. He said abductions are declining in the war-shattered region.
Shortly after Alkhanov's statement, however, residents of Prigorodnoe, a village near the republican capital, Grozny, told Reuters that six young men had been abducted, allegedly by pro-Moscow forces.
It is not clear why masked gunmen seized these six men, five of whom lived in the village. But rights groups point to what they call a hidden side of the war -- that is, relatives of rebels who are kidnapped, ostensibly by pro-Moscow forces.
Last week, relatives of rebel leader Aslan Maskhadov, who was killed by Russian troops in March, were released after nearly six months in captivity. It was still unclear who had held the seven and where. Russian authorities have denied accusations that they were behind the abductions.
Usam Baysaev is a representative of the Memorial human rights center in Ingushetia, an autonomous republic bordering Chechnya. He says people are being kidnapped in Chechnya just as before.
"According to the data we have, 74 people were kidnapped during the first four months of this year," Baysaev said. "Thirty-four of them were set free or released for ransom. Four were killed. Thirty-six disappeared after being detained or kidnapped."
Baysaev says Memorial's figures do not cover the whole republic because human rights activists are only allowed to work in some 50 percent of Chechnya's territory. He also says it is difficult to know the true figures because relatives often pay the ransoms and do not make their cases public.
Memorial's figures do not cover the whole republic because human rights activists are only allowed to work in some 50 percent of Chechnya's territory.
There is no way to verify the conflicting claims.
Baysaev says federal troops are only a small part of the problem. He says Memorial is more concerned about kidnappings conducted by pro-Moscow militia groups under the control of Ramzan Kadyrov, son of assassinated Chechen President Akhmad Kadyrov. Baysaev says Kadyrov's militia is armed and financed by federal authorities and that Alkhanov should treat it as an independently acting force.
Rights groups regularly accuse Russian troops, local security forces, as well as Chechen separatist fighters for the disappearances. But Baysaev says Memorial has little information to confirm that Chechen separatists are involved in the kidnappings.
He says Alkhanov's statement about a decline in abductions sounds cynical, since he is the head of the republic and should have the power to stop Kadyrov and others.
"It looks rather cynical, when they say that the numbers of kidnappings are going down," Baysaev said. "If these kidnappings were done by some criminal structures, and Alkhanov was saying they have managed to crush those organize criminal bands, it would have been possible to understand. It would have been something to boast about. But when a person says that we are kidnapping fewer people than we have kidnapped last year -- some percent less -- it doesn't look nice."
Tatyana Stanovaya, a senior analyst at the Center for Political Technologies, a Moscow-based think tank, says Alkhanov' s statement appears intended to boost his popularity ahead of parliamentary elections scheduled for the autumn.
"[Alkhanov] sends a message that he is an efficient leader who tries to deal efficiently with the situation and introduce order in the republic," Stanovaya said.
But RFE/RL correspondent Andrei Babitsky, who has traveled extensively in Chechnya, believes Alkhanov's statement has little to do with the elections. He says Alkhanov is only trying to help Moscow silence Western criticism over human rights violations in the republic.
Babitsky says the Kremlin will pick the Chechen parliament, as it did before, and that Chechen voters will have no say.