The state does not allow them to broadcast the adhaan via loudspeakers; what should they do?

Praise be to Allah.

Firstly: 

Mauritius is a small island in the middle in the Indian Ocean, approximately 500 miles from Madagascar. It is isolated and not many people know about it because of its small size and remoteness. It is located east of Madagascar and was discovered by the Arabs several centuries before the arrival of the Europeans. To the south west lies the island of Reunion. Despite this distance and remoteness, Islam reached it in 923 AH/1510 CE. The island was occupied by Britain until it won independence in 1388 AH/1968 CE. The number of its inhabitants in 2002 CE was 1,189,825. 

The ratio of Muslims is 20% of the population, i.e., 215,000; 55% are Hindu and 20 % are Christian. 

End quote. Summarized from Wikipedia. 

Secondly: 

In the answer to question no. 10078 we stated that the adhaan (call to prayer) is a communal obligation (fard kifaayah), and we quoted the scholars of the Standing Committee for Issuing Fatwas as saying that if one mosque gives the adhaan in a quarter or neighbourhood and all the people in that neighbourhood can hear it, then the other mosques do not need to do it, but it is better for all the mosques to give the adhaan, although it is not obligatory. 

Thirdly: 

There are several rulings pertaining to the adhaan: 

Al-Nawawi (may Allaah have mercy on him) said:  

The scholars stated four things about the wisdom behind the adhaan: manifesting the slogan of Islam and proclaiming the word of Tawheed; announcing that the time for prayer has begun; announcing where the prayer is to be offered; and calling people to pray in congregation. End quote. 

Sharh Muslim (4/77). 

Once you understand the great reasons for which the adhaan has been prescribed, this should motivate you to try to have the government ruling cancelled, so long as you use wise and appropriate means of doing so. It is also essential to avoid confrontation with the state or provoking any trouble with followers of other religions, let alone with other Muslims. What we advise you to do is the following: 

1.Continue to give the adhaan inside the mosque but avoid using loudspeakers.

2.Explain the ruling on the adhaan to your fellow Muslims and that you can do away with broadcasting the sound over loudspeakers. Tell them that the ritual of adhaan is being carried out, and that what is being disallowed is only broadcasting the sound over loudspeakers, not carrying it out.

3.Launch an appeal seeking annulment of the unjust ruling issued by that government.

4.If your appeal does not succeed, then present another appeal asking for the adhaan to be broadcast at other times which will not cause – according to the claim that has been made against you by this criminal – any disturbance to people. Do not waste the opportunity that lies before you. You could – in sha Allaah – get permission to broadcast the adhaan sometimes, so do not give that up, even if it is for only one time.

5.If you do not succeed in any of the things that we have mentioned to you, do not despair. Carry on trying. You are being forced to refrain from broadcasting the adhaan via loudspeakers, but you will never be sinning thereby.

Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allaah have mercy on him) said: Then if it so happens that they are banned from broadcasting the adhaan, then they are being forced (not to do it), and if they are forced then they are like those who do not do obligatory duties because they are prevented by force. End quote. 

Liqaa’aat al-Baab il-Maftooh (166/question no. 15). 

We ask Allaah to help you to serve Islam and unite you, and to benefit Islam and the Muslims through you. 

And Allaah knows best.